Appendix: Basics and Useful Relations from Linear Algebra

A.1 Inner Product

T hx, yiA = x Ay. (A.1) The index, written with font Times, indicates the matrix used in the bilinear form. In the T case of homogeneous vectors we have hx, yiA = x Ay. We omit the index when it is clear from the context.

A.2 Determinant

A.2.1 Deﬁnition of the Determinant

The determinant of an N × N matrix is a scalar function D = det(A) : IRN×N → IR with the following properties 1. The determinant is linear in the columns (or rows) of the matrix. That is, if the nth N column is an = αx + βy for any vectors x, y ∈ IR and some constants α, β, then

|(a1, ..., αx + βy, ..., aN )| = α|(a1, ..., x, ..., aN )| + β|(a1, ....., y, ..., aN )| (A.2)

2. When exchanging two rows or two columns, the sign of the determinant changes. 3. If N = 1, det([1]) = 1. We also write det A = |A| . (A.3) For N = 2, we have a11 a12 det = a11a22 − a12a21 . (A.4) a21 a22

A.2.2 Laplacian Development of a Determinant

The following theorem allows us to write the determinant of a matrix A as a sum of products of sub-determinants of a matrix. Let r = {r1, ..., rK } with r1 < ... < rK be a set of row indices rk ∈ N , and c = {c1, ..., cK } with c1 < ... < cK be a set of column indices ck ∈ N . The K × K submatrix only keeping the indices r and c is written as

S(A, r, c) . (A.5)

Ó Springer International Publishing Switzerland 2016 767 W. Förstner and B.P. Wrobel, Photogrammetric Computer Vision, Geometry and Computing 11, DOI 10.1007/978-3-319-11550-4 768 Appendix

The complementary (N − K) × (N − K) submatrix removing these indices is

S 0(A, r, c) . (A.6)

Then we have the Theorem A.2.9: Laplacian development theorem. Given the N ×N matrix A and two lists r = {rk} and c = {ck} of K row and column indices with 1 ≤ r1 < ... < rK < N and 1 ≤ c1 < ... < cK < N, the determinant can be expressed as X |A| = (−1)|c| (−1)|r||S(A, r, c)| |S 0(A, r, c)| , (A.7) r where |r| = r1 + ... + rK and |c| = c1 + ... + cK , and the summation is taken over all possible combinations of c with 1 ≤ c1 < ... < ck < ... < cK < N. Clearly, if the properties of the determinant hold for the submatrices S(A, r, c) and S 0(A, r, c), they also hold for the determinant of the matrix A, which allows the theorem to be proven by induction, as it holds for N = 2. The determinant of a quadratic submatrix is also called minor. Thus the Laplacian development theorem expresses the determinant of the matrix as a sum of products of minors. Two cases are of special interest. An important example is the development of a 4 × 4 matrix by the ﬁrst two columns. Thus we ﬁx c = (1, 2) and obtain X det A = (−1)1+2 (−1)r1+r2 |S(A, r, c)| |S 0(A, r, c)| (A.8) r = +|S(A, (1, 2), (1, 2))| |S 0(A, (1, 2), (1, 2))| −|S(A, (1, 3), (1, 2))| |S 0(A, (1, 3), (1, 2))| +|S(A, (1, 4), (1, 2))| |S 0(A, (1, 4), (1, 2))| +|S(A, (2, 3), (1, 2))| |S 0(A, (2, 3), (1, 2))| −|S(A, (2, 4), (1, 2))| |S 0(A, (2, 4), (1, 2))| +|S(A, (3, 4), (1, 2))| |S 0(A, (3, 4), (1, 2))| (A.9)

a11 a12 a33 a34 = + a21 a22 a43 a44

a11 a12 a23 a24 − a31 a32 a43 a44

a11 a12 a23 a24 + a41 a42 a33 a34

a21 a22 a13 a14 + a31 a32 a43 a44

a21 a22 a13 a14 − a41 a42 a33 a34

a31 a32 a13 a14 + . (A.10) a41 a42 a23 a24 As the minors referring to a set c of columns of a square matrix can be interpreted as the N−1 Plücker coordinates of the join of the points Xc in IP in these columns, the determinant of a matrix is the sum of the products of the Plücker coordinates of the columns c and of the columns not c, taking the correct signs into account. The second application of (A.7) is the following lemma. Lemma A.2.1: Development of a determinant by row. The determinant of an N × N matrix can be expressed as Appendix 769

N X 1+n 0 |A| = (−1) a1,n |S (A, 1, {2, ..., n})| . (A.11) n=1

This results from (A.7) by setting r = 1 and c = 2 : n. For example, take the determinant of a 3 × 3 matrix:

a b c e f d f d e d e f = a − b + c . (A.12) h i g i g h g h i

A.2.3 Determinant of a Block Matrix

The determinant of a block matrix is given by

A11 A12 −1 −1 = |A11| |A22 − A21A11 A12| = |A22| |A11 − A12A22 A21| . (A.13) A21 A22

A.3 Inverse, Adjugate, and Cofactor Matrix

The inverse A−1 of a regular square matrix A fulﬁls A−1A = AA−1 = I . We have the Woodbury identity, with correctly related matrices A, B, C,

(A CBC T)−1 = A−1 − A−1C(C TA−1C B−1)−1C TA−1 (A.14)

(see Petersen and Pedersen, 2012). We also have

A−1 + B−1 = A−1(A + B)B−1 ; (A.15)

(see Petersen and Pedersen, 2012, (144)). The inverse of a symmetric 2 × 2 block matrix is given by

−1 −1 −1 −1 −1 −1 −1 A11 A12 A11 + A11 A12C 2 A21A11 −A11 A12C 2 = −1 −1 −1 (A.16) A21 A22 −C 2 A21A11 C 2 −1 −1 −1 C 1 −C 1 A12A22 = −1 −1 −1 −1 −1 , (A.17) −A22A21C 1 A22 + A22 A21C 1 A12A22 with −1 −1 C 1 = A11 − A12A22 A21 , C 2 = A22 − A21A11 A12 , (A.18) assuming at least one of the two submatrices Aii to be regular. The cofactor matrix AO of a square, not necessarily regular, matrix is the matrix of the determinants of its submatrices

AO = [(−1)i+j|A(ij)|] , (A.19) where A(ij) is the matrix with row i and column j deleted. For a 2 × 2 matrix we have

a −a AO = 22 21 . (A.20) −a12 a11

For a general 3 × 3 matrix A = [a1, a2, a3] with column vectors ai, it can be shown that

O A = [a2 × a3, a3 × a1, a1 × a2] . (A.21) 770 Appendix

The adjugate matrix A∗ of a square matrix, which is not necessarily regular, is the transpose of the cofactor matrix,

A∗ = (AO)T = [(−1)i+j|A(ji)|] . (A.22)

It is closely related to the inverse by

A∗ = |A|A−1 , (A.23) and thus is proportional to the inverse, if A is regular. The determinant therefore can be written as 1 1 |A| = tr(A∗A) = tr((AO)TA) , (A.24) N N where trA is the trace of the matrix A. Finally, we observe for regular n × n matrices,

(A∗)∗ = |A|n−2A and (AO)O = |A|n−2A , (A.25) due to (A∗)∗ = (|A|.A−1)∗ = |A|n−1.|A|−1A = |A|n−2A.

A.4 Skew Symmetric Matrices

Skew matrices play a central role when representing rotations. An N × N skew symmetric matrix S has properties:

S = −S T , (A.26) trS = 0 . (A.27)

A.4.1 2 × 2 Skew Matrix

For a scalar x, we obtain the 2 × 2 skew-symmetric matrix

0 −x S = S(x) = (A.28) x x 0 with the following properties: • It is regular with determinant det(S(x))2 = x2 (A.29)

and eigenvalues √ λ1 = ix λ2 = −ix with i = −1 . (A.30) • Its square, its cube, and its fourth power are

2 2 3 3 4 4 S (x) = −x I 2 , S (x) = −x S(x) , S (x) = x I 2 . (A.31)

• If x = 1, then S(1) rotates a 2-vector

−b 0 −1 a a = = R ◦ (A.32) a 1 0 b 90 b

by 90◦ anti-clockwise. Appendix 771

• We have the rotation matrix cos x − sin x R(x) = exp(S ) = cos(x) I + sin(x) S(1) = (A.33) x 2 sin x cos x

using the matrix exponential, see Sect. (A.13), p. 781, which can be proven by using the deﬁnition of the matrix exponential and collecting the odd and even terms.

A.4.2 3 × 3 Skew Matrix

For a 3-vector x = [x, y, z]T, the 3 × 3 skew symmetric matrix is deﬁned as

 0 −z y  S x = S(x) =  z 0 −x  . (A.34) −y x 0

The matrix S(x) has the following properties: • The product with a 3-vector is identical to the anti-symmetric cross product of two vectors: S(x)y = x × y = −y × x = −S(y)x . (A.35)

Therefore, often S(x) is denoted by [x]×, leading to the intuitive relation x × y = [x]×y. We do not follow this notation since the vector product does not immediately generalize to higher dimensions. • Its right null space is x as x × x = 0. • If x 6= 0, the matrix has rank 2. Its eigenvalues are

λ1 = i|x| λ1 = −i|x| λ3 = 0 . (A.36)

• The matrix S(x) and its square S 2(x) are related to the dyad

T 2 Dx = xx with trDx = |x| (A.37)

by S x Dx = 0 (A.38) and 2 T 2 2 2 S x = xx − |x| I 3 with tr(S x) = −2|x| . (A.39) • The third and the fourth powers are

xxT S 3 = −|x|2S and S(x)4 = |x|4(I − ) . (A.40) x x 3 |x|2

• Therefore we have the relation, for any 3 × 3 skew matrix, 1 S S TS = tr(S S T)S . (A.41) x x x 2 x x x • The following relations hold for unit vectors r with |r| = 1:

2 Dr = Dr (A.42) 2 S r = −(I 3 − Dr) (A.43) 3 S r = −S r (A.44) 4 S r = I 3 − Dr . (A.45) 772 Appendix

The following relations between a skew-symmetric matrix and a regular matrix are useful. Lemma A.4.2: Product of skew symmetric matrix with regular matrix. For each regular matrix M and all vectors x and y ∈ IR3, we have

Mx × My = MO(x × y) (A.46) and S(Mx)M = MOS(x) , (A.47) with the cofactor matrix MO = |M| M−T . (A.48) For a rotation matrix, due to RO = R, we thus have

Rx × Ry = R(x × y) and S(Rx)R = RTS(x) , (A.49) and also S(x)R = RS(RTx) . (A.50)

Proof: For proving (A.46) we start from

< x × y, z >= (x × y)Tz = |x, y, z| . (A.51)

Thus we have, for arbitrary z,

(x × y)Tz = < Mx × My, z > (A.52) = |Mx, My, MM−1z| (A.53) = |M| |x, y, M−1z| (A.54) = |M| < x × y, M−1z > (A.55) = |M| (x × y)M−1z (A.56) = |M| < M−T(x × y), z > (A.57) = < |M| M−T(x × y), z > (A.58) = < MO(x × y), z > (A.59) = (MO(x × y))Tz . (A.60)

A.5 Eigenvalues

The characteristic polynomial of an N × N matrix A is given by

P (λ) = |A − λI N | . (A.61)

Its zeros are the eigenvalues. The eigenvectors result from the homogeneous equation system (A − λI N )x = 0 . (A.62) A symmetric matrix A can be decomposed into the following product

N T X T A = X ΛX = λnxnxn (A.63) n=1 where the orthogonal matrix X = [x1, ..., xn, ..., xN ] contains the eigenvectors xn as columns and the real eigenvalues are collected in the diagonal matrix Λ = Diag(λ1, ..., λN ). The eigenvectors are unique if all eigenvalues are distinct. The eigenvalue decomposition requires 2/3 N 3 + PN 2 for determining the eigenvalues and P eigenvectors, thus 5/3 N 3 if Appendix 773 all eigenvectors are to be determined (see Bathe and Wilson, 1973, Table 1). It is realized in Matlab as [X,Λ]) = eig(A).

A.5.1 Eigenvalues of Matrix Products

The nonzero eigenvalues of the product AB of an m × n matrix A and an n × m matrix B are invariant to the sequence of the product:

λi(AB) = λi(BA) , i = 1,..., min(m, n) . (A.64)

Proof: We have from the determinant of the special matrix,

A.5.2 Eigenvalues of Sub-blocks of a Matrix and Its Inverse

Given a symmetric positive deﬁnite 2 × 2 block matrix

Σ Σ Σ = xx xs (A.68) Σsx Σss and its inverse, W = Σ−1, W W W = xx xs . (A.69) W sx W ss The two block matrices can be interpreted as the covariance matrix and the normal equation matrix for the U unknowns x and the P additional parameters s, in all cases omitting the hats on the estimated parameters. The block diagonals of these two matrices are closely related. Theorem A.5.10: Relative diﬀerence of covariance and weight matrices. The largest min(P,U) eigenvalues of the two matrices

−1 −1 K = (Σxx − W xx )W xx , L = (W ss − Σss )Σss (A.70) U×U P ×P coincide, i.e., assuming the eigenvalues sorted in decreasing order, we have

λi (K) = λi (L) i = 1,..., min(U, P ) . (A.71)

This theorem can be exploited to determine the eigenvalues of K by calculating the eigenvalues of L, which is numerically less complex if U P . Proof: Using

2 2 2 2 W xx = D1 , W ss = D2 , Σxx = X 1 , Σss = X 2 , W xs = B , (A.72) let us simplify the notation and indicate the diagonal blocks of the inverse, see (A.18), p. 769, 774 Appendix

2 2 2 −2 T −1 D1 B X 1 . (D1 − BD2 B ) . W = T 2 Σ = 2 = 2 T −2 −1 . (A.73) B D2 . X 2 . (D2 − B D1 B) Thus we need to compare the eigenvalues of the two matrices

2 −2 2 −1 K = (X 1 − D1 )D2 := (Σxx − W xx )W xx (A.74) and 2 −2 2 −1 L = (D2 − X 2 )X 2 := (W ss − Σss )Σss . (A.75) We use the relation for the U × P matrix B from the Woodbury identity (A.14), p. 769,

T −1 T −1 T (I U − BB ) = I U + B(I P − B B) B , (A.76) and the eigenvalue relation from (A.64), p. 773,

λi(UV ) = λi(VU) . (A.77)

Let us now express both matrices as a function of the elements of W using the =∼ sign to indicate the largest eigenvalues are identical:

2 −2 2 K = (X 1 − D )D1 (A.78) U×U 1 (A.77) ∼ 2 −2 = D1(X 1 − D1 )D1 (A.79) X2=(D2−BD−2BT)−1 1 1 2 2 −2 T −1 −2 = D1((D1 − BD2 B ) − D1 )D1 (A.80) −1 −2 T −1 −1 = (I U − D1 BD2 B D1 ) − I U (A.81) B=D−1BD−1 1 2 T −1 = (I U − B B ) − I U (A.82)

(A.76) T −1 T = B(I P − B B) B . (A.83) | {z } U×U

Similarly, we obtain

2 −2 2 L = (D2 − X )X 2 (A.84) P ×P 2

X−2=(D2−BT D−2B) 2 2 1 2 2 T −2 2 T −2 −1 = (D2 − (D2 − B D1 B))(D2 − B D1 B) (A.85) T −2 2 T −2 −1 = B D1 B(D2 − B D1 B) (A.86) (A.77) ∼ −1 2 T −2 −1 T −1 = D1 B(D2 − B D1 B) B D1 (A.87) | {z } U×U −1 −1 −1 T −2 −1 −1 −1 T −1 = D1 BD2 (I P − D2 B D1 BD2 ) D2 B D1 (A.88) B=D−1BD−1 1 2 T −1 T = B(I P − B B) B . (A.89) | {z } U×U

Therefore the largest eigenvalues of K and L are identical.

A.6 Idempotent Matrices

A matrix P is called idempotent in case P2 = P. We have the following properties:

• The eigenvalues λn of an N × N idempotent matrix are

λi ∈ {0, 1} , (A.90)

which can easily be proven using the eigenvalue decomposition. • For an idempotent matrix P, we have

trP = rkP , (A.91) Appendix 775

where rkP is the rank of the matrix P. • If P is idempotent, also I − P is idempotent. • In case the N × U matrix A and the N × N matrix W have full rank, the matrices

P = A(ATWA)−1ATW and Q = I − P (A.92)

are idempotent with ranks U and N − U.

A.7 Kronecker Product, vec(·) Operator, vech(·) Operator

The Kronecker product and the vec(.)-operator are important for deriving trilinear relations between geometric entities and their transformations. For symmetric matrices, it is also useful to know the vech(.) operator. The Kronecker product collects all products of the elements of two matrices in one matrix. Deﬁnition A.7.28: Kronecker product. Let A = (aij ) be an m × n matrix and B = (bij ) be a p × q matrix. Then the Kronecker product A ⊗ B of A and B yields the mp × nq matrix   a11B ··· a1nB  ..  A ⊗ B =  ......  . (A.93) am1B ··· amnB The vec(.)-operator transforms a matrix into a vector by stacking its column vectors. Deﬁnition A.7.29: vec(.) operator. Let A = (aij ) be an m × n matrix, then vec(A) is a mn × 1 vector: T vecA := (a11, ··· , am1, a12, ··· , amn) . Especially, we have for two m × n matrices A and B,

T T X tr(A B) = vec A vecB = aij bij , (A.94) ij which is the sum of the products of corresponding elements of both matrices. The vec(.)-operator and the Kronecker product are intimately related. The basic relation refers to vectorizing the product of three matrices (see Koch, 1999, Eq. (1.147)). Given are three matrices, the m × n matrix A, the n × p matrix B and the p × s matrix C. Then we have vec(ABC) = (C T ⊗ A)vecB . (A.95) From this expression we can ﬁnd a set of useful relations by assuming the matrices to have a special forms. For example, if either A or C is a vector, due to vec x = vec xT, we obtain the relation vec(aTBC) = (C T ⊗ aT)vecB = (aT ⊗ C T)vecBT . (A.96) The vech(.)-operator assumes a symmetric matrix and stacks the columns of the lower left triangular matrix into a vector. Deﬁnition A.7.30: vech(.)-operator. Let A = (aij ) be a symmetric n × n matrix, n then vechA is a 2 × 1 vector:

T vechA := (a11, ··· , an1, a22, ··· , an2, a33, ··· , ann) .

776 Appendix

A.8 Hadamard Product

The Hadamard product is the elementwise multiplication of two matrices having the same size, M × N: A B = [aij bij ] . (A.97)

It forms an Abelian i.e., commutative group, with the unit element 1 = [1MN ]. We need the following result: Given are diagonal matrices U = Diag(u) and V = Diag(v) and square matrices A and B, all of the same size; then,

tr(UAV B) = uT(BT A)v . (A.98)

Proof: With P = UAV B or X pim = uij ajkvklblm , (A.99) jkl we have X X X trP = pii = uij ajkvklbli = uiiaikvkkbki (A.100) i ijkl ik X X T T = uiaikvkbki = uibkiaikvk = u (B A)v . (A.101) ik ik

A.9 Cholesky and QR Decomposition

Cholesky Decomposition. The Cholesky decomposition of a symmetric positive deﬁ- nite N × N matrix A is unique and given by

A = C TC , (A.102) where C is an upper triangular matrix, with cij = 0, i > j, and positive diagonal elements. The Cholesky matrix C = C −TA can be determined efficiently with approximately N 3/3 operations (see Golub and van Loan, 1996). It can be used to solve the linear equation system Ax = b by first calculating C, then solving C Ty = b for y, and finally solving Cx = y for x.

QR Decomposition. The QR decomposition of an M × N matrix is deﬁned as

A = Q R , (A.103) M×N M×M M×N with the orthonormal matrix Q, with QT = Q−1, and the upper triangular matrix R, with rij = 0, i > j. Since Q is regular, the rank of R is identical to the rank of A. The QR decomposition is only unique (up to a multiplication with a diagonal matrix with entries 1) if A has full rank and M ≤ N. If we require the diagonal elements of R to be positive, the decomposition is unique. If M > N we have the partition   R1 N×N A = [ Q1 , Q2 ]   = Q1R1 , (A.104) M×N M×N M×(M−N) 0 (M−N)×N

T −1 T with some arbitrary matrix Q2 fulﬁlling Q2 Q2 = I M−N and Q1 Q2 = 0, since Q is orthonormal. Calculating the QR decomposition requires 4N(M 2−MN+N 2/3) operations 2 for (Q, R) and 2N (M − N/3) operations if only R1 is required (see Patel, 2002, Lemma 1.11.1). Appendix 777

The QR decomposition can be used for solving a linear equation system. This is a factor two slower than with Cholesky decomposition but is numerically more stable. It also can be used for determining the null space of a matrix, see below. Observe, if we have the QR decomposition of the design matrix A = QR of a linear Gauss–Markov model (with R having positive diagonal elements) and the Cholesky decomposition of the corresponding normal equation matrix ATA = C TC, we see that R = C, i.e., we can directly determine the Cholesky matrix C from A, without having to build the normal equation matrix.

A.10 Singular Value Decomposition

The singular value decomposition of an M × N matrix A, with M ≥ N, is deﬁned as

N T X T A = U S V = snunvn , (A.105) M×N M×M M×N N×N n=1 with the orthogonal matrices

UT = U−1 V T = V −1 (A.106) not necessarily having determinant +1, and the rectangular matrix   Diag(sn) S = N×N . (A.107)  0  (M−N)×N

This partitioning requires 4 M 2N + 22 N 2 operations (see Golub and van Loan, 1996, p. 254) and is generated in Matlab using [U, S, V] = svd(A). The often very large matrix U may be split into two parts,

U = [ U1 | U2 ] , (A.108) M×M M×N M×(M−N) where only the left part is relevant, as

T A = U1Diag(sn)V . (A.109)

This more economic partitioning is generated in Matlab using [U1, S, V] = svd(A,0 econ0). It requires 6 MN 2 + 20 N 3 operations (see Golub and van Loan, 1996, p. 254).

A.11 The Null Space and the Column Space of a Matrix

Null space. Given an M × N matrix A with rank R its null space, also called kernel, is deﬁned as the set of vectors x with Ax = 0:

kernel(A) = {x ∈ IRN | Ax = 0} . (A.110)

The dimension of the null space is N − R. The null space may be empty. The null space usually is given by an N × (N − R) matrix null(A) = N such that any vector x in the null space is a linear combination of its column vectors:

x = Nb for some arbitrary b ; (A.111) therefore 778 Appendix

AN = A null(A) = 0 . (A.112) In particular, we have for any N-vector y the null space of its transposed vector,

null(yT) = J with yTJ = 0 T , or JTy = 0 . (A.113) N×(N−1)

The null space of a nonzero column vector is empty: null(y) = ∅. We always interpret null(.) as an orthonormal matrix with basis vectors as columns, see (A.106).

Relation to SVD. The null space is closely related to the singular value decomposition of the matrix A. Let " #" #" #T Diag(sn) 0 R×(N−R) A = U1 | U2 R×R V 1 | V 2 . (A.114) M×N M×R M×(N−R) 0 (M−R)×R 0 (M−R)×(N−R) N×R N×(N−R)

The rank of the matrix is R and the null space is

null(A) = V 2 . (A.115)

Relation to QR Decomposition and Eﬃcient Computation. The null space of an M × N matrix A with M < N and full row rank is also related to the QR decomposition of its transpose. Let the QR decomposition of the transpose of the M × N matrix be   R1 T M×M A = Q R = [ Q1 , Q2 ]   = Q1R1 , (A.116) N×M N×N N×M N×M N×(N−M) 0 (N−M)×M where QT = Q−1, and R is an upper triangular matrix. We partition the two matrices Q T T and R after their ﬁrst M columns and rows, respectively. Then we have AQ2 = R1 Q1 Q2 = 0 since Q is orthogonal. Thus the null space is identical to

null(A) = Q2 . (A.117)

For nearly square matrices this procedure for determining the null space is approximately 20 times faster than using the SVD.

Efficient Computation of null(xT). The null space null(xT) of a N-vector xT efficiently can be determined from a partitioning of the rotation matrix representing the minimal (N) (N) rotation R = Rab eN , x = [J, x] (see (8.76), p. 340) from eN to the vector x. Then the T T (N) null space null(x ) is the left N ×(N −1) submatrix J of R. This follows from R x = eN , T (N) (N) thus x R = [0, ..., 0, xN ]. If x = −eN , the rotation matrix R (eN , x is not defined. Therefore we use the negative vector −x if xN < 0 thus the rotation matrix (N) R = Rab eN , x = J , x , (A.118) N×(N−1) N×1 and obtain the null space as null(xT) = J . (A.119) The rotation matrix can be given explicitly if the vector is spherically normalized and s T T partitioned x = [x0 , xN ] .

T T I N−1 − x0x0 /(1 + xN ) I N−1 − x0x0 /(1 − xN ) J(xN > 0) = T , J(xN ≤ 0) = T . −x0 x0 (A.120) Appendix 779

This explicit way of determining the null space of a row vector is faster than when using the QR decomposition, since it only requires 2(N − 1)2 operations. For example, the s s s s T s normalized 3-vector x = [x1, x2, x3] with x3 > 0 has null space  xsxs xsxs  1 − 1 1 − 1 2 1 + xs 1 + xs  s s 3 s 3s  J(x3 > 0) =  x1x2 x2x2  . (A.121)  − s 1 − s   1 + x3 1 + x3  s s −x1 −x2

Column space The column space of a matrix is the set of all vectors spanned by its columns: span(A) = {x | x = Ab for some b} . (A.122) The column space also is called the image of A. It can be given by an N × R matrix span(A), whose columns form a basis of the column space of A. The column space can be determined using the singular value decomposition of A:

span(A) = V 1 . (A.123)

Therefore, also the matrix span(A) is orthonormal, see (A.106).

A.12 The Pseudo-inverse

The pseudo-inverse A+ of a possibly rectangular M × N matrix A is an N × M matrix which satisﬁes the following relations:

AA+A = A , A+AA+ = A+ , (AA+)T = AA+ , (A+A)T = A+A . (A.124)

Thus AA+ and A+A are symmetric idempotent matrices, see A.6. A matrix fulﬁlling these constraints is unique and also called the Moore–Penrose inverse, see (Penrose, 1954; Moore, 1920).

A.12.1 The Pseudo-inverse of a Rectangular Matrix

For an M × N matrix A with singular value decomposition A = USV T, its pseudo-inverse is given by N + + T X + T A = VS U = sn vnun , (A.125) n=1 with M ≥ N and + + + 1/sn, if s 6= 0 S = [Diag(sn ) | 0 N×(M−N)] sn = . (A.126) N×M 0, else

Thus S + is the transpose of S with all nonzero elements replaced by their inverses.

A.12.2 Pseudo-inverse of a Singular Symmetric Matrix

Let the U × U matrix A be symmetric and singular with rank Q < U and null space N , U×Q thus 780 Appendix

T AN = 0 , N N = I U−Q . (A.127) Then the pseudo-inverse may be determined from

−1 A+ N AN = . (A.128) NT 0 NT 0

It explicitly reads as A+ = (A + NNT)−1 − NNT . (A.129) Proof: Let the singular value decomposition of A be

T T A = [U | N] Diag([S, 0]) [U | N] with [U | N][U | N] = I U . (A.130)

Then the pseudo-inverse is given by

A+ = [U | N] Diag([S+, 0]) [U | N]T , (A.131) which allows us to prove (A.128).

+ + T AN A N AA + NN AN I U 0 T T = T + T = . (A.132) N 0 N 0 N A N N 0 I U−Q

A.12.3 The Pseudo-inverse of a Rectangular Matrix Having Full Rank

If the rectangular M × N matrix with M > N has rank N, its pseudo-inverse is given by

A+ = (ATA)−1AT . (A.133)

If M < N then A+ = AT(AAT)−1 . (A.134)

A.12.4 The Weighted Pseudo-inverse of a Rectangular Matrix Having Full Rank

With the positive symmetric M ×M weight matrix W and V , the weighted pseudo-inverse of a rectangular M × N matrix A with M > N and rkA = N is deﬁned as

+ T −1 T AW = (A WA) A W . (A.135) It fulﬁlls the following relations:

+ + + + + T + + T AAW A = A , AW AAW = AW , (W AAW ) = W AAW , (AW A) = I . (A.136) This is a special case of the doubly weighted Moore–Penrose inverse of an arbitrary matrix used in Pepić (2010). The weighted pseudo-inverse can be used to determine the minimum + T xb = AW b of (Ax − b) W (Ax − b) when A has full column rank. Appendix 781

A.13 Matrix Exponential

The exponential of a matrix occurs naturally with rotation matrices and is frequently used when updating linear transformations. The exponential of an n × n matrix A is deﬁned as 1 1 eA = I + A + A2 + A3 + ... . (A.137) n 2! 3!

In the case of a diagonal matrix D = Diag(di), we have eD = Diag edi . (A.138)

Taking the decomposition of a symmetric A,

A = UDU−1 , (A.139) we therefore obtain the following relations:

eA = UeDU−1 , (A.140) A −A e e = I n , (A.141) eαAeβA = e(α+β)A , (A.142) AT T e = eA , (A.143) det eA = etrA . (A.144)

For skew symmetric matrices S, see Sect. 8.1.1, we get

R = eS , (A.145) a proper rotation matrix with |R| = 1. In general, the product is not commutative,

eAeB 6= eB eA , (A.146) but if the matrices commute, we have

AB = BA ⇒ eAeB = eB eA = e(A+B) , (A.147) which can be shown by expanding the product series and collecting terms of the same order. λ λ As the multiplication of a matrix with a scalar e is the multiplication with e I n, which commutes with all matrices, we have, from (A.147),

eλeA = eλI n+A . (A.148)

The inverse relation to the matrix exponential is called the matrix logarithm. For regular, symmetric A we have, with (A.139),

ln A = U ln DU−1 , (A.149) with ln Diag(di) = Diag(ln di) . (A.150) As for any complex number z = reiθ we have 782 Appendix

ln z = ln r + i(θ + 2πk) , (A.151) with some arbitrary integer k ∈ ZZ; the logarithm of matrices is not unique.

A.14 Tensor Notation

Tensor notation simplifies the derivation of multilinear forms. The elements of vectors and matrices are represented using their elements together with the indices. The indices are assumed to run through a fixed sequence. Coordinates of points x are written with upper-script indices: xi, the index i ∈ {1, 2, 3}. The coordinates of lines l are written with lower-indices, thus li. They are tensors of first- order. The inner product w = xTl is written as

3 i i . X i w = x li = lix = x li . (A.152) i=1 The sum is taken over indices with the same name, one index being an upper one, the other a lower one. Matrices are represented by two indices. They are second-order tensors. For example, j the homography matrix H is represented by hi , which allows us to write the projective transformation as 3 j j i . X j i x = hi x = hi x j = 1, 2, 3 . (A.153) i=1 We also have matrices with two lower or two upper indices, e.g., when determining the quadratic form 2 i j . T s = x x wij = x Wx , (A.154) with the weight matrix wij . The Jacobian a = (∂s/∂x) of a scalar s and a vector x has indices which must satisfy the relation

i ds = aidx . (A.155)

However, in case we want to express b = ∂t/∂l, we have

i dt = b dli . (A.156)

The index of a vector may be exchanged by multiplication with the unit matrix:

j j i x = δi x . (A.157)

ij The transpose of a matrix alm is the matrix aml. The inverse b of a matrix ajk must fulﬁll ij i b ajk = δk . (A.158) The skew symmetric matrix S(x) depending on a vector x uses the fully anti-symmetric third-order tensor   1, if (ijk) is an even permutation εijk = −1, if (ijk) is an odd permutation (A.159)  0, if (ijk) is no permutation, thus contains an index at least twice, as i sjk = εijkx . (A.160) Appendix 783

A.15 Variance Propagation of Spectrally Normalized Matrix

Given an uncertain n × n-homography (H, Σhh) with Σhh = D(vecH), spectral normalization leads to H M = sign(|M|) = sign(|H|) . (A.161) abs|H|1/n

This section shows that the covariance matrix Σmm of vecM is

T Σmm = JmhΣhhJmh , (A.162) with the Jacobian 1 1 J = I − hiT (A.163) mh abs|H|1/n n n and h = vecH m = vecM i = vec(H−T) . (A.164) Proof: In the following we assume the determinants to be positive. We use the derivative of the determinant of a general regular matrix X (see Petersen and Pedersen, 2012),

d|X | = |X |tr(X −1dX ) . (A.165)

We have tr(AB) = vec(AT)T vecB ; (A.166) thus, we obtain d|X | = |X |vec(X −T)T vec(dX ) . (A.167)

We can now determine the Jacobian Jmh of m w.r.t. h. From

|H|1/n M = H , (A.168) we obtain the diﬀerential d(|H|1/n) M + |H|1/n dM = dH . (A.169) For y = xa we have dy = d(xa) = axa−1dx. Thus the diﬀerential is

1 |H|−(n−1)/nd|H| M + |H|1/n dM = dH , (A.170) n and therefore, using M from (A.161),

1 |H|−(n−1)/n|H|vec(H−T)T vec(dH) M + |H|1/n dM = dH . (A.171) n Vectorization yields

1 vec(H−T)T vec(dH) vecH + |H|1/n vec(dM) = vec(dH) . (A.172) n Solving for vec(dM) gives −1/n 1 −T T vec(dM) = |H| I n − vecH vec(H ) vec(dH) . (A.173) n

With h = vecH m = vecM i = vec(H−T) , (A.174) this ﬁnally leads to −1/n 1 T dm = J dh , with J = |H| I n − hi . (A.175) mh mh n This proves the claim. Observe −1 −T T T n = trI n = tr(H H) = vec(H ) vecH = i h . (A.176) Therefore T i Jmh = 0 Jmhh = 0 , (A.177) O O and thus the null space of Σmm is i = λvec(H ), where H is the cofactor matrix of H. References

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Cn-continuous, 738 2D affinity, 252 through two points, 221 L1-norm minimization, 150 fixed line of a., 274 transformation of l., 258 ML-type estimation as L1., 148 2D autocollineation, perspective a., 277 uncertain Hessian form to L12-norm minimization, 148, 755, 764 2D block adjustment homogeneous coordinates of S1, 214, 215 free b., 663 l., 375 S2, 199, 200, 242, 243 functional model of b., 651 uncertain Hessian parameters of S3, 242, 333 gauge constraints in b., 665 l., 376 S5, 243 gauge transformation in b., 668 uncertain homogeneous coor- I I , 300 inner precision of b., 666 dinates to Hessian form, I I (x), independent rows and columns, mathematical model of b., 651 377 319 sparsity of matrices, 655 uncertain l., 373–377 I I , 301 stochastical model of b., 652 2D model block adjustment, 650–674 I , 219 theoretical quality of b., 670–674 2D motion, 251 I (x), independent rows, 319 2D circle, 237 fixed entities of m, 274 I , 233 2D homography, 250, 253 2D point, 206 Ω, 84 algebraic solution for h., 389 at infinity, 206 χ distribution, 34 closed form solution of h., 406 closest to conic, 295 2 χ -square distribution, 33 degrees of freedom of h., 321 covariance matrix of spherically δ0, 131 fixed entities of h., 274 normalized p., 393 for multi-dimensional test, 68, 130 from uncertain point pairs, 425 degrees of freedom of p., 207 for one-dimensional test, 66, 128 minimal parametrization of h., direct least squares solution of 00 000 `(l , l ), 625 385 intersection p., 401 D, 227 minimal solution of h., 321 direct solution for intersection p., IP, 231 orientedness of transformed 401 0 IP , 215, 216, 231 entities, 355 distinct lines of p., 318 IP1, 214 uncertainty of points mapped with dual of p., 204 IP2, 200, 203, 206, 207, 211, 215, 231 h., 387 homogeneous coordinates of p., IP3, 210, 231 2D line, 207 206 IP5, 231 at infinity, 207 null space of covariance matrix of IPn, 215 centroid representation of l., 374 p., 368, 393 IP∗2, 209, 209 direction of l., 294, 348 oriented p., 345 IP∗3, 212, 231 distinct points of l., 318 Plücker coordinates of p., 223 Tn, 345 from two points, 292 reduced coordinates of p., 370 T∗2, 346 from uncertain centroid to Hessian sign of intersection p., 353 T∗3, 347 form of l., 374 spherical normalization of S(x), independent rows, 318 Hessian normal form of l., 207, uncertain p., 368 ρ-function, 144 374 uncertain p., 366–372 table of ρ., 149 Hessian normal form of uncertain 0 000 2D rotation, 251 ℘2(x , l )), 631 l., 374 0 00 fixed entities of r., 274 ℘3(x , l )), 631 homogeneous coordinates of l., x-parallax, 566, 589, 601–604 207 2D scaling, 251 y-parallax, 561, 566, 589, 590, 592, 602 null space of covariance matrix of 2D shear, 252 21/2D surface, 729 l., 375, 392 2D similarity, 252 1D homography, 257 optimal closed form solution for l., 2D translation, 251 fixed entities of h., 276 397 fixed entities of t., 274 1D point, 214 oriented l., 346 3D affinity, 255 at infinity, 214 Plücker coordinates of l., 223 3D autocollineation, 280 homogeneous coordinates of p., point at infinity of l., 209, 294 3D circle, 241 214 point-direction form of l., 209 3D conics, 241

Ó Springer International Publishing Switzerland 2016 801 W. Förstner and B.P. Wrobel, Photogrammetric Computer Vision, Geometry and Computing 11, DOI 10.1007/978-3-319-11550-4 802 Index

3D homography, 256 absolute orientation, 549, 552, 607, 609 Gauss–Markov model with closed form solution of h., 406 direct solution of a. with three constraints, 108 fixed entities of h., 275 points, 515 Gauss–Markov model, linear, 91 minimal solution of h., 322 of calibrated cameras, 552 homography from point pairs, 389 3D line of uncalibrated cameras, 552 model with constraints between and spatial triangle, 351 redundancy of a., 613 the observations only, 171 approximating 6-vector of l., 381 within two-step procedure, 612 optimal P from image points, 498 at infinity, 219 absolute points, 241 RANSAC, 156 coplanarity of two l., 304 acceptability reweighing constraints, 169 covariance matrix of reduced of a covariance matrix, 120 robust a. for Gauss–Helmert coordinates of l., 381 of bundle adjustment result, 688 model with constraints, 168 degrees of freedom of l., 216, 227 of configuration, 495, 516 sequential similarity transforma- direct solution of l., 412 of precision, 117, 120 tions, 710 directed l., 606 accuracy, 116, 490 sequential spatial resection, 709 direction of l., 348, 353 empirical a., 117 triangulation, 600 distinct planes of l., 319 identification a., 117, 490, 706 algorithmic complexity, 452 distinct points of l., 319 of the mean, 116 analysis model, 7, 448 dual l., 233 adaptive least Kth order squares, 146 angle from two planes, 220 additional parameters, 123, 464 between 2D direction and moment vector of l., 218, 219, choice of a., 509, 684 coordinate axes, 206 220, 227 elimination, 695 between 3D direction and oriented l., 348 evaluation, 693 coordinate axes, 210 parameters of l., 216 evaluation of a., 699 between two 2D lines, 298 Plücker coordinates of l., 218, 226 adjacency graph, 654, 660 direction a, 298 point at infinity of l., 220 adjugate matrix, 770 parallactic a., 420, 421 point-direction form of l., 220 adjustment, see estimation tilt a. of plane, 212 projection, 480 adjustment, block a., 643 zenith a., 210 reconstruction from two images, ADS 80, Leica, 443, 445, 446 antiparallel, 343 605 aerotriangulation, 707, 718, 721 antipodal reduced coordinates of l., 380–381 affinity line, 348 spherical projection of l., 481 2D a., 252 plane, 347 through four 3D lines, 302 3D a., 255 point, 344–346 through two points, 300 chirality of transformed configura- approximate residuals, 164 transformation, 259 tions, 357 approximate values, 145, 452 two-point form of l., 220 minimal solution of a., 320 for bundle adjustment, 707–715 uncertain l., 379–381 sign of entities transformed with for bundle adjustment, direct 3D model block adjustment, 649 a., 357 solutions, 711 3D motion, 255 algebraic solution, 178 for bundle adjustment, sequential fixed entities of m., 276 covariance matrix of a., 180 solutions, 708 3D point, 210 for 2D homography, 389 for relative orientation, 583 at infinity, 210 for 2D line, 396 for relative orientation, normal degrees of freedom of p., 211 for 2D line intersection, 401 case, 589 direct least squares solution of for 3D line intersection, 401 AR-process intersection p., 402 for plane, 396 AR(P ), 52 direct solution for line intersection for projection matrix, 494 AR(1), 53 p., 401 with eigenvalue decomposition, AR(2), 53 distinct planes of p., 319 179 for modelling profiles, 748 from several images, 602 with SVD, 179 integrated A., 54, 752 from two rays, normal case of algorithm observed A., 749 image pair, 601 K, R, Z from P, 500 area camera, 444 homogeneous coordinates of p., b, R from essential matrix, 583 area of triangle, 222 210 3D circle from its image, 536 astigmatism, 379 oriented p., 346 direct LSE 2D line from points, asymmetric weight function, 756 Plücker coordinates of p., 225 401 attitude of camera, 456 quality of p., 526 direct LSE 3D line from points, autocollineation, 248, 256 triangulation of p. from several 414 2D a., 277 images, 602 direct LSE 3D similarity from fixed elements of a., 272 uncertain p., 372–373 point pairs, 411 perspective a., 277 3D points, collinearity of p., 306 direct LSE mean axis, 405 automatic image matching, 563 3D rotation, see rotation direct LSE mean direction, 404 autoregressive process, see AR process 3D similarity, 255 direct LSE point from lines, 403 axis and angle 3D translation, 255 direct LSE rotation from point from rotation matrix, 331 fixed entities of t., 275 pairs, 408 axis, closed form solution of mean a., DLT for projection matrix, 496 405 a posteriori probability density, 77 Gauss–Helmert model with a priori probability density, 76 reduced coordinates, 416 back projection, 482 Index 803 bandwith of matrix, 662 for image triplet, 638 ideal c., 465, 561 barycentric coordinates, 213, 349 for relative orientation, 585 ideal perspective c., 468 base line, 551, 563 functional model of b., 675 ideal unit c., 465 base vector, 553, 578, 582 gauge constraints in b., 665 line c., 444 estimation from 2 points, given linear estimation of rotations in matrix, 281 rotation, 578 b., 713 metric c., 460, 460 base-to-height ratio, 604 linearized model of b., 676 model, 441, 445, 479 basis functions MAP estimate for b., 648 model of central c., 468 monomials as b., 733 nonlinear model of b., 675 model of perspective c., 462, 464, of collocation, 735 outlier detection, 707–715 470 radial b., 735 projective, 676 model of real c., 461 splines as b., 736 projective b., 649 model of spherical c., 462, 468 trigonometric b., 734 redundancy of image pair, 611 moving c., 568 Bayer pattern, 444 self-calibrating b., 450, 674–696 normalized c., 465, 472, 713 Bayesian estimation, 76–78 sensitivity of b., 701 orientation, 449 in Gauss–Markov model, 93 spherical camera b., 686 partially calibrated c., 460 surface reconstruction as B., 742 variance component estimation in perspective c., 248, 446, 456, 460, Bayesian factor, 64 b., 679 464, 607 Bayesian information criterion, 139, 686 view planning for b., 715–722 perspective c. for the image pair, Bertrand’s paradox, 22 with lines, 676 550 best linear unbiased estimate, 79 with points, 610, 649 perspective c. for the image best unbiased estimate, 79 bundle of rays, 461, 559 triplet, 622 bi-partite graph, 646 perspective c. with distortions, bias, 116 calibrated camera, 460, 555–557, 607, 464 induced by linearization, 44 622 pinhole c., 253, 281, 464, 465 of estimated variance factor, 137 for the image pair, 556 planes, 554 of estimates, 79, 141 calibration point c., 444 of normalization, 45 laboratory c., 697 pose, 456 of product, 44 of camera, 449 principal planes of c., 474 of the mean, 45 self-c., 697 ray, 469 of the variance, 45 stability of c., 702 real c., 456 of variance of image points and test field c., 697 relative orientation, 552 lines, 493 with unknown testfield, 698 spherical, 582 bicubic interpolation, 738 calibration matrix, 471 spherical c., 446 bilinear interpolation, 738 differential c., 501 spherical c. for the image pair, bilinearity of coplanarity constraint, from projection matrix, 499 555, 556 553 camera, 460 spherical c. for the image triplet, binomial distribution, 28 absolute orientation of calibrated 622 bivector, 235 c., 552 stellar calibration of perspective block, 643 absolute orientation of uncali- c., 496 block adjustment, 643 brated c., 552 systems, 488 2D model b., 651 affine c., 464 uncalibrated, 461 adjacency graph in b., 660 area c., 444 uncalibrated c., 452, 490, 550, 622 bundle adjustment as b., 648 calibrated c., 452, 460, 607, 622 with affine sensor, 470 free b., 663 calibrated perspective c., 555–557 with distortion, 462 linear planar b., 711–714 calibration, 449 with Euclidean sensor, 531 mathematical model of b., 647 calibration, bundle adjustment for Canon PowerShot A630, 443, 445 model b., 645 c., 696 catadioptric camera, 487 block matrix, inverse of b., 769 calibration, laboratory c., 697 catadioptric optics, 446 blunder, see outlier calibration, test field c., 697 caustic, 445, 446 boundary of region, 442 catadioptric c., 487 Cayley representation of rotation, 336 break down point, 145 central c., 446, 456, 465, 622 central camera, 446, 456, 465 bundle adjustment, 450, 609, 648 computational c., 444 model of, 468 acceptability of accuracy of b., 692 coordinate system, 463, 465, 602 central limit theorem, 30 acceptability of design of b., 691 coplanarity constraint for images central projection, 456, 467, 481, 485, acceptance criteria for results of of calibrated c., 555–557 487, 490 b., 688 coplanarity constraint for images centroid representation, 490, 492 approximate values, 707–715 of spherical c., 556–557 of 2D line, 374 as block adjustment, 648 digital c., 465 of plane, 377 Cramer–Rao bound of b., 683 distortions of perspective c., 505 Chasles’ theorem, 272 empirical accuracy of b., 682 essential matrix of normalized c. , check of linearization, 104 Euclidean b., 649 559 check points, 683, 700 evaluation of b., 687 Euclidean c., 464 checkability, 117, 453 factorization method for b., 714 fish-eye c., 485 of coordinates in absolute for camera calibration, 696 generic c., 446, 460 orientation, 411 for image pair, 610 geometric c. model, 443 of parameters, 133 804 Index checking the implementation, 139 conic, 236 coplanarity chirality, 357 central form of c., 237 condition, 228 effect of affinity, 357 closed form solution for c., 182 of 3D points, 602 effect of homography on c., 356 dual c., 239 of four 3D points, 307 of 2D line and point, 349 general form of c., 236 of two 3D lines, 304 of four 3D points, 350 orientation of c., 348 coplanarity constraint, 550, 564 of plane and point, 350 parametric form of c., 237 for normal case, 561, 562 of three 2D points, 349 point closest to c., 295 from projection rays, 554 of two 3D lines, 350, 355 point of symmetry of c., 237 of images of calibrated cameras, choice of additional parameters, 509, tangent at c., 238 555–557 684 transformation of c., 260 of images of spherical cameras, Cholesky decomposition, 86, 661, 776 conjugate 556–557 circle, 237 rotation, 281–282, 321 of images of uncalibrated cameras, 2D, 237 transformation, 278 552, 553 3D, 241 translation, 279 table, 562 circle fitting, 177 consensus set maximization, 143, 157 corrections, 82 close range applications, 452 constraints correctness, 116 closed form estimation, 176–183 between corresponding image of covariance matrix, 140 closed form solution, see direct solution features, 451 of variance factor, 140 clustering, 157 coplanarity c. for image of correlation, 248 cofactor matrix, 369, 556, 769 calibrated cameras, 555 coefficient, 31, 37 as dual transformation matrix, coplanarity c. for images from function, 50 259 uncalibrated cameras, 552 matrix, 38 for line conic, 239 coplanarity c. for normal case, 562 of parameters of spatial, 522 for plane quadric, 240 crisp c., 96 projective c., 282 for polarity, 238 epipolar c. for image triplet, 639 singular, 564 for transformation of hyperplanes, for essential matrix, 557 correspondence, 548 258 for fundamental matrix, 553 correspondence problem, 9 collinearity for groups of observations, 167 corresponding equations, 470 for three image points, 623 image lines, 568 of 3D points, 306, 602 gauge c., 110 image points, 561, 563, 568, 569 of projection centres, 622 weak c., 102 points and lines, 621 of three 2D points, 296 constructions covariance, 37 projective c. equations, 472 in 2D, 292–295 function, 49, 121, 736, 739, 741 collineation, 247, 277 in 3D, 300–304 intersection, 98 1D c., 257 continuous random variables, 26 operator, 38 perspective c., 248, 277, 278 continuous, Cn-c., 738 covariance matrix, 37 projective c., 248 control acceptability of c., 120 collocation, 174–176 feature, 647 basis functions of c., 735 correctness of c., 140 full c. point, 608 colour image, 569 effect of wrong c., 135 horizontal c. point, 609 complanarity, see coplanarity eigenvalue decomposition of c., line, 493, 548, 609 complete search, 151 667 complete space U, 231 plane, 609 empirical c., 118 complex numbers, 651 planimetric c. point, 609 evaluation of c., 669 computational camera, 444 point, 450, 493, 527, 548 metric for c., 121 concatenation points of image pair, 608 of algebraic solution, 180 of displacements, 262 stochastic c. point, 610 of centroid, 410 of homographies, 261 convolution, 42 of estimated observations, 86 of transformations, 261 Cook’s distance, 127 of estimated parameters, 86, 96 concurrence coordinate axes, 244 of estimated residuals, 87 of planes, 307 coordinate system of five-point solution, 588 of three 2D lines, 296 camera c., 461, 463, 465, 602 of homography from four points, condition number, 118, 286, 537, 574, elements of c., 243 388 659 image c., 463 of image coordinates, 569 conditional probability, 23 normalized camera c., 466 of matrix, 32 conditioning, 286 object c., 462, 465 of mean direction, 403 effect of c. on normal equations, of photogrammetric model, 559 of parallaxes, 569 657 scene c., 462 of parameters, relative orientation, of point coordinates, 321, 465, sensor c., 463 591 494, 571, 603, 606 coordinate transformation, 109, 262–266 of projection matrix, 495 of projection matrix, 537 interpretation of c., 249 of quaternion, 383 confidence coordinates of quaternion from directions, 408 ellipse, 32, 369 barycentric c., 213, 349 of reduced 2D line coordinates, hyperbola, 374 homogeneous c., 45, 195, 205 376 conformal geometric algebra, 236 image c., 469 of reduced coordinates xr, 371 Index 805

of reduced coordinates of 3D line, of 3D elation, 281 3D line from points, 414 381 of 3D homography, 255 3D similarity from point pairs, 411 of rotation, 435 of 3D homology, 281 mean axis, 405 of rotation from directions, 407 of 3D line, 216, 227, 264 mean direction, 404 of rotation matrix, 435 of 3D point, 211, 373 point from lines, 403 of the residuals, relative of 3D rotation, 327, 382 rotation from point pairs, 408 orientation, 591 of collineations, 285 direct solution, 176, 178, 452 reference c., 517 of essential matrix, 556 minimal d., 178 singular c., 33 of fundamental matrix, 553 minimum norm solution, 179 sparsity of c., 663 of general homography, 249 of 2D intersection point, 401 specification of c., 121 of general rotation, 326 of 3D homography, 406 theoretical c., 517 of image pair, 550 of 3D intersection point, 401, 402 Cramer–Rao bound, 86, 118, 648 of plane, 212 of 3D line, 411 of bundle adjustment, 683 of projection matrices from of absolute orientation with three crisp constraints, 96 fundamental matrix, 594 points, 515 criterion matrix, 120 of projection matrix, 472 of algebraic surface, 183 critical configuration, 452, 452 of test w.r.t. ground truth, 119 of best fitting 2D line, 397 DLT with points, 495 of test on bias, 141 of best fitting mean axis, 405 estimation of trifocal tensor, 636 of test on correctness of covariance of best fitting mean direction, 403 image pair, 614 matrix, 141 of best fitting plane, 400 line prediction, 626 of test on groups of outliers, 129, of best fitting rotation from prediction in image triplet, 635 129, 131 directions, 406 spatial resection, 515, 521 of test on noise level, 140 of best fitting similarity, 408 critical cylinder of spatial resection, of test on systematic errors, 134 of bundle adjustment, 711 515, 517 of tests on geometric relations, of circle fitting, 177 critical surface 393, 395 of conic, 182 estimation of essential matrix, 588 of transformations, 253 of ellipse fitting, 182 estimation of fundamental matrix, of trifocal tensor, 622 of estimation a 3D line, 412 571 of variance factor, 90, 98 of homography, 406 relative orientation, 588 Delaunay triangulation, 732 of quadric fitting, 183 cross ratio delta function, 26 of spatial resection, 513 of collinear points, 268 Denavit–Hartenberg parameters, 264 of spatial resection with > 3 of concurrent lines, 270 density points, 518 cumulative distribution, 26 a posteriori d., 77 of trifocal tensor, 636 cumulative distribution, inverse c., 40 function, 26 directed curvature dependent images, 558, 595 3D line, 606 as fictitious observations, 747 general parametrization, 558 prediction of d. image lines in as prior, 745 design matrix, 82 image triplet, 627 for regularization, 747 reduced d., 95 direction curve, flatness of c., 739 detectability angle, 298 Cuthill–McKee algorithm, 662 ellipse, 130 angle between 2D d. and factor, 125, 129, 131 coordinate axes, 206 d.o.f., see degrees of freedom of gross errors, relative orientation, angle between 3D d. and datum, see gauge 591 coordinate axes, 210 decision theory tasks, 19 of groups of gross errors, 129 closed form solution for mean d., decomposition of single gross errors, 125 403 Cholesky d., 661, 776 detectable gross error, 125 cosine matrix for rotation, 328 LU d., 661 detectable outliers during relative interpolation of d., 341 of projection matrix, 498 orientation, 592 of 2D line, 294 QR d., 776 diagnostics, 142 of camera ray, 467, 469 definition accuracy, 117 external d., 115, 119 of intersection of two planes, 353 degenerate configuration, 517 internal d., 115, 115, 117, 118 of join of two points, 352 degrees of freedom differential of line segment, 352 of 2D point, 207 angles, 337, 338 of lines and planes, 346–348 degrees of freedom, 360 calibration matrix, 501 vector of image point, 469, 553, 3D rotation, 327 rotation, 336 556 in χ2-distribution, 33 rotation vector, 337 discrete random variables, 26 in t-distribution, 35 similarity transformation, 111 dispersion operator, 38 in Fisher distribution, 35 differential GPS, 683, 721 displacements, 262 in noncentral χ2 distribution, 34 DigiCAMx4, IGI, 683 concatenation of d., 262 in Wishart distribution, 34 digital elevation model, 728 distance of 2D elation, 278 digital surface model, 728 between two 2D points, 298 of 2D homography, 251, 253, 321 direct linear transformation (DLT), between two 3D lines, 310 of 2D line, 375 247, 249 between two 3D points, 309 of 2D perspectivity, 278 direct LS estimation from origin in 2D, 297 of 2D point, 368 2D lines from points, 401 Mahalanobis d., 84 806 Index

of 2D entities, 297–298, 310 of 2D point, 204 error, quasi-systematic, 667 of 2D point from 2D line, 298 oriented projective plane, 346 essential matrix, 556–557, 562, 575–583, of 2D point from line, 298 oriented projective space, 347 613, 623 of 3D entities, 308 Plücker coordinates, 233 degrees of freedom of e., 556 of 3D line from origin, 308 Plücker matrix, 233 dependent images, 558 of 3D line to origin, 218 projective plane IP∗2, 209, 209 from ≥ 7 points, 575 of 3D point from line, 309 projective space IP∗3, 212, 231 from 2 points, given rotation, 578 of 3D point from origin, 308 transformation, 259 from 4 coplanar points, 577 of 3D point from plane, 310 duality, 203, 229–236, 283 from 5 points, 575 of plane from origin, 308 of 2D point and 2D line, 234 normalized cameras, 559 of two covariance matrices, 121 of 3D lines, 235 parametrizations of e., 557 signed d., 354 of 3D point and plane, 234 projection matrices from e., 595 distinct entities defining of transformations, 259 singular values of e., 557 2D line, 318 dualizing matrix, 227 estimable quantities, 109, 666 2D point, 318 estimate 3D line, 319 effect Bayesian e., 76–78, 93 3D point, 319 of intrinsics and extrinsics on best linear unbiased e., 79 plane, 319 image coordinates, 502 best unbiased e., 79 distortion of random errors on estimation, least squares e., 79 lens d., 464, 507 117 maximum a posteriori e., 77 nonlinear d., 452, 477 of wrong covariance matrix, 135 maximum likelihood e., 78 of perspective mapping, 479 ego-motion determination, 644 estimated radial, 506 eigenvalue covariance matrix of e. observa- radial d., 506–508 decomposition of covariance tions, 86 tangential d., 506 matrix, 667 covariance matrix of e. parameters, distortion model, 476 generalized e. problem, 517 86 phenomenological d., 508 eigenvalues, 773–774 covariance matrix of e. residuals, physical d., 506 elation 87 distribution, 24, 28–35 definition of e., 278 observations, 86 χ d., 34 elementary rotation, 328 parameters, 84 χ2-square d., 33 angles from rotation matrix, 330 residuals, 84 t-d., 35 concatenation of e., 329 size of gross errors, 131 binomial d., 28 ellipse, 237 size of group of gross errors, 128 cumulative d., 26 confidence e., 32 size of single gross error, 124 exponential d., 29 detectability e., 130 variance factor, 89 Fisher d., 35 fitting, 182 estimation Gaussian d., 29 sensitivity e., 130 Bayesian e., 93 inverse cumulative d., 40 standard e., 31, 366, 369 Bayesian e. in Gauss–Markov Laplace d., 29 empirical model, 93 mixed d., 143 accuracy, 117, 118 bias of e., 141 multi-dimensional normal d., 31 accuracy of bundle adjustment, evaluation of e., 117 normal d., 29 682 in Gauss–Helmert model with quantiles of d., 40 covariance matrix, 118 constraints, 163–170 Rayleigh d., 29 precision, 117, 118 in Gauss–Markov model with Student’s t-d., 35 sensitivity, 126, 130, 134 constraints, 99–102 uniform d., 28 standard deviation, 118 in linear Gauss–Markov model, Wishart d., 34 empty projective space IP0, 231 81–102 DLT, 247, 249, 480, 622 endlap, 700, 718, 721 in model with constraints between algebraic solution for d., 494 epipolar observations only, 170 direct estimation of d., 494 axis, 563, 564 in nonlinear Gauss–Markov model, explicit form of D., 472 constraints, 639 102–107 for uncalibrated cameras, 472 geometry, 562–565 of 2D intersection point, 417 from 3D lines, 504 line, 248, 563–565, 623 of 3D similarity transformation, precision of d. compared to spatial line, curved e., 564 607 resection, 523 line, oriented e., 564 of variance components, 91–93, theoretical precision of d., 522 plane, 563 493 two d. for image pair, 611 epipolar line, 573–574 on curved manifolds, 415 DMC, Intergraph, 683 epipole, 563, 565, 594 robust e., 141 double points, 523, 590, 592 equidistant projection, 487 sequential e., 96 doubly integrated white noise process, equisolid projection, 487 statistically optimal e., 452 53 equivalence of uncertain homogeneous stochastical model of e., 76, 83 driving process, 52, 749 vectors, 390 with implicit functions, 160 dual equivariant function, 267 with reduced coordinates, 415 3D line, 233 error in variables model, 161 with two group, 96 conic, 239 error propagation, see variance estimation theory, 75–81 entities, 231 propagation tasks, 19 Index 807

Euclidean field of view, 445, 446, 458, 459, 468, of block adjustment, 647 bundle adjustment, 649 484 of bundle adjustment, 675 camera, 464 field-based representation, 8 table with f., 171 normalization, 196, 198 field-based scene description, 442 fundamental matrices of image triplet, normalization of matrix, 285 fill-in, 662–665 623 normalization of vector, 199 filtering, 730, 736, 762 fundamental matrix, 553–555, 570–574, Euclidean part Kalman f., 96, 98 612, 622, 629 of 1D point, 214 Wiener f., 93 as singular correlation, 564 of 2D line, 207 FinePix REAL 3D W1, Fuji, 443, 445 degrees of freedom of f., 553 of 2D point, 206 fish-eye, 444 from ≥ 7 points, 571 of 3D line coordinates, 218 camera, 485 from ≥ 8 points, 570 of 3D point, 210 lens, 459, 485 from camera planes, 554 of plane, 211 optics, 478 from projection matrices, 554 Euler’s rotation theorem, 326 Fisher from projection matrix for lines, evaluation distribution, 35 564 w.r.t. groups of gross errors, 128 information matrix, 86 singular values of f., 554 w.r.t. single gross errors, 124 fitted observations, 86 w.r.t. systematic errors, 133 fixed entities, 272–277 Gamma-matrix, 219 of acceptability of precision, 120 of 1D homography, 276 dual G., 233 of additional parameters, 699 of 2D affinity, 274 gauge, 108, 703 constraints, 110 of block adjustment, 662 of 2D homography, 274 constraints in bundle adjustment, of bundle adjustment, 687 of 2D motion, 274 665 of calibration model, 684 of 2D rotation, 274 definition of g. of coordinate of checkability of parameters, 133 of 2D translations, 274 system, 109 of covariance matrix of block of 3D affinity, 276 definition of g. of covariance adjustment, 669 of 3D homography, 275 matrix, 109 of detectability of groups of gross of 3D motion, 276, 282 in bundle block, 645 errors, 129 of 3D translation, 275 linear g. constraints, 112 of detectability of single gross of planar motion, 275 minimal control in block errors, 125 of spatial motion, 276 adjustment for g., 664 of effect of errors, 122 flatness nonlinear g. constraints, 111 of empirical accuracy, 118 of a curve, 739 transformation, 108–114 of empirical precision, 118 of a surface, 739–741 transformation in block adjust- of estimation, 115 flight direction, 558 ment, 668 of theoretical precision, 117 flight plan, 452 transformation, regular g., 112, of uncertain relations, 393 flying height, 605 121 expectation, 36, 38 focal length, 256, 461 transformation, singular g., 112 of function of stochastic vector, 44 focal point, 461 unspecified g., 669 operator, 38 foot point Gauss–Helmert model, 160, 162, 162, exponential ρ-function, 149 of 2D origin on line, 295 163, 174, 414, 415 exponential distribution, 29 of origin on 3D line, 323 for 2D intersection point, 418 exterior orientation, 460, 460, 629, 634 forward motion, 589 for homography estimation, 424, model of e., 465 image pair, quality, 593 425 of camera systems, 488 free adjustment, 109 for relative orientation, 586 of image pair, 550, 610 inner precision of f., 666 for total least squares, 161 of image triplet, 622, 623 minimum trace solution of f., 111 Gauss–Helmert model with constraints, exterior parameters, 460 of block, 663 163, 174 external diagnostics, 115, 119 reduced normal equations of f., estimation in G., 163–170 external precision, 116 114 linear G., 163 extrapolation during transformation, free network, 109 nonlinear G., 163 389 Fuji FinePix REAL 3D W1, 443, 445 normal equations of G., 165 extrinsic parameters, see exterior function redundancy of G., 165 parameters ρ-f., 144 robust algorithm for G., 168 from spatial resection, 513 delta f., 26 Gauss–Markov model, 162, 173, 414, density f., 26 415, 497 factor graph, 654, 659, 661 multi-dimensional probability f., Bayesian estimation in G., 93 factorization 27 for homography estimation, 424, of matrix, see decomposition of a random variable, 40 427 factorization for bundle adjustment, of two random variables, 42 for image triplet, 638 714 separable f., 28 for self-calibrating bundle feature step f., 25 adjustment, 675, 678 control f., 647 functional model, 75, 490 for surface reconstruction, 743, image f., 646 algebraic structure of f., 161 746 scene f., 646 invertibility of f., 144 linear G., 81–102 fictitious observations, 78, 750 nonlinear – linear f., 161 nonlinear G., 102–107 for profile reconstruction, 746 of 2D block adjustment, 651 nonlinear G. with constraints, 104 808 Index

Gauss–Markov model with constraints, Hessian matrix, 44, 106 homology, see also perspective 99–102, 162, 173 of log-likelihood function, 78 autocollineation, 277 estimation in G., 100 of surface function, 741 horizon, 208, 458, 468, 474, 482 linear G., 162 Hessian normal form, 207 as 3D control line, 494, 504 nonlinear G., 162 uncertain, 374 horizon line, 458, 459 Gauss–Newton method, 103, 105 hierarchy of transformations, 285 horizontal view, 456 Gaussian distribution, 29 homogeneous horopter, 521 general weighted least squares, 80 uncertainty, 396 Hough transformation, 158, 283 generative model, 83 homogeneous entities, notation of h., Huber estimator, 148, 150 generic camera, 446, 460 196 human stereo vision system, 561 relative pose, 581 homogeneous coordinates, 45, 195, 490 hyperbola, 237 geometric algebra, 236 of 1D point, 214 standard h., 374 geometric image model, 447 of 2D line, 207 hyperplane, 221, 224, 226, 283 geometric relations of 2D point, 206 at infinity, 215 transformation of h., 258 in 2D, 295–299 of 3D point, 210 in 3D, 304–308 of plane, 211 geometry ideal homogeneous part camera, 465 epipolar g., 562–565 of 1D point coordinates, 214 image pair, 549–568 image point, 463 of 2D line coordinates, 207 lens, 256 image triplet, 622–632 of 2D point coordinates, 206 of single image, 456 perspective camera, 468 of 3D line coordinates, 218 single image, 488 unit camera, 465 of 3D point coordinates, 210 global test, 90 ideal point, see point at infinity of plane coordinates, 211 bundle adjustment, 689 identification accuracy, 117, 706 homogeneous representation, 195 GPS, 450, 452, 493, 647, 653 of features, 490 homogeneous stochastic process, 51 identity differential G., 683, 721 homogeneous uncertain vectors, of two 2D entities, 296 Gram–Schmidt orthogonalization, 511 equivalence of h., 390 of two 3D entities, 306 graph homogeneous uncertainty, 121, 371 IGI DigiCAMx4, 683 adjacency g., 654, 660 homogeneous vectors, normalization of ill-posed, 82 bi-partite g., 646 h., 198, 241 image factor g., 654, 659, 661 homography, 249, 247–249, 253, 254, coordinate system, 463 graph surface, 729, 739 256 coordinates, 469 Grassmann–Cayley algebra, 234 1D h., 257 coplanarity constraint of i. from gross error, see outlier 2D h., 250, 253 uncalibrated cameras, 552 detection, 452 2D h. between images, 567, 578 distortion model, 476 estimated size of g., 124, 131 distortions, 505 lower bound for detectable g., 125, 3D h., 255, 256, 613, 622 feature, 646 131 concatenation of h., 261 geometry, 456 model of g., 123 constraint of two 3D lines, 316 geometry of nadir i., 459 test statistic for g., 131 cross ratio, 268 geometry of slanted i., 459 gross errors, 452 depth and parallax map, 602 matching, 402, 563 ground sampling distance, 457, 590, effect of h. on chirality , 356 model, 7, 441 683, 716, 720 fixed entities of 1D h., 276 orientation of single image, 489 ground truth, 115, 119, 429, 683 fixed entities of 2D h., 274 oriented i. line, 482 groups of observations, 86 fixed entities of 3D h., 275 pair, see image pair from point pairs, 389 constraints for g., 167 perspective i., 456 from uncertain point pairs, 425 detectable gross errors of g., 129 point, see image point general h., 248 diagnostics of g., 130 pyramid, 491 image to map plane, 526 evaluation of g., 128 rectified i., 477 image to scene plane, 524, 525 in sequential estimation, 96 reduced i. coordinates, 468, 470 in variance component estimation, invariants of h., 268 relative orientation, 552 91, 92 minimal parametrization of scale, 457 normal equations for g., 96 uncertain h., 384, 426 sequence, 647, see image strip outlier model of g., 123 quasi-affine h., 357 straight line-perturbing i. errors, sensitivity factor of g., 130 table of constraints with h., 316 464 Gruber uncertain h., 384–386 theoretical precision of i. block, points, 590 uncertainty of h. from uncertain 673 position, 590 points, 387 theoretical precision of i. strip, GSD, see ground sampling distance uncertainty of points mapped with 670–673 h., 387 triplet, see image triplet Hadamard product, 137, 776 vector form of h., 315 two-step procedure of i. harmonic homology, 280 homologeous, see corresponding orientation, 549 harmonic points, 270 homology, 277, 280, 284, 568 image pair hat function, 736 between images, 567 bundle solution for i., 610 hat matrix, 86 harmonic h., 280 comparison of procedures for i., Helmert point error, 366 singular values of h., 568 614 Index 809

control points of i., 608 interpolation, 730 of spherical normalization of 3D critical configuration of i., 614 bicubic i., 738 point, 373 degrees of freedom of i., 550 bilinear i., 738 within estimation, 161 exterior orientation of i., 550 linear i., 737 within variance propagation, 43 geometry of i., 549 precision of i., 734 join interior orientation of i., 550 interpolation during transformation, of 3D point and line, 302 normal case of i., 465, 561 389 of three 3D points, 302 object points, 608 interpolation of directions, 341 of two 2D points, 202, 293 orientation of i., 549, 608 interpolation of rotations, 341 of two 3D points, 300 triangulation for normal case of i., interpretation, 9 601 interpretation model, 7, 448 K-transformation, 109 triangulation for perspective i., interpretation of line drawings, 523 Kalman filter, 96, 98 600 intersection, 549 keypoint detector two-step procedure for i., 612 of 2D line and conic, 293 uncertainty, 491 image point of 3D line and plane, 301 Kronecker product, 137, 555, 775 direction, 556 of three planes, 302 ideal i., 463, 469 of two 2D lines, 202, 292 laboratory calibration, 697 observable i., 461, 463 of two planes, 220, 301 Ladybug 3, Pointgrey, 443–445 uncertainty of i., 491 intersection of planes, direct LS solution lag, 49 image triplet of i., 403 Laplace distribution, 29 geometry of i., 622 intrinsic parameters, see interior Laplacian development theorem, 768 law of cosines, 514 nonlinear observation equations parameters least squares for i., 638 intrinsic parameters of a camera, 463 estimate, 79 orientation of i., 632 invariant, 266 general weighted l., 80 predicting directed lines in i., 627 number of independent i., 271 ordinary l., 80 relative orientation of i., 633, 636 of affinities, 268 weighted l., 79 images, dependent i., 595 of five 2D points, 272 with regularization for profile implicit functions, estimation with i., of homography, 268 reconstruction, 747 160 of perspective mappings, 268 leave-one-out test, 124, 128 implicit variance propagation, 43, 516 of polygon, 267 Legoland scene, 529 IMU, 653 of projective mapping, 266 Leica ADS 80, 443, 445, 446 incidence of rectangle, 266, 271 lens of 2D entities, 295 inverse cumulative distribution, 40 distortion, 461, 464, 507 of 2D line and 2D point, 295 inverse depth, 257 fish-eye l., 459 of 3D line and plane, 305 inverse perspective, 489, 523 narrow-angle l., 459 of 3D point and line, 306 inversion of transformation, 261 normal l., 459 of 3D point and plane, 304 invertibility of functional model, 144 thin l. projection, 256 isocentre, 459 of two 3D lines, 304 ultra-wide angle l., 459 isometric parallel, 459 independence wide-angle l., 459 isotropic stochastic process, 51 stochastic i., 28 zoom l., 459 isotropic uncertainty, 121 independent events, 23 levels of reasoning, 7 independent images, 558, 581, 589 of directions, 367, 371, 403, 413 leverage point, 127 independent random variables, 31 of points, 368, 396, 399, 406, 408, Lie group, 284 influence function, 147 412 likelihood function, 77 table of i., 149 iterative estimation, 92, 103, 414, 452 line information matrix, 86 of spatial resection, 520 antipodal l., 348 inner on curved manifolds, 415 at infinity, 203, 345 geometry, 110 iterative solution, 452 direction of l. segment, 352 precision, 110, 666 segment, 352 precision of free block, 667 Jacobian vanishing l., 529 inner product, 767 of DLT, 501 line at infinity, 208 INS, 452, 647 of DLT for nadir view, 501 of plane, 208, 212, 220 integrated AR-process, 54 of Hessian form to homogeneous line camera, 444 integrated white noise process, 53 2D line, 375 line drawing interpretation, 523 Intergraph DMC, 683 of homogeneous to Euclidean 3D line segment, uncertainty of l., 492 interior and exterior orientation coordinates, 373 linear substitute model, 103 from projection matrix, 500 of homogeneous to Euclidean linearization, check of l. within interior orientation, 460, 464, 610, 629 coordinates, 371 Gauss–Markov model, 104 of image pair, 550 of homogeneous to Hessian form linearized model of bundle adjustment, of image triplet, 622 of 2D line, 377 676 interior parameters of camera, 460 of reduction of 3D line coordinates, linearized models, table with l., 171 interior parameters of camera system, 380 loop closing, 672 488 of reduction of point coordinates, lower bound internal 370 for detectable deviation, 66 diagnostics, 115, 115, 117, 118 of spherical normalization, 368, for detectable gross error, 125, 131 precision, 116 376 LS, see least squares 810 Index

LU-decomposition, 661 of uncertain homography, 384 with constraints between the of uncertain motion, 383 observations only, 173 M-estimation, 609 of uncertain quaternions, 383 model block adjustment MAD, 40, 146 of uncertain rotation, 382 2D, 650 Mahalanobis distance, 69, 84, 361 of uncertain similarity, 383 2D m., 651–674 Manhattan scene, 529 of uncertain transformations, 381 3D m., 649 MAP estimate, 77 minimal representation projective m., 649 bundle adjustment, 648 of 2D uncertain point, 369 modified weights, 147 profile reconstruction, 744 minimal solution, 178 moment vector of a 3D line, 218, 219, Maple, 522 of 2D affinity, 320 220, 227 mapping, 644 of 2D homography, 321 moments, 36 affine m., 357 of 3D homography, 322 central m., 37 as coordinate transformation, 249 of basis of image pair, 578 general m., 36 as displacement, 249 of essential matrix, 575 of normal distribution, 39 general m. 3D to 2D, 479 of fundamental matrix, 571 mono-plotting, 526 of general lines, 484 of image orientation, 489 monomials, 733 of quadrics, 484 of projection matrix, 494 motion perspective m., 277 of relative orientation from mirror 2D m., 260 quasi-affine m., 357 images, 579 3D m., 260 matching, 9 of relative orientation of three fixed entities of planar m., 275 fixed entities of spatial m., 276 matrix images, 636 forward m., 589 block, inverse of b., 769 of relative orientation, iterative from object to camera, 466 correlation m., 38 m., 585 from structure, 449 covariance m., 37 of spatial resection, 513 planar, 251 Euclidean normalization of m., of trifocal tensor, 636 rigid body m., 255 285 with QR decomposition, 179 rotational m., 337 exponential, 781 minimum norm solution, 179 sideward m., 588 exponential for homography, 384 minimum trace solution, 111 spatial m., 255 exponential for motion and minimum-volume estimate, 146 uncertain m., 383 similarity, 384 mirroring moving object, 568 exponential for rotation, 326, 326, at y-axis, 251, 279 moving camera, 568 337, 338, 382 at a plane, 281 multiple solutions, 452 exponential for transformations, transformation in 2D, 279 of relative orientation, E-matrix, 382 mixed distribution, 143 582 normally distributed m., 32 ML-type estimation, 147 of relative orientation, F-matrix, precision m., 83 L -norm minimization as, 148 1 571 product, eigenvalues of m., 773 model of spatial resection, 515 random m., 27 functional m., 75 representation of 2D entities, 312 generative m., 83 nadir representation of 3D entities, 313 geometric m. of camera, 443 direction, 528 skew symmetric m., 336 geometric m. of scene, 442 point, 346, 458 sparse m., 86 linear substitute m., 103 view, 456, 521 sparse structure of m., 655 mathematical m., 75 narrow-angle lens, 459 spectral normalization of m., 286 notion of m., 7 negative point, 344 Toeplitz m., 53 of analysis, 7 net area of model, 719 weight coefficient m., 89 of camera, 441, 445 Newton–Raphson method, 105 weight m., 83, 89 of constraints between ob- noncentral χ02 distribution, 34 maximum a posterior estimate, 77 servations only, 162, noncentrality parameter, 65, 131 maximum likelihood estimation, 78–79 162 nonlinear maximum likelihood type estimation, of distortion, 476 distortions, 452, 477 147 of image, 7, 441 Gauss–Markov model, 102 mean, 36 of interpretation, 7 Gauss–Markov model with accuracy of the m., 116 of projection, 449 constraints, 104 bias of m., 45 of scene, 7, 441 model of bundle adjustment, 675 of ratios, 46 of sensor, 7 nonmetric camera, uncalibrated n., 697 operator, 38 of world, 7 nonrejection region, 63 precision of the m., 116 phenomenological m. of distortion, normal case vector, 38 505 of image pair, 561 median, 40 photogrammetric, see photogram- of image pair, iterative relative median absolute difference, 40, 146 metric model orientation, 588 meta model, 6, 441 physical m. of distortion, 505 of single image, 465 method of modified weights, 147 stochastical m., 76, 83 triangulation for n. of image pair, metric camera, 460, 696 thin plate m., 741 601 metric for covariance matrices, 121 thin rod m., 741 normal cases, 453 minimal parametrization weak membrane m., 740 normal distribution, 29 of 2D homography, 385 weak string m., 739 in best unbiased estimation, 81 Index 811

multi-dimensional n., 31 observations detection in bundle adjustment, normal equation fictitious o., 78 707–715 components, 85 uncertain o., 490 model, 143 for groups, 96 observed AR-process, 749 for system for two groups of One Shot 360, 443, 445, 446 P3P problem, 513–518 observations, 96 operator panorama, 644 in Gauss–Helmert model with covariance o., 38 Panoscan Mark III, Rollei, 443, 445, constraints, 165 dispersion o., 38 446 in Gauss–Markov model, 84 mean o., 38 parabola, 237 in Gauss–Markov model with optical axis, 461 parallactic angle, 420, 421, 548, 549, constraints, 100 optical ray, 528 550, 596, 598, 599, 601, 604, partitioned n., 94 optics, 456 717 profile of n. matrix, 662 catadioptric o., 446 parallax reduced n., 94, 660 optimal estimation x-p., 569, 589, 601–604 reduced n. of free adjustment, 114 of intrinsics and extrinsics, 501 y-p., 561, 589, 590, 592, 602 sparsity of n. matrix, 657–661 of projection matrix, 496 accuracy of p., 569 table with n., 171 oracle, robust estimation as o., 142, 167 covariance matrix, 569 normal lens, 459 ordinary least squares, 80 map, 602 normal line orientation vertical p., 561 through 2D point, 294 absolute o., 549, 552, 607 parallel line through the 2D origin, 294 absolute o. within two-step through 2D point, 294 normalization procedure, 612 through the 2D origin, 294 bias of n., 45 also see direction, 352 parallel projection, 545 Euclidean n., 242 exterior o., 465, 610 parallelepiped, 553 Euclidean n. of matrix, 285 exterior o. of camera systems, 488 parallelism Euclidean n. of vector, 199 exterior o. of image pair, 550 of 2D lines, 297 of homogeneous matrices, 285 interior o., 610 of 3D line and plane, 308 of homogeneous vectors, 198, 241 interior o. of image pair, 550 of 3D lines, 307 spectral n. of matrix, 286 of cameras, 449 of planes, 307 spherical n., 45, 199, 242 of conics and quadrics, 348 parameters spherical n. of matrix, 286 of image pair, 549, 608 additional p., 123, 464 normalized extrinsic p., 460 of image triplet, 632 camera, 465, 472, 713 interior p. of camera system, 488 of join of 3D line and point, 353 camera coordinate system, 466 intrinsic p. of a camera, 460 of plane, 354 definition of n. residuals, 144 parametrization of relative orientation, parameters from essential matrix, residuals, 152, 170 557–559 581 trifocal tensor, 628 singularity, 559 quality of o. procedures, 453 variance of n. residuals, 145 partially calibrated camera, 460 relative o., 450, 551 notation of homogeneous entities, 196 partitioning of normal equation matrix, relative o. of image triplet, 633 null space 94 relative o. within two-step for estimating homography, 389, PCA of covariance matrix, 667 procedure, 612 496 pencil of planes in epipolar geometry, orientation-preserving transformation, of covariance matrix of 2D line, 563 355 375, 392 percentile, 40 oriented of covariance matrix of 2D point, perspective 2D line, 346 368, 393 2D autocollineation, 277 using QR decomposition, 179 2D point, 345 3D autocollineation, 280 numerical differentiation, 453 3D line, 348 autocollineation, 277 NURBS, mapping of N., 484 3D point, 346 calibrated p. camera, 555–557 epipolar line, 564 camera, 248, 446, 456, 460, 464, object image line, 482 607, 622 coordinate system, 462, 465 plane, 347 camera for the image pair, 550 moving o., 568 point, 344 camera model, 462, 470 point, 563 projective geometry, 343 collineation, 248, 277, 277, 278 points, image pair, 608 projective plane, 345 distortion, 503 object-based representation, 8 projective space, 345 distortions of p. camera, 505 object-based scene description, 442 orthogonal image, 456 oblique view, 456 projection, 487 inverse p., 523 observable image point, 463 orthogonality mapping with distortions, 479 observation equations, 82 of 2D lines, 297 model of p. camera, 464 linearized o. for normal case of of 3D line and plane, 308 projection, 467, 470, 486 relative orientation, 589 of 3D lines, 307 projection of 3D line, 480 nonlinear o., 173 of planes, 307 perspectivity, 277, 278 nonlinear o. for image triplet, 638 outlier, see gross error, 609 phenomenological distortion model, observation process, 25 asymmetric distribution of o., 755 505, 508 observational errors, 83 detection, 142 photo collection, 644 812 Index photogrammetric model, 549, 612–613, quadric, 240 prediction operator 708–711, 719 slope of p., 212 `(l 00, l 000), 625 0 000 absolute orientation of p., 607 three-point representation of p., ℘2(x , l )), 631 0 00 coordinate system definition, 559 213 ℘3(x , l )), 631 coplanarity constraint, 551 transformation of p., 258 principal net area of p., 719 uncertain p., 377–379, 403 distance, 457, 462, 464, 465, 471 of image triplet, 622–633 planes, concurrent p., 306 distance from two vanishing of images of calibrated cameras, PnP problem, 513–521 points, 531 552 point line, 458 of images of uncalibrated cameras, antipodal p., 344–346 plane of optics, 256, 461 552 control p. of image pair, 608 planes of camera, 474 parameters of p. for given in tetrahedron, 351 point, 458, 462, 464, 465, 470, 471 projection matrix P0, 560 negative p., 344 point from projection matrix, 475 scale of p., 559, 607, 634, 714 of symmetry, 462 point from three vanishing points, photogrammetric models oriented p., 344 532 mutual scale of p., 621 positive p., 344 prior physical distortion model, 505, 506 tie p. of image pair, 608 for profiles and surfaces, 745–748 Pi-matrix, 300 uncertainty of transformed p., 387 variance component estimation of Pi-matrix, dual P., 301 vanishing p., 210, 529–534 p., 750 pinhole camera, 253, 257, 281, 464, 465 point at infinity, 472, 493, 550 prior, see also a priori, 76 pixel distance, 465 1D p., 214 probability Plücker 2D p., 206 a posteriori p. density, 77 constraint, 218 3D p., 210 a priori p. density, 76 enforcing P. constraint, 381 of 2D line, 209, 294 axiomatic definition of p., 22 matrix, 219 of 3D line, 220 axioms of p., 23 Plücker constraint, 227 point camera, 444 conditional p., definition, 23 Plücker coordinates, 221–229, 768 point of symmetry density function, 26 definition of P., 223 of conic, 237 distribution, 24 dual P., 233 of quadric, 240 notion of p., 21 Euclidean part of P. of 3D line, Pointgrey Ladybug 3, 443–445 total p., 23 218 polar of a point, 233 von Mises’ definition, 22 homogeneous part of P. of 3D line, polarity, 233, 283, 285 process 218 at conics, 238 doubly integrated white noise p., of 2D line, 223 at the unit circle, 233 53 of 2D points, 223 on the sphere, 200 integrated white noise p., 53 of 3D line, 218, 226 pole of a line, 233 stochastic p., 49 of 3D line from points, 217 polycamera, 445 profile of normal equation matrix, 662 of 3D points, 225 pose, 456 profile reconstruction of plane, 225 of camera, 6, 456, 460 fictitious observations for p., 746 Plücker matrix, 227 theoretical precision of p., 522, LS with regularization, 747 dual P., 233 523 MAP estimate for p., 744 planar homography positive definite function, 50 outlier detection in p., 755 fixed entities, 274 positive point, 344 projection planar motion, 251 power function, 65 central p., 481, 485, 487 planar object power of test, 62 equidistant p., 487 critical configuration of DLT, 495 PowerShot A630, Canon, 443, 445 equisolid p., 487 plane, 211 pre-image of line at infinity, 355 line, 483, 564 antipodal p., 347 precision, 116, 453 matrix, see projection matrix at infinity, 212 acceptability of p., 117 model, 449 centroid representation of p., 377 and accuracy, 116 not straight line-preserving p., 564 degrees of freedom of p., 212 empirical p., 117 of 3D lines, 480 distinct points of p., 319 external p., 116 of lines, 564 homogeneous coordinates of p., inner p., 110 orthogonal p., 487 211 internal p., 116 parallel p., 545 horizon of p., 208 matrix, 43, 83, 367 perspective p., 467, 470, 486 intersection, 301 of interpolation, 734 perspective p. of 3D line, 480 joining 3D point and line, 302 of the mean, 116 plane, 482, 483 line at infinity of p., 208, 212, 220 singular p. matrix, 367 ray, 445 optimal direct solution of p., 400, theoretical p., 117 spherical p., 467 436 prediction, 730 spherical p. of 3D line, 481 orientation of p., 348, 354 in image pair, image point, stereographic p., 346, 487 oriented p., 347 562–565 thin lens p., 256 parameters of p. through three in image triplet, points and lines, uncertainty of p. ray, 524 points, 225 623–625 projection centre, 248, 254, 457, Plücker coordinates of p., 225 of points, lines and planes, 451 460–466, 474–475, 563 point-direction form of p., 213 prediction errors, 97 collinear, 622 Index 813

from projection matrix, 498 transformation of q., 260 relative r., 145 quality of p. from spatial resection, quality table with r., 171 516, 517 checkability of the observations, reference covariance matrix, 517 projection matrix, 472, 607, 629 609 refraction, 477 algebraic solution for p., 494 criteria, 609 regression model, 81 covariance matrix of p., 495 of 3D point from two images, 603 regularization, 82, 747 decomposition of p., 498 of 3D points, 526 regularizing observations, 746 direct estimation of p., 494 of parameters, 452 rejection region, 63 DLT with p., 494 of relative orientation, 590 relative orientation, 450, 551, 634 for 3D lines, 480, 626 precision, 609 epipolar geometry of r., 562 for 3D points, 626 quantile, 40 iterative r., 585–594 from essential matrix, 595 quasi-affine projective mapping, 357 iterative r. for ideal forward from fundamental matrix, 594 quasi-systematic errors, 667 motion, 593 general p., 479 quaternion, 332–335 iterative r. for normal case, 588 interior and exterior orientation as hyper-complex numbers, 333 of image pair, 622 from, 500 covariance matrix of q., 383 of image triplet, 636 optimal estimation of p., 496 uncertain q., 383 of images of calibrated cameras, proper p., 468, 474, 482, 483 552 properties of p., 473 R (software package), 115 of images of generic cameras, 581 properties of p. for 3D lines, 481 radial basis function, 735 of images of uncalibrated cameras, uncertainty of p., 475 radial distortion, 506–508 552 projective random planar object, 567 bundle adjustment, 649 matrix, 27 quality of r., 590 collination, 248 number generation, 55 theoretical precision of r., 590 correlation, 282 vector, 27 theoretical reliability of r., 590 line IP, 231 random perturbations, 452 with known plumb line, 581 model, 480 random sample consensus, see within two-step procedure, 612 model block adjustment, 649 RANSAC relative redundancy, 145 oriented dual p. plane, 346 random variables, 24–28 relief displacement, 459 oriented p. plane, 345 continuous r., 26 representation of uncertain points and oriented p. space, 345 discrete r., 26 lines, minimal r., 369–371 2 plane IP , 200, 203, 206, 231 independent r., 31 residual of coplanarity constraint, 555, 3D points at infinity, 211 transformation of r., 41 562 partioning of p., 215 uncorrelated r., 31 residuals, 82 representation as unit sphere, 215 RANSAC, 153–157, 609 approximate r., 164 0 point IP , 215, 216, 231 ray direction, 467, 469, 514 covariance matrix of r., 87 1 space IP , 214 sign of r., 492 normalized r., 144, 152, 170 3 space IP , 210, 231 uncertainty of r., 492 standardized r., 125 n space IP , 215 Rayleigh distribution, 29 variance of normalized r., 145 transformation, 611 reasoning levels, 7 resolution, 442 projective bundle adjustment, 676 reconstruction reverse Cuthill–McKee algorithm, 662 projectivity of point on plane, 524 rho-function, see ρ-function 2D p., 253 of points and lines, 596–606 rigid body motion, 255 3D p., 256 quality of r. procedures, 453 RMSE, 119 pure p., 254 rectangle, invariants of r., 271 robust estimate singular p., 473 reduced of standard deviation of propagation of uncertainty, 386 design matrix, 95 normalized residuals, 146 proper normal equations, 660 of variance factor, 146 projection matrix, 468, 474, 482, reduced coordinates, 393 robust estimation, 141–185 483 covariance matrix of r., 371 as oracle, 142 rotation matrix, 499 of 2D point, 370 maximum likelihood-type pseudo-inverse, 101, 779 of 3D line, 380–381 estimation, 147 rectangular matrix, 779 redundancy, 82, 609 of variance factor, 145–146 symmetric matrix, 779 matrix, 87, 669 strategies, 158 pseudo-likelihood function, 144 numbers, 88, 145 with L1-norm minimization, 150 numbers, relative orientation, 592 with clustering, 157 QR decomposition, 776 of absolute orientation, 613 with complete search, 151 for minimal solution, 179 of bundle adjustment image pair, with RANSAC, 153 for null space, 179 611 robustness, 142 quadratic variation, 741, 760 of Gauss–Helmert model with evaluation of r. with influence quadric, 239 constraints, 165 function, 147 fitting, 183 of Gauss–Markov model, 82 Rollei Panoscan Mark III, 443, 445, 446 mapping of q., 484 of Gauss–Markov model with root mean square error, 119 orientation of q., 348 constraints, 100 rotating slit camera, 485 point of symmetry of q., 240 of two DLTs, 611 rotation, 325, 460 tangent plane at q., 240 of two spatial resections, 612 2D r., 251 814 Index

3D r., 255 scene values of homology, 568 averaging, 713 coordinate system, 462 vector, right s., 571 axis and angle from r. matrix, 331 feature, 646 skew matrix, 336, 770 axis angle representation of r., 331 field-based s. description, 442 product with matrix, 772 Cayley representation of r., 336 geometric model of s., 442 properties, 770 closed form solution for r. from model, 7, 441 SLERP, see spherical linear interpola- directions, 406 object-based s. description, 442 tion concatenation of elementary r., reconstruction, 450 slope of curve, 739 329 Schur complement, 660 slope of plane, 212 concatenation of r. with search, complete s., 151 smoothness quaternions, 337 segment, line s., 352 of a function, 740 conjugate r., 281, 321 selecting independent constraints, 317 of a surface, 740–742 differential equation for r., 337 self-calibrating bundle adjustment, 450, SO(n), 326 differential r. vector, 337 674–696 solution direction between planes, 353 self-calibration, 492, 697–699 direct s., 452 direction cosine matrix for r., 328 self-diagnosis, 452 iterative s., 452 eigenvalues of r. matrix, 327 sensitivity, 115 minimal direct s., 178 elementary r., 328 w.r.t. groups of outliers, 130 space Euler’s theorem for r., 326 w.r.t. single outliers, 126–128 complete s., 231 interpolation, 341 w.r.t. systematic errors, 134–135 empty s., 231 matrix, see rotation matrix analysis, 592, 609, 695, 699 sparse minimal r. between two vectors, ellipse, 130 covariance matrix, 86, 663 340 factor, 126–128, 130, 132, 692–694, design matrix, 656, 657 quaternion representation of r., 701, 705–706 matrix, 86 332, 335 of bundle adjustment, 691, 693, normal equation matrix, 657–661 relations between representations 701 reduced normal equation matrix, for r., 338 of relative orientation, 589 660 representations in 3D: overview, sensor coordinate system, 463 structure of matrices, 655 326 sensor model, 7 spatial resection, 513–521, 533, 623 Rodriguez representation of r., 335 separable function, 28 critical configuration of s., 515 singularity of r., 330 sequential estimation, 96 direct solution of s., 513 skew-symmetric matrix represen- shear, 464, 470, 471 direct solution with > 3 points, tation of r., 336 shear in 2D, 252 518 uncertain r., 382–383 sidelap, 718, 721 iterative solution, 520 vector, 336–338 sideward motion, 588 quality of projection centre from with Euler angles, 328 sign s., 516 rotation matrix, 326, 466 constraints for points in image theoretical precision of s., 523 as local coordinate system, 378 triplet, 624 direction cosine r., 328 effect of affinity on s. of entity, 357 two s. for image pair, 612 exponential form of r., 326, 337 of distance, 354 with observed scene lines, 521 from corresponding vectors, 339 of intersection of 3D line and spectral normalization from projection matrix, 499 plane, 353 of matrix, 286 from three image lines, 531 of intersection point of two 2D variance propagation, 783 representation, 327 lines, 353 spherical rotational motion, 337 of ray direction, 492 linear interpolation, 341 rounding error, 26, 29, 39 significance level, 62 normalization, 45, 198, 199, 242 significance number, 62 normalization of matrix, 286 S-matrix, 112 similarity normalization of uncertain 2D S-transformation, 109, 113 2D s., 252 point, 368 in block adjustment, 668 3D s., 255, 613, 622 projection, 467 regular S-, 112, 121 closed form solution of s., 408 projection of 3D line, 481 singular S-, 112 minimal solution of s., 320 spherical camera, 446, 456, 462, 468, sampling transformation, 611 555, 582 in RANSAC, 154 uncertain s., 383 bundle adjustment, 686 of distributions, 56 simulating data, 55 calibrated s. for the image pair, scale single image, normal case of, 465 556 factor, 604 single viewpoint, 445 triangulation for s., 597 mutual s. of photogrammetric singular spherically normalized homogeneous models, 621, 639 correlation, 564 coordinates, 490 number, 457, 591, 604, 720 dual conic, 241 spline, 736 of photogrammetric model, 551, line conic, 241 one-dimensional s., 736 559, 607 precision matrix, 367 two-dimensional s., 737 scale difference, 464 projectivity, 473 SPSS (software package), 115 of image coordinate system, 471 value decomposition, 777 stable configuration, 452 scale transfer, 634, 639 values of essential matrix, 557 standard deviation, 37 scaling in 2D, 251 values of fundamental matrix, 554 empirical s., 118 Index 815

of exterior orientation with spatial for estimation of projection of relative orientation, 590 resection, 523 matrix, 495 of spatial resection, 523 of height, 604 for partitioning of essential of strips and blocks, 670–674 of image coordinates, 527 matrix, 581 of surface interpolation, 761 of parameters for partitioning of fundamental theoretical reliability relative orientation, 591 matrix, 595 of absolute orientation, 411 robust estimator of s. of of essential matrix, 557 of relative orientation, 590 normalized residuals, 146 of fundamental matrix, 553 thin plate model, 741 standard ellipse, 31, 366, 369 swing angle, 459 thin rod model, 741 standard hyperbola, 374 symbolic image description, 448 tie points of image pair, 608 standardized residuals, 125 symbolic level of world model, 7 tilt angle, 459 stationary, 49 systematic error, 116 tilt angle of a plane, 212 statistically best fitting in bundle adjustment, 683 Toeplitz matrix, 53 2D line, 397 model of s., 123 torsion, 741 mean axis, 405 quasi-s., 667 total least squares, 161 mean direction, 403 systems of cameras, 488 total probability, 23 plane, 400 traffic light decision, 63 rotation from directions, 406 taking position, 457, 469, 550, 563 transformation, 247 similarity, 408 tangent 1D t., 257 Steiner’s theorem, 37, 116 line at conic, 238 2D affinity, 252 stellar calibration, 496, 533 plane at quadric, 240 2D homography, 253 step function, 25 tangent space, 370 2D mirroring at y-axis, 251 stereo vision, human s., 561 tangential distortion, 485, 506 2D projectivity, 253 stereographic projection, 243, 346, 487 telelens, 459 2D rotation, 251 stitching, 644 tensor notation, 782 2D scaling, 251 stochastic independence, 28 test 2D shear, 252 stochastic process, 48–55 for 3D line through triangle, 351 2D translation, 251 homogeneous s., 51 for coplanarity constraint, 555 3D affinity, 255 isotropic s., 51 for estimated parameters, 133 3D homography, 256, 552, 613 stationary s., 49 for gross error, 131 3D homography, image pair, 611 stochastical model, 76 for point in tetrahedron, 351 3D projectivity, 256 for surface reconstruction, 743 of point in triangle, 349 3D rotation, 255 of 2D block adjustment, 652 of prediction errors, 98 3D similarity, 255, 551, 613, 622 of block adjustment, 647 one-sided t., 67 3D similarity, image pair, 611 of estimation, 83 suboptimal t. for gross errors, 126 3D translation, 255 straight line segments, 480 two-sided t., 65 concatenation of t., 261 straight line-preserving, 249, 470 within sequential estimation, 98 conjugate t., 278 structure from motion, 450, 568 test field calibration, 697, 698 coordinate t., 262 structure tensor, 402, 569 testability, 66 direct solution of similarity t., 408 Student’s t-distribution, 35 of mean, 65 dual t., 259 suboptimal test for gross errors, 126 of mean vector, 67 extrapolation during t., 389 substitute model, 103 of observations, relative orienta- group, 284 substitute parameters, 177 tion, 592 hierarchy, 285 subsymbolic level of real world, 7 testing, 393 interpolation during t., 389 surface, 442 approximate t., 393 inversion of t., 261 21/2D s., 729, 733–742 geometric relations in 2D and 3D, of 2D line, 258 flatness of s., 739–741 393 of 2D point, 250 graph s., 729, 739 uncertain relations, 393 of 3D line, 259 reconstruction, 727, 730 tetrahedron of 3D point, 255 reconstruction as Bayesian chirality of t., 350 of conic, 260 estimation, 742 point in t., 351 of coordinates w.r.t. fixed reconstruction, outlier detection sign of volume of t., 225 reference, 263 in s., 755 volume of t., 225 of coordinates w.r.t. transformed smoothness of s., 740–742 theoretical reference, 264 surfel, 442 precision, 117 of hyperplane, 258 SVD sensitivity, 127, 135 of object w.r.t. fixed reference, 262 for algebraic solution, 179 theoretical covariance matrix, 517 of object w.r.t. transformed for consistency of essential matrix, theoretical precision reference, 263 575 of 3D point, 526 of oriented entities, 355 for consistency of rotation matrix, from two images, 603 of plane, 258 531 of bundle adjustment, 683 of quadric, 260 for estimation of 3D line, 606 of DLT, 522 of random variable, 41 for estimation of 3D point, 603 of image blocks, 673 orientation-preserving t., 355 for estimation of base vector, 579 of image strips, 671 planar motion, 251 for estimation of fundamental of point on horizontal plane, 527 similarity t., 252 matrix, 571 of pose estimation, 521 spatial motion, 255 816 Index translation isotropic u., 121, 396 vec operator, 775 2D t., 251 isotropic u. of directions, 367, vech operator, 775 3D t., 255 371, 403, 413 vector representation conjugate t., 279 isotropic u. of points, 368, 369, of conics and quadrics, 316 triangle 399, 406, 408, 412 of transformations, 315 area of t., 222 of directions in the camera, 492 vertical parallax, 561 chirality of t., 349 of estimated 3D line, 413 vertical view, 456 spatial t. and 3D line, 351 of extrinsics and intrinsics from Vexcel Ultracam, 443–445, 683 triangular irregular network, 732 projection matrix, 500 view triangulation, 549, 552, 595, 596 of feature identification, 490 horizontal v., 456 algebraic t. from multiple images, of image points, 491 nadir v., 456 602 of key point, 491 oblique v., 456 Delaunay t., 732 of line segments, 492 perspective v., 459 for normal case of image pair, 601 of projection matrix, 475 vertical v., 456 for perspective image pair, 600 of projection ray, 524 zenith v., 456 for spherical camera, 597 of ray direction, 492 view planning, 715–722 trifocal matrices, 625–626 of transformed points, 387 for flat areas, 718 trifocal plane, 623 propagation, 386 of buildings and rooms, 721 trifocal tensor, 622, 625, 625, 629 uncorrelated random variables, 31 rules of thumb for v., 716 degrees of freedom of t., 622 uniform distribution, 28 viewing angle, 371, 458 direct estimation for t., 636 unit camera, 465 viewing direction, 458, 462 from projection matrices, efficient unit circle from projection matrix, 475 computation, 626 S1 in IR2, 214, 215 normal case of image pair, 561 iterative solution for t., 637 unit sphere viewing position, 457, 469, 550 minimal solution of t., 636 2 3 S in IR , 199, 200, 242, 243 viewing sphere, 423, 447 normalized t., 628 3 4 S in IR , 242 viewline, 445 trigonometric basis functions, 734 5 6 S in IR , 243 viewpoint of camera, 445 true value, 79 unscented transformation for variance visual odometry, 644 truncated L -norm minimization, 148 2 propagation, 47 volume of tetrahedron, 225 Tschebyscheff polynomials, 510 twisted cubic, 495 vanishing line, 529 weak configuration, 698 ultra-wide lens, 459 vanishing point, 208, 210, 459, 529–534 weak constraints, 102 Ultracam, Vexcel, 443–445, 683 estimation, 417 weak membrane model, 740 uncalibrated camera, 461, 490, 622 variance, 37 weak string model, 739 uncertain bias of v., 45 weight, 83 2D line, 373–377 of normalized residuals, 145 coefficient matrix, 89 2D point, 366–372 of ratio, 46 matrix, 43, 83, 89 3D line, 379–381 of residual of coplanarity table of w. functions, 149 3D point, 372–373 constraint, 555 weight function Euclidean coordinates, 371 of scale of similarity from points, asymmetric w., 756 geometric entities, 359 410 weighted geometric relations, 359 variance component estimation, 91–93, least squares, 79 Hessian parameters of 2D line, 376 493 sum of the squared residuals, 84 homogeneous coordinates, 367, bundle adjustment, 679 weights 375 profile reconstruction, 750 in least squares estimation, 81 variance factor, 716 homography, 384–386 modified w., 147 correctness of v., 140 matrix, 32 whitening, 41 estimated v., 89, 680–685, 700–701 minimal representation of u. 2D wide-angle lens, 459 initial, 89 point, 369 Wiener filter, 93 interpretation of v., 498 motion, 383 Wishart distribution, 34 robust estimation of v., 145–146 observations, 490 Woodbury identity, 769 test of v., 90 plane, 377–379, 403 world model, 7 quaternion, 383 variance propagation, 42, 42–48 rotation, 382–383 implicit v., 43, 154, 180, 516 rotation matrices, 382 nonlinear v., 43 Yule–Walker equations, 184 scene points and lines, 493 of bilinear forms, 387 similarity, 383 of linear functions, 42 zenith spherical normalization of u. 2D of nonlinear functions, 43 angle, 210 point, 368 of spectral normalization, 783 point, 346 uncertainty unscented transformation, 47 view, 456 homogeneous u., 121, 371, 396 with weight matrices, 43 zoom lens, 459