Dudgeon, D. E. and Mersereau, R. M. Multidimensional Digital Signal Processing, Prentice-Hall, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....if and only if X does not vanish and has a continuous branch of the logarithm on T . If the conditions are satisfied, then F log x F x = log X: 3) Equation (3) sheds new light on the cepstrum and homomorphic transform. Though the definitions of the cepstrum in literature vary (see e.g. [14], 3] they are essentially equivalent to the following definition: Definition 1 If x and there exists a sequence such that F #x = F x ; 4) is called the cepstrum of x. By Theorem 3 the cepstrum of x and log x are equivalent and a sufficient and necessary condition for ....

....given by (5) Theorem 7 describes not only the homomorphisms of the Banach algebra 1 .I P but also the Gelfand transform 0. x = X k#I P ; where P 1 m : 6) The Gelfand transform is therefore equivalent to the discrete Fourier transform of n dimensional periodic sequences [14]. Obviously, X k are the samples of the Fourier transform X: X k X.j2 k P 1 m : From Proposition 1 we get the well known invertibility condition in P P 1 m for k I P T ; then x is invertible in P if and only if X k The logarithmic function in P , however, ....

D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing. (Englewood Cliffs, NJ: PrenticeHall, 1984).

....fact quarterplane causal, which by definition means that there is a solution to equations (2.1) of the form k,l#0 Thus we see that it is important to develop criteria for when a polynomial of two variables is stable. Conditions for stability have been extensively investigated [23] 10] 12] [13] and more recently in [2] In [15] a set of three one dimensional tests were developed that characterize whether or not a two variable polynomial is stable. The bivariate autoregressive filter design problem is the following. Given are autocorrelation elements c k,l = E(x k,l x 0,0 ) k, l) ....

D. E. Dudgeon and R. M. Merserau, Multidimensional Digital Signal Processing, Prentice-Hall, Englewood Cli#s, 1984.

....incrementally as the kernel is moved from one position to another across the image. Keywords: Digital convolutions, correlations, digital morphology, digital geometry, approximation algorithms. 1 Introduction A fundamental problem in image processing is that of computing discrete convolutions [9, 12, 5]. Consider an image, which is given as a 2 dimensional m n array I of real numeric values, and a q r image array K, called the kernel (sometimes called a template or structuring element in the literature) The discrete convolution [9] of I with K, denoted by I K, is de ned to be (I K) x; ....

D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Processing. Prentice-Hall, Englewood Clis, NJ, 1984.

.... 14 VI. Non rectangular Sampling for IBR Chan and Shum were the first authors who realized that the plenoptic sampling problem is basically a multidimensional sampling problem [7] It is well known that the rectangular sampling strategy is not necessary optimal for multidimensional signals [17]. Instead, the generalized periodic sampling theory with arbitrary geometry should be applied. We refer the details of the theory to [17] 18] In [7] the authors mentioned that it is also possible to determine the minimum sampling densities for the quincunx and hexagonal sampling lattices. We ....

....basically a multidimensional sampling problem [7] It is well known that the rectangular sampling strategy is not necessary optimal for multidimensional signals [17] Instead, the generalized periodic sampling theory with arbitrary geometry should be applied. We refer the details of the theory to [17][18] In [7] the authors mentioned that it is also possible to determine the minimum sampling densities for the quincunx and hexagonal sampling lattices. We have performed some experiments on non rectangular sampling for IBR. We will briefly go over our experiments and give some comments. ....

D.E. Dudgeon and R.M. Mersereau, Multidimensional Digital Signal Processing, Prentice-hall signal processing series, 1984.

....signal model given in [7] s(x, t) l A l e i# l (t k x) 1) where x is the observation location in three dimensional space, t is time and A l is the amplitude of frequency component # l . Vector k is the propagation vector and it is closely related to the wavenumber vector # defined in [11] k = kx ky kz ] 2) An estimate for propagation vector k can be obtained using four sensors as follows. First, estimate time delays between all sensor pairs to form the time delay vector t. Time delays can be estimated reprint republish this material for advertising or promotional ....

D. E. Dudgeon and R. M. Merserau, Multidimensional digital signal processing, Prentice-Hall, 1984.

....) s 0 ,t 0 ) light ray z Object Focal plane Camera plane Figure 1 Lightfield parameterization. Since the number of images taken in lightfield is often huge, it is very important to know the minimum sampling requirement for a specific scene. In this paper, we apply generalized sampling (GS) [7] to lightfield. We adopt the spectrum analysis results by Chai et al. 5] and show that with generalized sampling the sampling efficiency can be improved by a factor of 2 compared with rectangular sampling (RS) However, when we use GS in practice, we find many limitations. In this paper, we will ....

....can be improved by a factor of 2 compared with Figure 2 (b) which means we only need 50 of the samples. The reconstruction filter is marked in bold contour in Figure 3 (a) which is a tilted fan like filter. For detailed information about the generalized sampling theory, we refer the reader to [7]. a) b) Figure 3 Generalized sampling of lightfield. a) The most compact way to pack the lightfield spectrum. b) Reduce the sampling rate such that GS has the same efficiency as RS in Figure 2 (b) There are several problems to be concerned for the GS approach. First, ....

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D.E. Dudgeon and R.M. Mersereau, Multidimensional Digital Signal Processing, Prentice-hall signal processing series, 1984.

....by the same set of basis vectors and (where each is an integer) and a different choice of origin. If point belongs to a certain partition (sublattice) so do points and . Hence, the partitioning function exhibits the so called generalized periodicity (in other words, it is doubly periodic [9]) Let us define the periodicity matrix as is typical in the signal processing literature [9] 1) The matrix as defined above is the transpose of the sublattice generator matrix (in lattice theory [6] basis vectors are usually written as row vectors) Throughout the paper, we will use the term ....

....of origin. If point belongs to a certain partition (sublattice) so do points and . Hence, the partitioning function exhibits the so called generalized periodicity (in other words, it is doubly periodic [9] Let us define the periodicity matrix as is typical in the signal processing literature [9] (1) The matrix as defined above is the transpose of the sublattice generator matrix (in lattice theory [6] basis vectors are usually written as row vectors) Throughout the paper, we will use the term periodicity matrix bearing in mind that this is just the short way of saying transpose of ....

D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1983.

....form: 3:2) The restriction of fE (e) to the basic cell P 0 is fE (e) For a brief summary of the relation between multidimensional Fourier series and Fourier transform see Appendix A. For further details of multidimensional Fourier series representation, the reader is referred to [24]. Theorem 1. The quantization error of a lattice quantizer Q(P 0 ; V) is uniform in P 0 , that is ; e 2 P 0 ; 3:3) if and only if the input vector x is Nyquist V, that is Phi X (Un) ffi(n) Proof. From the properties of Fourier series, we know that fE (e) in (3.2) is a constant for ....

....that our solution reduces to the scalar half whitening filter. 21 APPENDIX A The definitions of multidimensional Fourier transform, Fourier series and their interrelations are summarized here in a way most suited to our notations. Details can be found in many standard references, for example [24]. 1. The MD Fourier transform of f(x) is defined as We see that the characteristic function (2.7) is therefore Phi X ) F ) 2. f(x) is said to be periodic V; if f(x Vn) f(x) for every x 2 R and n 2 Z : Let P be a basic cell with respect to V, and let U be the matrix ....

D.E. Dudgeon and R.M. Mersereau, Multidimensional digital signal processing, New Jersey: Prentice Hall, 1984.

....in one dimension. This decoupling of the 2 D problem into sequential 1 D problems leads to a substantial reduction in computational cost without loss of asymptotic accuracy. Estimating the frequencies of two dimensional sinusoids has received considerable interest in the literature, see e.g. [12, 13, 21, 24, 28, 35, 44]. This problem finds applications in many fields such as sonar, radar, geophysics, radio astronomy, radio communications, and medical imaging. A typical application, that is the focus of this thesis, is to find the DOA and Doppler frequencies of multiple targets with a pulsed Doppler radar using ....

D. E. Dudgeon and R. M. Merserau. Multidimensional Digital Signal Processing. Prentice-Hall, Englewood-Cli#s, NJ, 1984.

....as .2, 2. N M j cn Jtm Xk, l = Xn, me , 2) n=lm=l where j = and (k, l) e ( 1,N] 1, M] are the discrete fre quencies. Using a 2D Fast Fourier transform (FFT) we can compute the DFT in O(NMlog(NM) operations, instead of O(N2M 2) operations required for the direct evaluation of Eq. 2 [5]. Fundamental for the implementation of the spectral filters described below is the convolution theorem. It relates a convolu tion x y of two signals x and y in the spatial domain with a multiplication in the spectral domain: F(x y) F(x) F(y) Instead of doing a computationally expensive ....

Dudgeon, D.E., Mersereau, R.M. Multidimensional Digital Signal Processing, Prentice-Hall, 1984.

....frequency donlain. In this case, among all periodic sampling schemes x(k) xc(Tk) with a sampling matrix T C IR 2x2 the hexagonal sampling T: tl, t2] with tl,2: 1, 4 q] T is the most efficient one, because there are fewest samples needed to describe the whole information of the continous signal [2]. And in fact, as shown in [18] the photoreceptors in the mammal fovea are nearly hexagonally arranged. The average cone spacing (csp) has been estimated to about 1 csp = tl,2[ 2.5. 2.8 m. The sampling theorem of SHANNON [2] then would postulate a maximal frequency of 53. 56 cpd in the ....

....needed to describe the whole information of the continous signal [2] And in fact, as shown in [18] the photoreceptors in the mammal fovea are nearly hexagonally arranged. The average cone spacing (csp) has been estimated to about 1 csp = tl,2[ 2.5. 2.8 m. The sampling theorem of SHANNON [2] then would postulate a maximal frequency of 53. 56 cpd in the signal leading to the presumption that the sampling in the fovea of the human eye nearly complies with the sampling theorem. Indeed this conclusion seems to be correct and gets along with WILLIAMS [16] Although the mathematical ....

Dudgeon, D.E., Mersereau, R.M.: Multidimensional Digital Signal Processing. Prentice Hall (1984)

....illustrate the clear computational gain in comparison to both the wellknown classical implementation and the method recently published by Liu et al. 1. INTRODUCTION The problem of two dimensional (2 D) high resolution spectral estimation has been widely studied in the past literature (see, e.g. [1, 2, 3]) as well as in more recent contributions such as [4, 5, 6] Applications occur in a wide variety of fields, such as geophysics, radio astronomy, biomedical engineering, sonar, and radar, to mention a few. In many of these applications, it is of key importance to obtain computationally efficient ....

D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing, Prentice Hall, Englewood Cliffs, N. J., 1984.

....representation of the function we want to visualize can be arbitrarily chosen since typically only two dimensional projections of the data set are examined. Since it has been shown that a bcc grid can represent isotropic, band limited data as accurately as Cartesian grids using 29.3 less samples [4], the advantages of using a bcc grid are significant. In this paper we will show how we can take advantage of hexagonal sampling in volume rendering. We outline and propose solutions for the inherent issues of re sampling of rectangular grids as well as interpolation and gradient estimation. The ....

....simplicity we will first describe our method to find an optimal sampling pattern in 2D before we directly delve into the mysterious structure of 3D Euclidian space. 2. 1 Optimal sampling density in 2D We will describe sampling as a mapping from indices to actual sample positions as introduced in [4]: x y = V i j (1) Here the integers i; j are the indices of the sample and x; y is it s corresponding sampling position. The matrix V , which is called sampling matrix, describes the mapping itself, e.g. Vrect2D = T1 0 0 T2 (2) is the matrix for rectangular sampling ....

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D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Processing. Prentice-Hall, Inc., Englewood-Cliffs, NJ, 1st edition, 1984.

....and a minimal amount of hardware, while the second (Ex tended) makes increased use of information derived from the code being executed and requires additional hardware for implementation. Consider the case of the Infinite Extent Impulse Response filter (IIR) below, originally described in [3]. H(z 1 ; z 2 ) 1 (1 P 2 n 1 =0 P 2 n 2 =0 c(n1;n2 ) z1 n 1 z2 n 2 ) This filter can be translated to: y(n 1 ; n 2 ) x(n 1 ; n 2 ) P 2 k1=0 P 2 k2=0 c(k 1 ; k 2) y(n 1 k 1 ; n 2 k 2 ) for k 1 ; k 2 6= 0. In simulations run on this filter in a five level memory system, the ....

D.E. Dudgeon and R.M. Mersereau. Multidimensional Digital Signal Processing. Prentice Hall, Englewood Hills, NJ, 1984.

.... among all periodic sampling schemes x(k) x c (T k) with a sampling matrix T 2 IR 2 Theta2 the hexagonal sampling T = t 1 ; t 2 ] T h 1 1 p 3 Gamma p 3 (1) is the most efficient one, because there are fewest samples needed to describe the whole information of the continous signal [1]. And in fact, as shown in [13] the photoreceptors in the mammal fovea are nearly hexagonally arranged. The average cone spacing (csp) has been estimated to about 1 csp = kt 1;2 k 2:5 : 2:8 m. The sampling theorem of SHANNON [1] then would postulate a maximal frequency of 53 : 56 cpd in ....

....needed to describe the whole information of the continous signal [1] And in fact, as shown in [13] the photoreceptors in the mammal fovea are nearly hexagonally arranged. The average cone spacing (csp) has been estimated to about 1 csp = kt 1;2 k 2:5 : 2:8 m. The sampling theorem of SHANNON [1] then would postulate a maximal frequency of 53 : 56 cpd in the signal leading to the presumption that the sampling in the fovea of the human eye nearly complies with the sampling theorem. Indeed this conclusion seems to be correct and gets along with the experiments of WILLIAMS [11] ....

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D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Processing. Prentice-Hall, 1984.

....The classification performed on the aggregated data might be performed by a human operator or automatically. Both the method of performing data aggregation and the classification algorithm are application specific. For example, acoustic signals are often combined using a beamforming algorithm [5, 17] to reduce several signals into a single signal that contains the relevant information of all the individual signals. Large energy gains can be achieved by performing the data fusion or classification algorithm locally, thereby requiring much less data to be transmitted to the base station. By ....

D. Dudgeon and R. Mersereau. Multidimensional Digital Signal Processing, chapter 6. Prentice-Hall, Inc., 1984.

....assumed to be random and independent of one another. The spatial power spectrum distribution (PSD) of the extended object field is referred to as its spatial image. The actual K D image vector u obtained with the imaging system can be expressed in the form of a linear image degradation equation [1, 2], u = F v n , with the symmetric blur matrix F corresponding to the system s point spread function (PSF) and the additive noise n with a priori unknown statistics. The desired PSD vector v is referred to as the original image, while the actually formed image u is referred to as the degraded ....

....v obtained with M different sensor systems, given the models of the PSFs F (m) of the systems but given no prior statistical models of noise n (m) m = 1, M , in the degraded images. It is well known that the PSFs F (m) are ill conditioned for practical image formation systems [1, 2], hence, the regularization based approach is needed when dealing with the formulated problem. III. MODEL BASED PROBLEM REGULARIZATION We impose maximum entropy (ME) H (v) K k k v 1 ln k v as a prior model of the desired image v that assumes the minimum of prior knowledge about the ....

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E. Dadgeon and R.M. Mersereau, Multidimensional Digital Signal Processing, Prentice Hall Inc., Englewood Cliffs, NJ, 1984.

....[13] and Thompson [14] In this paper we use the 1 D DFT array by Beraldin et al. 15] and derive a modular array structure for computing the 2 D DFT without transposition memory. The 2 D separable convolution has been studied extensively for many years (see for example Dudgeon and Mersereau [16]) and has been used as a basis for constructing general transfer functions. Abramatic et al. 17] has used separable FIR systems for constructing non separable IIR filters, and Treitel and Shanks [18] have used parallel separable filters for approximating nonseparable ones. Despite the limited ....

D. E. Dudgeon and R. M. Mersereau. "Multidimensional Digital Signal Processing". PrenticeHall, 1984.

....e mail: karam asu.edu. This work was supported in part by the Joint Services Electronics Program, Contract DAAH 04 93 G0027. 1 Introduction The term migration refers to a processing technique that is applied to seismic data sections to compensate for some undesirable effects of wave propagation [1, 2, 3]. The geometry of the seismic 2 D migration problem is illustrated in Fig. 1, which is a simplified view of a two dimensional earth. The variable x represents position on the surface, which is assumed to be flat, and the variable z represents depth. The objective is to define the boundaries of the ....

....at depth z and taking the Fourier transform of both sides of (1) we obtain a second order differential equation for Phi(K x ; z; Omega Gamma in the variable z. Given the initial condition at z = z o , the solution of the obtained differential equation for an upward propagating wave is given by [2] Phi(K x ; z o Deltaz; Omega Gamma = e j Deltaz s ( Omega 2 v 2 Gamma K 2 x ) Phi(K x ; z o ; Omega Gamma = A(K x ; Omega Gamma31 K x ; z o ; Omega Gamma (2) where the extrapolation operator A(K x ; Omega Gamma is a linear shift invariant analog filter. In practice, the wave ....

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D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1984.

....e.g. a spatial rectangle in the input over which the convolution should be applied, can be defined. Subsampling is user selectable and decreases the computational cost for the convolution. There are two basic classes of filters, FIR (finite impulse response) and IIR (infinite impulse response) [1]. Both types of filters are supported by the convolver. 2. Generalized convolution Convolution is defined as a product sum of a kernel K and input data I over a neghbourhood m: out(n) X m K(m) I(n Gamma m) 1) where m is the spatio temporal coordinate in the kernel and n is the ....

D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Processing. Prentice-Hall signal processing series. Prentice-Hall, 1984. ISBN 0-13-604959-1.

....of centers of four basis functions (Fig. 2(a) The display sample grid will be referred to as the interleaved grid. kb hx hexagonal packing of the Kaiser Bessel basis function with display samples on a square grid (Fig. 2(b) In 2 D, hexagonal packing has the property of maximum packing density [24, 25]. The centers of the basis functions are on a triangular grid defined by x = 2 p 3 i 1 p 3 j j Deltax ; 33) where i and j are integers (negative and positive) and Deltax is a (scalar) spacing parameter. The square grid used previously is generated by x = i Deltax j Deltax) ....

D.E. Dudgeon and R.M. Mersereau, Multidimensional Digital Signal Processing. New Jersey, U.S.A.: Prentice Hall, 1984.

....is de scribed by e(s; t) exp( j(w 0 t k 0 s) 1) where the plane wave is characterized by the wavenumber k 0 and the angular frequency w 0 . The time is t and s is the observation point in space. The function (1) is considered to be an elemental function in the 4D Fourier transform pair [3] p(s; t) 1 (2p) 4 Z ZZZ P(k;w)exp( j(wt k s) dkdw P(k;w) Z ZZZ p(s; t)exp( j(wt k s) dsdt; 2) where k is the wavenumber which determines the propagation direction and, thus, is perpendicular to any iso phase plane of the wave. Basically (2) shows that any function of space and ....

D. E. Dudgeon. Multidimensional Digital Signal processing. Prentice-Hall, first edition, 1984.

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Dudgeon, D. E. and Mersereau, R. M. Multidimensional Digital Signal Processing, Prentice-Hall, 1984.

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Dudgeon, D.E. and Mersereau, R.M. Multidimensional Digital Signal Processing, Prentice-Hall, 1984.

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D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing, Prentice Hall, 1984

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D.E. Dudgeon and R.M. Mersereau. Multidimensional Digital Signal Processing. Prentice-- Hall, Englewood Cliffs, New Jersey, 1984.

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D.E. Dudgeon and R.M. Mersereau. Multidimensional Digital Signal Processing. Prentice--Hall, Englewood Cliffs, 1984.

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Dudgeon, D. E., 1994. Multidimensional Digital Signal Processing. Prentice-Hall.

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D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing, Prentice Hall, NJ, 1984.

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Dan E. Dudgeon and Russell M. Mersereau. Multidimensional Digital Signal Processing. Prentice-Hall, 1984.

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D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Processing. Prentice-Hall, Inc., Englewood-Cliffs, NJ, 1st edition, 1984.

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Dan E. Dudgeon and Russell M. Mersereau, Multidimensional Digital Signal Processing, Prentice Hall, Englewood Cliffs, NJ, 1984.

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Dudgeon, D. and R. Mersereau: 1984, Multidimensional Digital Signal Processing. Prentice Hall.

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D. E. Dudgeon and R. M. Merserau, Multidimensional digital signal processing, Prentice-Hall, 1984.

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D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing. Prentice Hall, 1984.

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D.E. Dudgeon and R.M. Mersereau. Multidimensional Digital Signal Processing. Prentice--Hall, Englewood Cliffs, 1984.

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Dudgeon, D.E. and R.M. Mersereau, Multidimensional Digital Signal Processing. 1984, Englewood Cliffs, New Jersey: Prentice-Hall.

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D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1984.

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D. E. Dudgeon and R. M. Merserau, Multidimensional digital signal processing, Prentice-Hall, 1984.

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D. E. Dudgeon and R. M. Merserau, Multidimensional Digital Signal Processing, Prentice-Hall, Englewood Cli#s, 1984.

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D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1984.

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D. Dudgeon and R. Mersereau, Multidimensional Digital Signal Processing. Englewood Cli#s, NJ: Prentice-Hall, 1984.

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D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Processing. Prentice-Hall signal processing series. Prentice-Hall, 1984. ISBN 0-13-604959-1.

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Dudgeon, D. E. and Mersereau, R. M. (1984). Multidimensional Digital Signal Processing.

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Dudgeon, Dan E. and Mersereau, Russell M., Multidimensional Digital Signal Processing, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984.

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D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Processing. Prentice-Hall signal processing series. Prentice-Hall, 1984. ISBN 0-13-604959-1.

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D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Processing. Prentice-Hall signal processing series. Prentice-Hall, 1984. ISBN 0-13-604959-1.

No context found.

D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Processing. Prentice-Hall signal processing series. Prentice-Hall, 1984. ISBN 0-13-604959-1.

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D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Processing. Prentice-Hall signal processing series. Prentice-Hall, 1984. ISBN 0-13-604959-1.

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Dudgeon, D.E. - Mersereau, R.M.: Multidimensional Digital Signal Processing. Prentice -Hall, Englewood Cliffs, NJ 1984.

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