Appendix A Estimating the AR Coefficients for the 3DAR Model This appendix presents the least squared solution (the Maximum Likelihood solution may be found in [149]) for the coefficients of the three dimensional autoregressive model as outlined in Chapter 2. The model is best discussed in its prediction mode. The prediction equation is as below where i (i, j, n) is the predicted value of the pixel at (i, j, n). N I(i, j, n) = L ak!(i + qk(x) + SXn,n+qk(n), j + qk(y) + SYn,n+qk(n), n + qk(n)) k=l (A.1) The task then becomes to choose the parameters in order to minimize some function of the error, or residual, t:(i,j,n) = I(i,j,n)- I(i,j,n) (A.2) The parameters of the model are both the AR coefficients a= [a 1 ,a2 ,a3 .. aN], and the displacement dk,l = [sxk,l SYk,l OJ. This section is concerned only with coefficient estimation given an estimate for the displacement. The coefficients are chosen to minimize the squared error, ;:(), above. This leads to the Normal equations [11, 32, 71]. The derivation is the same as the one dimensional case and the solution can be determined by invoking the prin­ ciple of orthogonality. E[t: 2 (i,j,n)] is minimized by making the error t:(i,j,n) orthogonal to the signal values used in its generation [71]. Therefore, E[t:(i, j, n)I(i + qm (x) + SXn,n+qm(n), j + qm(Y) + SYn,n+qm(n)' n + qm(n))] = 0 (A.3) 274 Appendix A. Estimating the AR Coefficients for the 3DAR Model Where m = 1 ... N. Defining q 0 = [0, 0, OJ and a0 = 1.0, then N E(i, j, n) = 2:.>kl(i + qk (x) + SXn,n+qk(n), j + qk(y) + SYn,n+qk(n), n + qk(n)) k=O (A.4) Note that the ak are now reversed in sign to allow for this simpler formulation. To continue, the following notation is introduced. X [i j nJ (A.5) [qk(x) qk(y) qk(n)J (A.6) [sxn,n+qk(n) SYn,n+qk(n) OJ (A.7) Substituting for E() in equation A.3 gives, N L akE[I(x + qk + dx,x+qk)I(x + qm + dx,x+q,JJ 0 (A.8) k=O Vm l..N The expectation can be recognized as a term from the autocorrelation function of the 3-D signal J(x). Matters may be simplified therefore by redefining the equation as N LakC(q~,q;,_,) = 0 (A.9) k=O Where q~, q;,_, are both motion compensated vector offsets as defined impli­ citly in the previous equation. However, a0 has already been defined to be 1.0. Therefore, letting a = [at a2 .. aNf (A.lO) C(q~' q~) C(q~,q;) C(q~' q~) C(q;, qD C(q;,q;) C(q;,qN) c C( q~, q1') C(q~,q;) C(q~, q~v) (A.ll) c (qN, q~) c (qN, q~) c (qN, qN l c [C ( qo , q~ ) C ( qo , q;) ... C ( qo , qN) J (A.l2) [0 0 0] (A.l3) the parameters, a can be determined by solving Ca= -c (A.l4) It must be pointed out that although C is symmetric, it is not Toeplitz in the multidimensional case. This is due to the fact that along a diagonal, the differences between the offset vectors that define each correlation term are not Appendix A. Estimating the AR Coefficients for the 3DAR Model 275 necessarily parallel or the same magnitude. Consider the diagonal of matrix C, consisting of terms at locations (2, 1](3, 2](4, 3] ... [N, N -1], where the top left element of Cis at position (1, 1]. Then vector v~ = [q~- qU is not necessarily equal to v2 = [q~- q~] or v3 = [q~- q~] or any other such difference vector along the diagonal. The support vectors q may be chosen to allow this to occur by choosing vectors that lie along a line in the support volume. In general, however, when the support set delineates some volume, the vectors do not allow C to be Toeplitz. Therefore, it is difficult to exploit the structure of this matrix for computational purposes. In the book, the equation A.14 is solved exactly. That is to say that no approximations about the autocorrelation function are made in estimating C or c. The expectation operator in equation A.9 is taken to be the mean operation. Note that in order to calculate the required autocorrelation terms from a block of data of size N x N in the current frame n say, the offset vectors q require that data outside this block is necessary. The extent of this extra data is explained next. Figure 2.5 shows a support set of 5 vectors. Calculation of C(qo, q2), say, requires the following sum of products, where q2 = (-1, 0, -1]. L I(x + qo)I(x + q 2 ) (A.15) xEB1 Block B1 is of size N x N as stated before, and this yields data for I(x + qo). The term I(x + q2) requires data from a block, B2, which is in the previous frame and the same size, but offset by q2 in that frame. In this case therefore, to solve for the AR coefficients exactly in blocks of size N x N involves data from a block of size (N + 2) x (N + 2) in the previous frame centred at the same position. Appendix B The Residual from a Non-Causal AR Model is not White This section investigates the nature of the residual sequence from an AR model given a least squared estimate for the coefficients of the model. The analysis shows that unlike the causal AR model, the error or residual sequence of a non­ causal model is not white but coloured (See [71, 149, 193, 192]). The model is considered in its 3D form as introduced in Chapter 2. The model equation is as follows (see Chapter 2). N I(x) = ~ akl(x + qk) + c:(x) (B.1) k=l This form of the model does not allow for any motion of objects between frames. Incorporation of this movement makes the expressions more cumbersome but does not affect the result. Typical support sets of N = 9 and N = 1 vectors defined by different qk are shown in Figure 3.8. In solving for the coefficients using the least squared approach (see Ap­ pendix A), the error, c:(x) is made orthogonal to the data at the locations pointed to by the support vectors, qk. This implies that E[c:(x)I(x + qn)] = 0 for n = 1 ... N (B.2) The goal of this analysis is to find an expression for the correlation function of c:(x). That is R .. (x, qn) = E[c:(x)t:(x + qn)] (B.3) Multiplying equation B.1 by c(x + qn) and taking expectations gives N E(I(x)c(x + qn)] = ~ akE[I(x + qk)E(x + qn)] + E[c(x)t:(x + qn)] k=l (B.4) 278 Appendix B. The Residual from a Non-Causal AR Model is not White Let the variance of E(x) be (J;c Then from B.4, when qn = [0 0 0], E[J(x)t:(x)] = a;, (B.5) The summation term disappears because of equation B.2, since x :j:. (x + qk)· When the qn refer to other positions within the support of the model then the following simplifications may be made E[J(x)t:(x + qn)] 0 by B.2 (B.6) N ~ akE[I(x + qk)E(x + qn)] (B.7) k=l These simplifications can be substituted into B.4 to give the correlation term for non-zero vector lags. From this substitution it can be seen that the correlation structure of t:(x) is not white and it depends on the model coefficients. The final result then is for qn = [0 0 OJ (B.S) for n = 1 ... N Appendix C Estimating Displacement 1n the 3DAR Model The three dimensional autoregressive model incorporating motion is defined as below (from Chapter 2). I(i,j,n)= N I>kJ(i + qk(x) + SXn,n+qk(n), j + qk(y) + SYn,n+q, (n), n + qk (n)) + E(i, j, n) k=l (C.l) The parameters of the model are both the AR coefficients a= [a 1 , a 2 , a3 .. aN], and the displacement dk,l = [sxk,l syk,l 0]. This section is concerned only with Least Squares displacement estimation given an estimate for the coefficients. For a Bayesian approach see Chapter 7. In order to gain an explicit relation for E( ·), in terms of d, the approach used by Biemond [17] and Efstratiadis [37], was to expand the image function, I(-), in the previous frames, as a Taylor series about the current displacement guess. This effectively linearizes the equation for E(-) and allows a closed form estimate for d. It is this solution that is used for estimating d in this work. The derivation given here is for a general non-causal model which involves support in both the past and future frames as well as the current frame. It is necessary first of all to separate the support region for the AR model into three parts. 1. The support in the frames previous to the current one, i.e. qk ( n) < 0. This is the temporally causal support. 2. The support in the current frame, qk(n) = 0.