Brian L. Evans, Jíurgen Teich, and Ton A. Kalker. the resampling.

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1 In 199 Proc. of IEEE Asilomar Conf. on Signals, Systems, and Compu ters FAMILIES OF SMITH FORM DECOMPOSITIONS TO SIMPLIFY MULTIDIMENSIONAL FILTER BANK DESIGN Brian L. Evans, Jíurgen Teich, and Ton A. Kalker Dept. of Electrical Engineering and Computer Sciences University of California, Berkeley, CA USA Abstract Imposing structure on the Smith form of an èintegerè periodicity matrix N = U æv leads to eæcient m-d DFT implementations. For resampling matrices, i.e. non-singular rational matrices, we introduce Smith form decomposition algorithms to generate æ matrices whose diagonal elements exhibit minimum variance and U matrices with minimum norm. Such structure simpliæes non-uniform m-d ælter bank design. 1. Introduction In m-d signal processing, a sampling matrix describes the uniform sampling of m-d data because its column vectors form the basis vectors of the sampling grid ë1ë. Once discretized, the sampled m-d data may be resampled onto a diæerent sampling grid. Resampling the sampled signal in the analog domain requires a conversion of the data to analog form and a resampling with a diæerent scheme èwhich may possibly require additional hardwareè. Just as in the 1-D case, performing the resampling in the digital domain avoids the introduction of noise from digital-to-analog and analog-todigital conversion and does not need additional hard- B. L. Evans was supported by the Ptolemy project. J. Teich was supported by a grant from the German government èdfgè, and received supplemental support from the Ptolemy project. B. L. Evans can be reached by phone at and by fax at The Ptolemy project is supported by the Advanced Research Projects Agency and the U.S. Air Force èunder the RASSP program, contract F61-9-C-117è, Semiconductor Research Corporation èproject 9-DC-008è, National Science Foundation èmip-90160è, Oæce of Naval Research èvia Naval Research Laboratoriesè, the State of California MICRO program, and the following companies: Bell Northern Research, Dolby, Hitachi, Mentor Graphics, Mitsubishi, NEC, Paciæc Bell, Philips, Rockwell, Sony, and Synopsys. World Wide Web information about the Ptolemy project is located at URL T. A. Kalker is employed by Philips Research Laboratories, Eindoven, Prof. Holstlaan, 66 AA, The Netherlands. ware because a digital computer can eæciently perform the resampling. In the digital domain, resampling from one grid to another is carried out by mapping the grid coordinates by a rational non-singular matrix known as a resampling matrix. Since a rational matrix can be written as a rational number times an integer matrix, resampling matrices include integer non-singular matrices such as periodicity matrices ë1, ë and upèdownsampling matrices ë, ë. When the resampling matrix is diagonal, the resampling grid is rectangular and the resampling operation is separable. In general, however, resampling matrices are not diagonal, and a transformation is required to decouple the m-d dependencies. One common transformation is known as the Smith form decomposition ë,,,ë. The Smith form decomposes a resampling matrix S into a product of three ësimpler" resampling matrices U æ V, and each component matrix has the same dimensions as S èe.g., æ for -D resamplingè. U and V are regular unimodular integer matrices because their determinant isæ1, and æ is a diagonal integer matrix for which j det æj = j det Sj. The resampling grid associated with S is formed by a linear mapping of samples on an integer grid ærst by V, then by æ, and ænally by U. The linearly mapping of samples by a regular unimodular matrix èor its inverseè corresponds to ashuæing or lossless rearrangement of samples. Thus, downsampling by S can be decomposed into ashuæing of input samples by U, followed by aseparable downsampling by æ, and followed by reshuæing of samples by V ëë. One method for designing m-d non-uniform ælter banks ë6ë ærst decomposes the resampling matrix R associated with each channel into Smith Form U R æ R V R. Then, by cascading the analysisèsynthesis sections, the downèupsampling associated with V R cancels. To simplify the design of the analysisèsynthesis ælters, the skewing from the indexing of the data by each U R matrix and the non-uniform separable resampling by each

2 æ R matrix should be minimized. This is equivalent to making the columns of U R be as close to orthogonal as possible and making the diagonal elements of each diagonal matrix æ R be as close to being equal as possible. More formally, wewant to design U R such that its lower and upper frame bounds, F min and F max respectively, are as close to 1 as possible, where F min and F max are deæned as the minimum and maximum, respectively, of the set ë7ë fku R fk: kfk=1g This paper introduces a new algorithm to decompose a resampling matrix into a Smith form that meets the above requirements, which we will call the minimized Smith equalized form. Section reviews computing Smith form decompositions and imposing structure on them. Section establishes the minimization problem being solved and gives a simple example of ænding Smith equivalent form. Section gives an eæcient algorithm to compute the minimized Smith equivalent form and states the relevant theorems underlying the procedure. Section summarizes the results in the paper, which are not conæned to any particular number of dimensions.. Smith Form Decompositions The Smith form decomposition of a non-singular integer matrix S = UæV was ærst reported by H.J.S. Smith in An algorithm to ænd it iteratively multiplies the given integer matrix on the left and right by elementary èregular unimodularè matrices until the matrix is reduced to a diagonal form ë8ë. The initial step in the decomposition of an n æ n integer matrix S sets U = I næn,æ=s, and V = I næn. In the ærst iteration, an element of æ is pivoted to the è1; 1è position by multiplying on the left to interchange two rows and on the right tointerchange two columns. Each component in the ærst row and column, except the è1,1è element which isnow the pivot, is then reduced modulo the pivot by subtracting a multiple of the pivot. Then, a new pivot is chosen and the process is repeated until all of the components in the ærst row and column, except for the è1,1è position, are zero. The pivoting is then performed at the è,è position and so on until æ is diagonalized. The only degree of freedom in this algorithm is the criteria to choose the pivot at each iteration. In ë8ë, the pivot is chosen to be the element that is smallest absolute value. The algorithm above assumed that the matrix to be decomposed contains only integer components. Smith forms of rational matrices can be computed by ærst factoring out 1=d where d is the least common multiple of all of the denominators of the matrix. The resulting integer matrix is then decomposed into its Smith form. Each diagonal element of the æ matrix is divided by d to produce the Smith-McMillan form èthe m-d analog of rational sampling rate changersè ëë. Even though æ becomes a non-singular rational matrix, U and V remain square regular unimodular integer matrices. Smith forms S = U æv are, however, not unique. The n æ n matrix S is being mapped into two full n æ n matrices èsuch that their determinants are æ1è and one diagonal n æ n matrix èsuch that the product of the diagonals equals the determinant of Sin absolute valueè. The Smith form has many more degrees of freedom than the original matrix. For example, alternate Smith forms can be generated by any pair of regular unimodular matrices X and Y for which the product XæY is a diagonal matrix. Because the inverse of a regular unimodular matrix is regular unimodular and the product of two regular unimodular matrices is regular unimodular, UæV = U, X,1 X æ æ, YY,1æ V =, UX,1æ èxæyè, Y,1 V æ = U æ V è1è One application of this equation is to map a Smith form into its canonical form ë8ë. Another important application of this equation is to derive the conditions to pivot factors along the diagonal elements and the matrices to perform the pivoting ëë. The matrix product XæY can move aninteger factor i from the lth diagonal entry to the kth diagonal entry ëë. X and Y èwith det X = æ1 and det Y = æ1è are each an identity matrix except for æ submatrix, thereby reducing XæY = æ to xkk x kl k 0 ykk y kl = x lk x ll 0 i l y lk y ll èè ik 0 0 l After multiplying both sides by X,1 and dividing both sides by æ = gcdè k ; l è, ^k 0 0 i ^ l ykk y kl 1 det X = y lk y ll xll èè,x kl i^k 0,x lk x kk 0 ^l where ^ k = k =æ and ^ k = l =æ. Without loss of

3 generality, detèxè is set to 1, and after some algebra, x kk = iy ll x lk =,^ l y kk = ix ll y lk = ^ k x kl = ^ k x ll is ëfree" y kl =,^ l y ll is ëfree" èè where,, x ll, and y ll are integers constrained by the fact X and Y are regular unimodular integer matrices. Without loss of generality, det Y = det X =1: det X = det Y = x ll y ll i + ^ k^l = èx ll y ll è i +èèè^ k^l è = æi + æq =1 èè The condition æi + æq = 1 is the Bezout identity ë8ë which has a solution if and only if i and q = ^ k^l are relatively prime. Theorem 1 Given a Smith form decomposition of integer matrix S = U æv, the regular unimodular matrices X and Y applied to the Smith form decomposition according to the equation è1è and deæned by equation èè can move a factor from one diagonal entry of æ to another if and only if i and ^ k are relatively prime and i and ^ l are relatively prime. Euclid's algorithm can compute the solution to the Bezout identity eæciently ë9ë. The Bezout numbers æ and æ are not unique. All solutions to the Bezout identity, ^æand ^æ, can be written in terms of an integral parameter t as ^æ = æ + tq and ^æ = æ, ti.we can use this degree of freedom to compute æ and æ that will yield the U with the smallest norm ëë.. The Minimization Problem For næn resampling matrix S in Smith form S = UæV, we can redistribute the factors of the diagonal elements of æ more evenly using the elementary matrices discussed in the previous section. As mentioned in the previous section, j det Sj = jdet æj = æ ii = i=1 i=1 where the i terms are the diagonal values of the diagonal matrix æ. To minimize the variance, we minimize the arithmetic mean under a given constant geometric mean g = j det Sj 1 n. If the i terms were allowed to take real values, then we could set the arithmetic mean equal to geometric mean, so each diagonal element would be made equal to the geometric mean. i All of the entries of the above matrices, however, must assume integral values. Rational matrices, as previously mentioned, can be rewritten as a rational number times an integer matrix, and this approach would be applied to the integer matrix. In order to minimize the variance of the diagonal entries of æ in the Smith form S = UæV,we can formulate the following minimization problem: min 0 i subject to nx i=1 0 i 0 i = i i=1 i=1 0 i Z for i =1:::n S = U 0 æ 0 V 0 è6è In other words, we are trying to ænd a new set of diagonal elements 0 i such that their sum is minimized while their product èi.e., the determinantè remains constant. Even though the cost function is linear in the free variables 0 i, the overall minimization is highly non-linear because of the constraints. The constraints require that any mapping from the original set of i values to another set of allowable 0 i values must follow Theorem 1 in order to preserve the Smith form decomposition. However, there are many questions concerning a reduced form. The basic problems to be solved are discussed in the next section. Example 1 As an example of the resulting minimal variance form, we will begin with the diagonal entries of some æ matrix as f1; ; 90g. The representation of each entry into a product of prime numbers raised to an integer power is: 1 = 1 = 1 0 = = = 90 = The geometric mean is g =æ10 1 6:6, and the initial arithmetic mean is 9 = 1 1. At each step in the rearrangement, a factor from one of the diagonal elements whose value is above the geometric mean should be distributed to a diagonal element whose values is below the geometric mean if the resulting arithmetic mean is smaller. Table 1 shows all of the legal pivots èas deæned by Theorem 1è of factors of to the other two diagonal elements 1 and and the impact of pivoting on the arithmetic mean. Assuming that we choose the pivot that gives us the smallest arithmetic mean, we pivot the factor from the third element to the ærst element, thereby producing the new diagonal entries f9; ; 10g. In the next iteration, the smallest arithmetic mean occurs when the factor of from the

4 Factor Pivot New Diagonal Arithmetic Mean to 1,, 16 to 1, 6, 17 1 to 1,, 0 1 to 1, 9, to 1 9,, to 1,, 18 8 to 1, 1, Table 1: Rearrangement of Factors of the Diagonal Entries f1; ; 90g third diagonal entry is moved to the second. The ænal diagonal values are f9; 6; g, and the associated arithmetic mean is 6:667 which is very close to the geometric mean of 6:6.. Algorithm For an n æ n integer resampling matrix S, this section will present an algorithm to compute the minimal variance Smith form of S = U æ V. First, this section will give the representation of æ used by the algorithm. Using the representation, we prove that the minimal variance Smith form is unique up to a permutation of the diagonal elements of æ. Then, we state the algorithm. Our algorithm relies on a representation of the prime factorization of the diagonal elements of the n æ n diagonal matrix æ. The representation takes the form of an n æ m matrix, where m is the number of prime factors p appearing in the diagonal elements of æ. The entries of the matrix, say æ, are the exponents of the prime factors: i =æ ii = p æji j ; for i =1:::n è7è In Example 1, the diagonal elements of æ= = contain the four prime factors p = f1; ; ; g,som=. The diagonal elements of æ written as a matrix æ of prime factor powers would be æ = è1 æji è è æji è è æji è è æji è è1è èè è90è Notice that the prime factor exponents have been sorted down each column. This is in fact the necessary and suæcient condition for æ to be in canonical form, as more formally stated next. Let æ and æ 0 be two diagonal matrices. Let fp1; æææ;p m g be the set of all prime factors occurring in any of the diagonal entries of æ or æ 0. Now deæne the n æ m matrices æ and æ by and i =æ ii = p æji j ; for i =1:::n 0 i =æ 0 ii = p æji j ; for i =1:::n That is, æ and æ are matrices containing the powers of the prime factors of the diagonal elements of æ. Using this form, we can easily verify whether or not æ and æ 0 have the same Smith canonical form. Lemma 1 The matrices æ and æ 0 have the same Smith canonical form if and only if there exists permutations j S m such that æ ij = æ jèièj. So, as a particular application of the lemma above, the Smith canonical form of the matrix æ can be found by sorting the columns of the matrix æ in ascending order. Given a resampling matrix S, we ænd one of its Smith forms S = U æ V. Next, we removeany negative signs on the diagonal entries of æ by multiplying æ on the right bydand by multiplying V on the left by D,1 = D, where D is the diagonal matrix D ii = sgnèæ ii è. By factoring the diagonal elements of the new æ, we ænd the matrix æ according to equation è7è. Next, we consider all possible sets of permutations = f j g and ænd a set of permutations 0 that gives the minimum arithmetic mean: min X i Y j p æ j èièj j It is well-known that the set of permutations is generated by the set of two-element permutations. For every such elementary permutation of 0, Theorem 1 provides a pair of unimodular matrices X and Y realizing that permutation. That is, permuting p k j and p l j is equivalent to pivoting p j raised to the èk, lèth or èl,kèth power, whichever is positive. Therefore, by decomposing the permutations of 0 into elementary permutations, we can constructively ænd a pair of regular unimodular matrices X and Y such that æ 0 = XæY is diagonal, and such that the sum of the diagonal entries of æ 0 is minimal among all diagonal matrices with the same Smith canonical form as æ. From Section, we can set the free parameters of each X and Y pair so as to minimize the norm of U. The ænal result will be a

5 Smith form decomposition in the form we desire: the diagonal elements of æ will be as close to each other as possible, and the columns of U will be as orthogonal as possible. A dual of this algorithm exists to minimize V. Figure 1 shows an example of computing the Smith form which simultaneously has minimal variance in æ and minimal norm in either U or V.. Conclusion This paper presents an eæcient algorithm to ænd the Smith form of an integer resampling matrix S = UæV so that the diagonal elements of æ have minimal variance. The algorithm ærst computes a Smith form so that the diagonal elements of æ are positive and ordered. Then, at each iteration, the algorithm applies a square integer regular unimodular matrix to the left and right of æ and the inverse to U and V, respectively, to achieve the minimal variance form. The algorithm takes advantage of the two degrees of freedom in each matrix transformation to minimize U with respect to a given norm measure. The ænal form imposes structure on the Smith form decomposition to simplify the design of non-uniform m-d ælter banks ë6ë. A preliminary version of these algorithms is available for the Mathematica computer algebra environment ë10ë asthe standalone package LatticeTheory.m on the FTP site gauss.eedsp.gatech.edu èip è è in the directory pubèmathematica. Another version of the lattice theory package is embedded in the signal processing packages which isavailable on the same FTP site. 6. References ë1ë D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing. Englewood Cliæs, NJ: Prentice-Hall, Inc., 198. ëë T. Chen and P. P. Vaidyanathan, ëthe role of integer matrices in multidimensional multirate systems," IEEE Trans. on Signal Processing, vol. 1, pp. 10í 107, Mar ëë B. L. Evans, R. H. Bamberger, and J. H. McClellan, ërules for multidimensional multirate structures," IEEE Trans. on Signal Processing, vol., pp. 76í 771, Apr ëë E. Viscito and J. Allebach, ëdesign of perfect reconstruction multidimensional ælter banks using cascaded Smith form matrices," in Proc. IEEE Int. Sym. Circuits and Systems, èespoo, Finlandè, pp. 81í8, June ëë B. L. Evans, T. R. Gardos, and J. H. McClellan, ëimposing structure on Smith form decompositions of rational resampling matrices," IEEE Trans. on Signal Processing, vol., pp. 970í97, Apr ë6ë T. R. Gardos, K. Nayebi, and R. M. Mersereau, ëanalysis and design of multi-dimensional, non-uniform band ælter banks," in SPIE Proc. Visual Communications and Image Processing, pp. 9í60, Nov ë7ë I. Daubechies, Ten Lectures on Wavelets. CBMS-NSF regional conference series in applied mathematics; 61, SIAM, 199. ISBN ë8ë A. Kaufmann and A. Henry-Labordere, Integer and Mixed Programming: Theory and Applications. New York: Academic Press, ë9ë J. H. McClellan and C. M. Rader, Number Theory in Digital Signal Processing. Englewood Cliæs, NJ: Prentice-Hall, ë10ë S. Wolfram, Mathematica: A System for Doing Mathematics by Computer. Redwood City, CA: Addison- Wesley, S = èaè Original resampling matrix 1, ,7,, æ æ,88,901, ,8,1168,08 èbè Smith Equalized Form with Minimized U, ,11 1 9, æ æ, ,10,1, ècè Smith Equalized Form with Minimized V The Smith Form routine returned the diagonal elements of f, 1, 60g. The equalized form is obtained by pivoting a factor of from the third to ærst and a factor of from the third to second positions. Figure 1: Examples of Smith Equalized Forms

Abstract. circumscribes it with a parallelogram, and linearly maps the parallelogram onto

Abstract. circumscribes it with a parallelogram, and linearly maps the parallelogram onto 173 INTERACTIVE GRAPHICAL DESIGN OF TWO-DIMENSIONAL COMPRESSION SYSTEMS Brian L. Evans æ Dept. of Electrical Engineering and Computer Sciences Univ. of California at Berkeley Berkeley, CA 94720 USA ble@eecs.berkeley.edu