A semi-separable approach to a tridiagonal hierarchy of matrices with application to image flow analysis

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1 A semi-separable approach to a tridiagonal hierarchy of matrices with application to image flow analysis Patric Dewilde Klaus Diepold Walter Bamberger Abstract Image flow analysis (e.g. as derived by Horn and Schunc) gives rise to matrices that are not semi-separable in themselves but have a hierarchical structure such that each level of the hierarchy bloc tridiagonal matrices appear, whose bloc entries contain themselves bloc tridiagonal matrices. In this paper we explore how typical algorithms to handle the inversion of semi-separable systems can be adapted to that hierarchical situation. At first it appears that the attractive property of linearity in the number of equations given the state dimension of non hierarchical semi separable inversion gets lost, due to a gradual increase of the semi separable complexity at a lower level of the hierarchy. We indicate how this increase can be ept in chec. This leads to hierarchical algorithms for a more general class than covered by the HSS class. 1 Introduction We follow the notation of the boo 2 on time-varying systems throughtout (it should be clear that semi-separable systems of equations are nothing else but time-discrete and time-varying systems). In particular, we utilize non-uniform vectors characterized by a sequence of dimensions, where particular entries may disappear when they have dimension zero. Finite vectors may always be thought embedded in doubly infinite sequences by augmenting them with empty entries acting as placeholders. Liewise, all matrices considered are of bloc-type and such that entries may disappear when the corresponding entries in the vectors disappear as well - the only additional property needed for matrix calculus with disappearing entries being that the product of a matrix of dimensions m 0 with a matrix of dimensions 0 n is a matrix with zero entries of dimensions m n. Furthermore, special matrices such as bloc-diagonal matrices and the unilateral shift Z are utilized, whereby (uz) = u 1 or (Zu) = u +1 is a unit matrix that shifts indices in the vectors it is applied to as indicated, Z will be its transpose. A semi-separable upper operator then has a representation T = D + BZ(I AZ) 1 C where A, B, C, D are bloc-diagonal matrices (presumably of low dimension), while a semi-separable representation for a bloc lower matrix is similarly given by Delft University of Technology Technical University of Munich T = D + BZ (I AZ ) 1 C. 1

2 The shift operator can equally well be applied to matrices, both left and right resulting in a shift of rows or of columns respectively. In addition we will need a shift along diagonals for which we will use the special notation A (1) = Z AZ to represent a unit shift in the South-East direction on the matrix A (A () will be a shift in the same direction over positions). The sequence of dimensions of the so called state transition operator A is called the degree of the realization (at each particular index), it is closely related to the computational complexity of the local operations needed to compute a matrix vector product with the corresponding semiseparable operator - in many cases a general statement can be made on their size. A general semi-separable operator will be the sum of an upper and a lower operator, each coded with its own semi-separable representation, often called a realization of the operator because of the system theoretic connotations. It is nown that in many cases a high degree semi-separable representation can be efficiently represented by a low degree one, given a certain accuracy requirement. There is a general theory available to derive such representations nown as model reduction theory for time-varying systems, originally due to 1. In 6 an algorithm is presented for exact Hanel norm approximation of a time-varying system, while in 3 an efficient approximation algorithm is given. A tri-diagonal matrix T generically has degree two: for each side of the representation one. We indicate this by writing #T = 1 1. If an invertible upper operator has degree n, then its inverse has also degree n, meaning that there is a realization for it of the same dimension as for the original. In particular, an invertible upper bi-diagonal matrix has an inverse that is a full matrix but still has semi-separable degree one. A general invertible operator will have the same overall degree for its inverse, but the numbers will be redistributed over the upper and the lower part. The degree of the sum and the product of two upper ss-operators is generically the sum of the degrees. Hence, it pays to write the operators and their inverses down in ss form resulting in more efficient calculations. The evaluation of the degree is indicative for (if not equal to) the complexity of the calculations. 2 Image flow analysis Image flow analysis consists in comparing two image pixel maps (an original and a target) and determining for each pixel optimal displacement vectors that when applied to the original yields an image that is optimally close (in the least squares sense) to the target 4. These vectors can then be used for optimal coding as is been done extensively by the MPEG standards. The Horn and Schunc method, as well as other methods for this problem produce matrices that have a strongly pronounced hierarchical form, they are not simply semi separable (in fact, they are not semi separable of low degree at all), but at each level of the hierarchy they exhibit a strongly semi separable structure, if the entries at that level were scalar, then the system would be semiseparable. Image flow analysis is prototypical for a class of system modeling and system inversion problems that are governed by partial differential equations woring in layered structures. They provide a simple but generic instance of a tough modeling problem for which efficient solution algorithms are very welcome. 2

3 3 Elementary inversion In this section we show the basic ideas on an elementary LU type factorization, executed on a (hiarchical) doubly tridiagonal matrix. This algorithm will only be applicable when the original can be LU factorized without pivoting. A more general algorithm capable of computing the Moore- Penrose inverse will be proposed in the next section. Let the original operator be given as T = Z L + M + UZ in which L, M, N are conformal bloc diagonal matrices consisting of tridiagonal blocs. Our goal is to factor T = (Z l+δ)(ɛ+uz). Let us introduce on diagonal matrices the notation A (1) = Z AZ which shifts blocs one notch in the south-east direction. The solution is of course classical: L = lɛ M = δɛ + (lu) (1) U = δu This defines a recursion, all -1 entries are zero, the first step gives: STEP 1: M 0 = δ 0 ɛ 0, l 0 = L 0 ɛ 1 0, u 0 = δ0 1 U 0. GENERIC STEP: δ ɛ = M l 1 u 1 l = L ɛ 1 u = δ 1 U It should be clear that due to the recursion the ss degree will increase by an order 2 at each step of the recursion, yielding degree n by the time the end is reached - an unacceptable (but mathematically correct) situation. The redemption comes from model reduction. It has to be executed at each step, in the first equation. This should eep the ss order of δ and ɛ within a given bound - say s. The model reduction step would require O(sn) operations. As there are n steps in the recursion, the overal LDU factorization complexity is O(sn 2 ), of the order of the number of non-trivial blocs in the matrix, except for a constant the minimal number possible. 4 Application to the Horn and Schunc case In the Horn and Schunc case 4, the general hierarchical appearance of the flow matrix is as follows: D1;1 R T = 1;1 (1) L 1;2 D 1;2 Where D 1;i stands for a bloc tridiagonal matrix of dimension N 2 N 2 assuming square images with N pixels per line: D 1;i = (2) 3

4 where each stands for a normal tridiagonal matrix, and the off diagonal blocs R 1,2 and L 1;2 are purely diagonal matrices (see the previous paper in these proceedings for more detailed information on these types of matrices). We shall play various games on these types of matrices to put them in a form that is amenable to efficient treatment. At this point we try to connect up with the elementary LU or Cholesy factorization theory as rudimentarily explained in the previous section. One attractive way of handling the off diagonal blocs (which are very simple indeed) is to reorder the matrix by interleaving bloc column 1 with bloc column N + 1, 2 with N + 2 etc... This produces a bloc matrix of the form: T = D 1 U 1 V 1 L 1 D 2 U 2 V 2 M 1 L 2 D 3 U 3 M 2 UN 2 V N 2 DN 1 U N 1 M N 2 L N 1 D N The bloc entries do have special structure: the D i, M i and V i are (scalar) tridiagonal of dimension N N, the L 2i+1 and U 2i+1 are diagonal and the L 2i and U 2i are simply zero. This special structure is to be exploited in the next hierarchical level down (i.e. at the matrix entry level). A direct extension of the LU or Cholesy factorization technique of the previous section (in the Horn and Schunc case the matrix is actually positive definite, here we only assume that all pivots are invertible for a little more generality) provides the factorization: δ 1 ɛ 1 u 1 v 1 l 1 δ 2 T = m 1 l 2 δ 3 ɛ 2 u 2 v 2.. ɛ 3 u.. 3 (4).. Woring the recursion out in the classical way gives: δ ɛ = D l 1 u 1 m 2 v 2 l = (L m 1 u 1 )ɛ 1 m = M ɛ 1 u = δ 1 (U l 1 v 1 ) v = δ 1 V It is easy to see that these recursions lead to an increase in complexity of the lower level matrices involved at each step, when operations are executed exactly. The recursion simply starts up with δ 1 ɛ 1 = D 1, and since D 1 is tridiagonal, we have δ 1 and ɛ 1 bidiagonal. Next we find l 1 = L 1 ɛ 1 1 which is now semi-separable of degree 1 since L 1 is diagonal and ɛ 1 1 is the inverse of a band matrix of degree 1 (i.e. a diagonal plus an adjacent diagonal). The situation with m 1 is a little more dramatic, we have m 1 = M 1 ɛ 1 1, which is a product of a degree 2 semi-separable matrix with one of degree one, hence an SS-matrix of degree three. That means that in the next step δ 2 and ɛ 2 will be of degree three, and the resulting v 2 and m 2 will both be of degree six. This process eeps escalating linearly if uncheced. This means that at a certain point (e.g. when degree 8 or 10 is reached), the matrices δ and ɛ have to be trimmed in degree, using a Hanel ran reduction method as proposed e.g. in 2, 3. As long as the degree is ept low this can be done efficiently. 4 (3) (5)

5 An alternative approach for this example is to include a second level of (recursive) hierarchy. Let us define (note the redefined quantities!) D2 1 U M = 2 1 V2 1 0 M2 1 0, U L 2 1 D =, L 2 0 V = 2 0 M 2 then T = M 1 U 1 L 1 M 2 U 2 L 2 M 3. (6) Now the recursions of the previous section are directly applicable. However, it is not clear that this scheme is more advantageous than the previous five bloc diagonals approach. 5 The Moore-Penrose case The algorithm given in 3 and which is derived from 6 can be applied in an adapted form. In the bloc tridiagonal case the first step: conversion to upper form is trivial, the main bloc diagonal is just displaced one bloc down. The crucial step is the inner-outer factorization for this case. This consists in a bloc-square root algorithm whose central ( th) step is as follows. Y A Q Y C Y+1 =. B D Here Q is a unitary matrix which brings the matrix it operates on in row independent form, i.e. er( D o ) = 0 and er( Y +1 ) = 0. Y +1 is a connecting matrix that is passed from the th stage to the next one, carrying the residue which might be involved further-on in the Moore-Penrose inversion. It plays a crucial role in eeping the computational complexity down. In our case, the {A, B, C, D } have a special form, namely (with the notation of section (3) 0 0 I A C = U 0 M. 0 I B D The equation to be solved at the th step is (1) the determination of an orthogonal transformation matrix such that (here we assume that the original system is invertible, if that is not the case, then D o can be rectangular and we have to require that it is right invertible, zero rows have to be collected separately and Y +1 must be required to be right invertible as well, we sip details, see 2) Y1 + Y Q 2 M 0 = L D o followed by the update of Y : B o D o L Y+1,1 Y +1,2 = Q Y2 U 0 Given that the ss-degree is limited to s, these update operations necessitate a degree s approximation of the resulting matrices Y and Q. The latter will be automatic if the former is ept in chec. 5

6 Because of the hierarchical nature of the matrices, the ss degree of the given data at a given recursive step is given by the ss degree of Y plus the ss degree of the base matrices {A, B, C, D }. In the QR factorization we now that we can eep the ss degree of the results the same, the degree is hence increased by two. Keeping it in chec requires a model reduction either at each step, or, preferably, after a number of steps. In the hierarchical case given by eq. (6), the realization specializes further to 0 0 I A C = U B D 0 M (7) 0 I L in which the significant entries are 2 2 blocs as defined in the previous section - their significant entries in turn are either tridiagonal or diagonal matrices of size N N. The connecting matrix Y is a 2 2 matrix. Given Y, Q and Y +1 are recursively defined by Q Y ;11 + Y ;12 M Y ;21 + Y ;22 M L 0 = in which D o is square non-singular (when the original matrix is invertible or right invertible when it is not) and Y;12 U Y +1 = Q 0. (9) Y ;22 U 0 Here again we observe the increase in degree at each step, a degree reduction is called for at regular intervals. D o (8) 6 The Horn-Schunc case revisited In the special case of the Horn Schunc algorithm, a further exploitation of the data is advantageous and leads to a simpler solution. We start out with the observation that the Horn Schunc equations can conveniently be summarized as follows: where ( α 2 C 0 0 C + I1 I 2 I1 I 2 ) x 1 x 2 = I1 I 2 I t (10) C is a positive definite, bloc tridiagonal matrix with blocs that are themselves tridiagonal. In addition, C is Toeplitz bloc Toeplitz with fixed (data independent) entries. It represents the negative of the Laplace operator; α is a scalar constant; I are diagonal matrices with light intensities as its entries; the unnowns x are the sought after components of the flow vector; 6

7 I t is also a matrix with intensities. To invert the factor between round bracets we first extract the bloc diagonals of C s, and utilize the inversion rule (I + ab T ) 1 = I a(i + b T a) 1 b T to obtain x1 x 2 = α 2 C 1 I 1 C 1 I 2 (I + α 2 (I 1 C 1 I 1 + I 2 C 1 I 2 ) ) 1 It. (11) Hence the problem is reduced to the computation of C 1 I 1 and C 1 I 2 in which C is a constant Laplace matrix, and the inversion of the braceted middle term. These two problems are substantially different in nature. The first is basically a 2D inverse filtering problem of data that has been passed through a nown direct filter, which has a doubly banded sparse Toeplitz bloc Toeplitz structure. While straight 2D filtering methods could be used here, we present a solution in which the inverse filter is realized as a low degree, hierarchically semi-separable system. The matrix C has the form C= M U U M U U M U. U..... U U M in which M and U indicate (equal) matrices that are all Toeplitz tridiagonal and positive definite. The inverse of this matrix is best characterized in terms of its Cholesy factors. If C = LL then we have L = m 1 1 µ 1 1 Um 1 2 µ 1 1 Um 1 m 1 2 µ 1 2 Um 1 in which the µ satisfy the (Cholesy-Riccati) recursion m 1 µ +1 = M Uµ 1 U with starting value µ 1 = M, and µ i = m i m i. L 1 and L exhibit a hierarchical semi-separable form which is to be exploited when computing its multiplication with a vector, and each of their entries itself will exhibit an orthodox semi separable form provided a semi separable approximant can be obtained for the µ i either by Hanel norm model reduction or Schur approximation, see the next paragraph for the latter, the treatment of these approximations has been done in the literature cited. A further step is taen by assuming that the Cholesy-Riccati recursion stabilizes quicly to a steady state value (that will be the case in the present example), in that case all µ can be taen equal and an invariant, a low complexity inverse filter will result which can be precomputed and burned into the architecture. For the inversion of the braceted term the situation is quite different as the entries of the matrix to be inverted are data dependent. There are two different avenues possible. The first has already been described in the previous sections, since the matrix has a triple band of blocs. The second consists 7

8 in using a hierarchical Schur interpolation method as proposed in 7, 8. We briefly introduce this latter method. Each of the non-zero submatrices is sampled along its main diagonal and a few side diagonals, in a symmetric fashion, producing a pattern of sampled data that loos lie The hierarchical Schur interpolation will produce a tridiagonal bloc tridiagonal approximate inverse of this matrix fitting exactly the same pattern, and using only the sampled data to compute it. It is based on the observation that the Schur interpolation technique is additive on the inverse, and that superblocs of 2 2 adjacent blocs can be reordered to yield banded matrices, for details we refer to the literature cited Generalizations It is an open question whether the algorithms presented here can be generalized to higher orders of hierarchy. We believe they can, provided a new notion of symbolic degree of semi-separability is defined. References 1 P. Dewilde and A.-J. van der Veen, On the Hanel-norm approximation of upper-triangular operators and matrices, Integral Eq. and Operator Theory, 17(1):1-45, P. Dewilde and A.-J. van der Veen, Time-Varying Systems and Computations, Kluwer 1998, 450 pp. 3 S. Chandrasearan, P. Dewilde, M. Gu, T. Pals, A.-J. van der Veen, Fast Stable Solvers for Sequentially Semi-separable Linear Systems of Equations, SIMAX to appear. 8

9 4 B.K.P. Horn and B.G. Schunc, Determining Optical Flow, Artificial Intelligence, 17, (1981) W. Hacbusch, A sparse algorithm bases on H-matrices. Part I: Introduction to H-matrices, Computing 62, pp , Inner-outer factorization and the inversion of locally finite systems of equations, Linear Algebra and its Applications, 313:53-100, H. Nelis, Sparse Approximations of Inverse Matrices, Ph. D. Thesis, Delft University of Technology, Inversion of partially specified positive definite matrices by inverse scattering. Operator Theory: Advances and Applications, 40: ,