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Appendix A Matrix-Vector Representation for Signal Transformation A set of numbers can be used to represent discrete signals. These numbers carry a certain amount of information and are subject to change by various kinds of trans- formations, called systems. For example, a one-dimensional linear time-invariant system can be expressed by its corresponding impulse response. The output of the system is then determined by the convolution of the impulse response and the input signal. Convolution equations, in general, are too complicated to efficiently express related theories and algorithms. Analysis and representation of signal transformations can be substantially sim- plified by using matrix-vector representation, where a vector and a matrix, respec- tively, represent the corresponding signal and transformation. A.1 One-Dimensional Signals and Systems Suppose a one-dimensional system has input signal x(n), n ¼ 0, 1, ..., N À 1 and impulse response h(n). The output of the system can be expressed as the one-dimensional convolution: XNÀ1 ynðÞ¼ hnðÞÀ q xqðÞ,forn ¼ 0, 1, ..., N À 1: ðA:1Þ q¼0 By simply rewriting Eq. (A.1), we have yðÞ¼0 hðÞ0 xðÞþ0 hðÞÀ1 xðÞþ1 hðÞÀ2 xð2 ÞþÁÁÁ yðÞ¼1 hðÞ1 xðÞþ0 hðÞ0 xðÞþ1 hðÞÀ1 xð2 ÞþÁÁÁ ðA:2Þ ⋮ yNðÞ¼À 1 hNðÞÀ 1 xðÞþ0 hNðÞÀ 2 xðÞþ1 hNðÞÀ 3 xð2 ÞþÁÁÁ: © Springer International Publishing Switzerland 2016 249 M.A. Abidi et al., Optimization Techniques in Computer Vision, Advances in Computer Vision and Pattern Recognition, DOI 10.1007/978-3-319-46364-3 250 A Matrix-Vector Representation for Signal Transformation If we express both input and output signals as N  1 vectors, such as x ¼ ½xðÞ0 xð1 Þ ÁÁÁ xNðÞÀ 1 T and y ¼ ½yðÞ0 yð1 Þ ÁÁÁ yNðÞÀ 1 T; ðA:3Þ then the output vector is obtained by the following matrix-vector multiplication y ¼ Hx; ðA:4Þ where 2 3 hðÞ0 hðÞÀ1 hðÀ2 Þ ÁÁÁ hðÞÀN þ 1 6 7 6 7 6 hðÞ1 hðÞ0 hðÀ1 Þ ÁÁÁ hðÞÀN þ 2 7 6 7 6 7: : H ¼ 6 hðÞ2 hðÞ1 hð0 Þ ÁÁÁ hðÞÀN þ 3 7 ðA 5Þ 6 7 4 ⋮⋮⋮⋱⋮5 hNðÞÀ 1 hNðÞÀ 2 hNðÀ 3 Þ ÁÁÁ hðÞ0 We note that H is a Toeplitz matrix having constant elements along the main diagonal and sub-diagonals. If two convolving sequences are periodic with period N, their circular convolu- tion is also periodic. In case, hðÞ¼Àn hNðÞÀ n , which results in the circulant matrix and can be expressed as 2 3 hðÞ0 hNðÞÀ 1 hNðÀ 2 Þ ÁÁÁ hðÞ1 6 7 6 7 6 hðÞ1 hðÞ0 hNðÀ 1 Þ ÁÁÁ hðÞ2 7 6 7 6 7: : H ¼ 6 hðÞ2 hðÞ1 hð0 Þ ÁÁÁ hðÞ3 7 ðA 6Þ 6 7 4 ⋮⋮⋮⋱⋮5 hNðÞÀ 1 hNðÞÀ 2 hNðÀ 3 Þ ÁÁÁ hðÞ0 The first column of H is the same as that of the vector h ¼ ½hðÞ0 hð1 Þ ÁÁÁ hNðÞÀ 1 T, and the second column is the same as that of the rotated version of h indexed by one element, such as ½hNðÞÀ 1 hð0 Þ ÁÁÁ hNðÞÀ 2 T. The remaining columns are determined in the same manner. Example A.1: One-Dimensional Shift-Invariant Filtering and the Circulant Matrix Consider a discrete sequencefg 1 2 3454321. Suppose that the corresponding noisy observation is given as x ¼ ½1:10 1:80 3:10 4:20 5:10 3:70 3:20 2:10 0:70 T. One simple way to remove the noise is to replace each observed sample by the average of the neighboring samples. If we use an averaging filter that replaces each sample by A Matrix-Vector Representation for Signal Transformation 251 the average of two neighboring samples, plus the sample itself, we have the output y ¼ ½1:20 2:00 3:03 4:03 5:00 4:00 3:00 2:00 1:30 T, where the first and the last samples have been computed under the assumption that the input sequence is periodic with period 9, because they are located at a boundary and do not have enough neighboring samples for convolution with the impulse response. The averaging process can be expressed as a one-dimensional time-invariant system whose impulse response is 1 hnðÞ¼ fgδðÞþn þ 1 δðÞþn δðÞn À 1 : ðA:7Þ 3 We can make the corresponding circulant matrix by using the impulse response as 2 3 110ÁÁÁ 1 6 7 6 7 6 111ÁÁÁ 0 7 1 6 7 6 ⋱ 7: : H ¼ 6 011 0 7 ðA 8Þ 3 6 7 4 ⋮⋮⋱⋱⋮5 100ÁÁÁ 1 It is straightforward to prove that y ¼ Hx. A.2 Two-Dimensional Signals and Systems In the previous section we obtained the matrix-vector expression of one-dimensional convolution by mapping an input signal to a vector and the impulse response to a Toeplitz or circulant matrix. In a similar manner, we can also represent two-dimensional convolution as a matrix-vector expression by map- ping an input two-dimensional array into a row-ordered vector and the two-dimensional impulse response into a doubly block circulant matrix. A.2.1 Row-Ordered Vector Two-dimensional rectangular arrays or matrices usually represent image data. Representing two-dimensional image processing systems, however, becomes too complicated to be analyzed if we use two-dimensional matrices for the input and output signals. Based on the idea that both vectors and matrices can represent the same data, only in different formats, we can represent two-dimensional image data by using a row-ordered vector. 252 A Matrix-Vector Representation for Signal Transformation Let the following two-dimensional M  N array represent an image 2 3 xðÞ0; 0 xð0; 1 Þ ÁÁÁ xðÞ0, N À 1 6 7 6 ; ; 7 6 xðÞ1 0 xð1 1 Þ ÁÁÁ xðÞ1, N À 1 7 X ¼ 6 7; ðA:9Þ 4 ⋮⋮⋱⋮5 xMðÞÀ 1, 0 xMðÀ 1, 1 Þ ÁÁÁ xMðÞÀ 1, N À 1 which can also be represented by the row-ordered MN  1 vector, such as  à T x ¼ xðÞ0; 0 xðÞÁÁÁ0; 1 xðÞ0, N À 1 xð1; 0 Þ ÁÁÁ xðÞ1, N À 1 ÁÁÁ xMðÞÁÁÁÀ 1, 0 xMðÞÀ 1, N À 1 : ðA:10Þ A.2.2 Block Matrices A space-invariant two-dimensional system is characterized by a two-dimensional impulse response. The output of the system is determined by two-dimensional convolution, expressed as MXÀ1 XNÀ1 ymðÞ¼; n hmðÞÀ p, n À q xpðÞ; q ; ðA:11Þ p¼0 q¼0 where y(m, n), h(m, n), and x(m, n), respectively, represent the two-dimensional output, the impulse response, and the input signals. Like the one-dimensional case, two-dimensional convolution can also be expressed by matrix-vector multiplication. Example A.2: Two-Dimensional Space-Invariant Filtering and the Block Circulant Matrix Suppose that an N  N image x(m, n) is filtered by the two-dimensional low-pass filter with impulse response: 8 9 δ δ δ <> ðÞþm þ 1, n þ 1 2 ðÞþm þ 1, n ðÞm þ 1, n À 1 => 1 hmðÞ¼; n þ2δðÞþm, n þ 1 4δðÞþm; n 2δðÞm, n À 1 : ðA:12Þ 16 :> ;> þδðÞþm À 1, n þ 1 2δðÞþm À 1, n δðÞm À 1, n À 1 The output is obtained by two-dimensional convolution as given in Eq. (A.11). We can also express the two-dimensional convolution by multiplying the block matrix and the row-ordered vector. If we assume that both the impulse response and the A Matrix-Vector Representation for Signal Transformation 253 input signal are periodic with period N  N, it is straightforward to prove that the matrix-vector multiplication y ¼ Hx; ðA:13Þ is equivalent to the two-dimensional convolution, where the row-ordered vector x is obtained as in Eq. (A.10), and the block matrix is obtained as 2 3 H0 HÀ1 0 ÁÁÁ H1 6 7 6 7 6 H1 H0 HÀ1 ÁÁÁ 0 7 1 6 7 6 7: : H ¼ 6 0 H1 H0 ÁÁÁ 0 7 ðA 14Þ 16 6 7 4 ⋮⋮⋮⋱⋮5 HÀ1 00ÁÁÁ H0 Each element in H is again a matrix defined as 2 3 420ÁÁÁ 2 2 3 6 7 6 7 210ÁÁÁ 1 6 242ÁÁÁ 0 7 6 7 6 7 6 121ÁÁÁ 0 7 6 7 6 7 6 7 6 7: H0 ¼ 6 024ÁÁÁ 0 7, and H1 ¼ HÀ1 ¼ 6 012ÁÁÁ 0 7 6 7 6 7 6 7 4 ⋮⋮⋮⋱⋮5 4 ⋮⋮⋮⋱⋮5 100ÁÁÁ 2 200ÁÁÁ 4 ðA:15Þ Any matrix A whose elements are matrices is called a block matrix, such as 2 3 A0, 0 A0,1 ÁÁÁ A0, NÀ1 6 7 6 7 6 A1, 0 A1,1 ÁÁÁ A1, NÀ1 7 A ¼ 6 7; ðA:16Þ 4 ⋮⋮⋱⋮5 AMÀ1, 0 AMÀ1,1 ÁÁÁ AMÀ1, NÀ1 where Ai,j represents a p  q matrix. More specifically, the matrix A is called an m  n block matrix of basic dimension p  q. If the block structure is circulant, that is, Ai,j ¼ AimodM,jmodN, A is called block circulant. If each Ai,j is a circulant matrix, A is called a circulant block matrix. Finally, if A is both block circulant and circulant block, A is called doubly block circulant. 254 A Matrix-Vector Representation for Signal Transformation A.2.3 Kronecker Products If A and B are M1  M2 and N1  N2 matrices, respectively, their Kronecker product is defined as 2 3 aðÞ0; 0 B ÁÁÁ aðÞ0, M2 À 1 B 6 7 A B 4 ⋮⋱ ⋮ 5; ðA:17Þ aMðÞ1 À 1, 0 B ÁÁÁ aMðÞ1 À 1, M2 À 1 B which is an M1  M2 block matrix of basic dimension N1  N2.
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  • 1 1.1. the DFT Matrix

    1 1.1. the DFT Matrix

    FFT January 20, 2016 1 1.1. The DFT matrix. The DFT matrix. By definition, the sequence f(τ)(τ = 0; 1; 2;:::;N − 1), posesses a discrete Fourier transform F (ν)(ν = 0; 1; 2;:::;N − 1), given by − 1 NX1 F (ν) = f(τ)e−i2π(ν=N)τ : (1.1) N τ=0 Of course, this definition can be immediately rewritten in the matrix form as follows 2 3 2 3 F (1) f(1) 6 7 6 7 6 7 6 7 6 F (2) 7 1 6 f(2) 7 6 . 7 = p F 6 . 7 ; (1.2) 4 . 5 N 4 . 5 F (N − 1) f(N − 1) where the DFT (i.e., the discrete Fourier transform) matrix is defined by 2 3 1 1 1 1 · 1 6 − 7 6 1 w w2 w3 ··· wN 1 7 6 7 h i 6 1 w2 w4 w6 ··· w2(N−1) 7 1 − − 1 6 7 F = p w(k 1)(j 1) = p 6 3 6 9 ··· 3(N−1) 7 N 1≤k;j≤N N 6 1 w w w w 7 6 . 7 4 . 5 1 wN−1 w2(N−1) w3(N−1) ··· w(N−1)(N−1) (1.3) 2πi with w = e N being the primitive N-th root of unity. 1.2. The IDFT matrix. To recover N values of the function from its discrete Fourier transform we simply have to invert the DFT matrix to obtain 2 3 2 3 f(1) F (1) 6 7 6 7 6 f(2) 7 p 6 F (2) 7 6 7 −1 6 7 6 .