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Simoncelli89-Reprint Non-Separable Extensions of Quadrature Mirror Filters to Multiple Dimensions Quadrature Mirror Filter (QMF) banks have been used in a variety perfect reconstruction filter banks based on a polyphase of one-dimensional signal processing applications, and have been matrix decomposition in the frequency domain [7]. Vetterli applied separably in two dimensions. As with most one-dimen- sional filters, separable extension to multiple dimensions pro- was the first to propose the use of QMFs for image decom- duces a transform in which the orientation selectivity of some of position [8]. He showed examples of both separable and the high-pass filters is poor. We describe generalized non-separa- non-separable non-oriented QMF decompositions in two ble extensions of QMF banks to two and three dimensions, in dimensions. Vaidyanathan established criteria for perfect which the orientation specificity of the high-pass filters is greatly improved. In particular, we discuss extensions to two dimensions reconstruction QMF banks for two-dimensional applica- with hexagonal symmetry, and three dimensional spatio-temporal tions [9]. Viscito and Allebach developed perfect recon- extensions with rhombic-dodecahedral symmetry. Although these struction multi-dimensional filter banks with arbitrarydeci- filters are conceived and designed on non-standard sampling lat- mation patterns [IO]. Woods and O’Neil used separable tices, they may be applied to rectangularly sampled images. As in QMFs for image data compression [Ill. Several other one dimension, these transformations may be hierarchically cas- caded to form a multi-scale “pyramid” representation. We design authors have used QMF pyramid transforms for data a set of example filters and apply them to the problems of image compression [12]-[14]. Mallat [3] related QMF pyramids to compression, progressive transmission, orientation analysis, and wavelet theory and proposed their use in machine vision. motion analysis. In a parallel development, there has been a great deal of interest in image representations that are tuned for ori- INTRODUCTION entation as well as scale. This is equivalent to requiring that Sub-band transforms have been successfully employed the frequency spectra of the basis functions of the repre- in many areas of signal processing. For many applications, sentation exhibit angular localization. A variety of argu- especially in image processing, researchers have advocated ments have been advanced in favor of such transforms, the use of sub-band transforms that divide the frequency based on properties of the human visual system and the spectrum into octave bandwidth pieces [I]-[4]. In such a statistics of images [15]-[18]. Daugman, and Porat etal. have transform the basis functions represent information at spa- explored two-dimensional Gabor transforms, in which the tial scales which are related by powers of two. basis functions are Gaussian windows modulated by si- A particularly useful one-dimensional orthogonal sub- nusoidal gratings [18], [19]. A related transform has been band transform is the Quadrature Mirror Filter (QMF) bank, described by Watson [16]. which was introduced by Croisier et al. [SI, [6]. These filters Most applications of QMFs to two or more dimensions are used in an analysis/synthesis system which decomposes have involved separable filters. A two-dimensional example a signal into high-pass and low-pass frequency sub-bands. is illustrated in Fig. 1: The frequency spectrum is split into They are also well-suited for octave band splitting, since low-pass, horizontal high-pass, vertical high-pass, and diag- they can be applied recursively to split the low-pass sub- onal high-pass sub-bands. The diagonal band contains band. Vaidyanathan developed a more general theory of mixed orientations. Adelson er a/. [I21 demonstrated that one could develop Manuscript received January 14,1989; revised June25,1989. This a non-separable decomposition based on hexagonally sym- work was supported in part by IBM Corporation, the National Sci- metric QMFs, thereby achieving an orthogonal transform ence Foundation, under grant NSF IRI 871-939-4, and the Defense in which the all of the basis functions are localized in space, Research Projects Agency, under grant DARPAIRADC F30602-89-C- spatial frequency, and orientation. This approach to pyr- 0022. The views expressed are those of the authors, and do not nec- essarily represent the views of MIT or the sponsors. amid construction allows smooth spatial overlap between E. P. Simoncelli is with the Vision Science Group, Media Lab- basis functions, and produces much better frequency tun- oratory, Dept. of Electrical Engineering and Computer Science, ing than does the blocked non-overlapping construction Massachusetts Institute of Technology,Cambridge, MA02139, USA. employed by Crettez and Simon [20] or Watson and Ahu- E. H. Adelson is with the Vision Science Group, Media Labo- ratory, Dept. of Brain and Cognitive Science, Massachusetts Insti- mada [21]. In the present paper we extend the Adelson et tute of Technology, Cambridge, MA 02139, USA. a/. concepts; we describe methods for designing hexagonal IEEE Log Number 9034602. QMFs, and we apply them to a variety of problems. We also 0018-9219/90/0400-0652$01,000 1990 IEEE 652 PROCEEDINGS OF THE IEEE, VOL. 78, NO. 4, APRIL 1990 Authorized licensed use limited to: IEEE Xplore. Downloaded on April 29, 2009 at 21:04 from IEEE Xplore. Restrictions apply. The boxes indicate thatthe sequence is subsampled by a factor of 2, and the boxes indicate that the sequence should be upsampled by insertingazero between each sample. .oX ..... Using the definition of the DTFT and some well known facts about the effects of upsampling and downsampling in the frequency domain, one can derive equations for the DTFT of the representation sequences y,[n]: Y;(w) = ;[F,(;)x(;) + F,(; + .)x(; + .)I (1) and the AIS system output is R(w) = Y0(2w)Go(w) + Y,(2W)G7(W). Combining these equations gives the overall system response of the filter bank: fb)= ;[Fob) Go(w) + F1 (U)Cl (w)lX(w) + i[Fo(~+ *)Go(w) + Fl(W + T)C~(W)]X(W+ *). Fig. 1. Idealized partition of the frequency domain by sep- arable application of two-band one-dimensional QMFs. (2) The first term is a linear shift-invariant (LSI) system response, describe three-dimensional generalizations with rhombic- and the second is the system aliasing. dodecahedral symmetry. The term QMF refers to a clever choice of filters that are related by spatial shifting and frequency modulation. We REVIEW OF ONEDIMENSIONAL QMF CONCEPTS define In this section, we give a brief review of Quadrature Mir- F&w) = Go(-w) = H(w) ror Filters in one dimension. A more thorough review may Fl(w)= Gl(-w) = e/"H(-w + T) (3) be found in [22], or more recently, [23] or [24]. The original QMF problem was formulated as a two-band critically Sam- for H(w) an arbitrary function of 0. This definition, which pled analysislsynthesis filter bank problem, as illustrated was proposed in [25], corresponds to the linear algebraic in the schematic diagram in Fig. 2. The purpose of the anal- notion of an orthogonal transform, and is a more general ysis section of the filter bank is to decompose the input definition than that originally provided by Croisier et al. In sequence x[n] into two half-density representation particular, the original definition does not contain an sequences yo[n] and y1 [n]. The synthesis section then explicit spatial (temporal) shift factor and is therefore valid recombines these sequences to form an approximation onlyfor even-length filters. The definition given above con- i[n]to the original sequence. The system is called "critically tains the original as a subcase, and is also valid for odd- sampled" becausethe sample input rate is equal tothe total length filters. sample rate of the intermediate sequences. The notation in With thechoiceof filtersgiven in (3), equation (2) becomes the diagram is standard for digital signal processing. The boxes indicate convolution of an input sequence with a filter with impulse response fi[n]and discrete time Fourier transform (DTFT) Fi(w) = fl[n]e-'"" n Analysis section Synthesis section A 2.1 2.1 Fig. 2. A two-band analysis/synthesis filter bank in one dimension. SIMONCELLI AND ADELSON: NON-SEPARABLE EXTENSIONS OF QUADRATURE MIRROR FILTERS 653 Authorized licensed use limited to: IEEE Xplore. Downloaded on April 29, 2009 at 21:04 from IEEE Xplore. Restrictions apply. 1 2.1’ w - Yo, [nl 1 - Fl(@ - 2.1 w Fig. 3. A non-uniformlycascaded analysislsynthesisfilter bank. The second (aliasing) term cancels, and the remaining LSI system response is 2(w) = i[H(w)H(-w) + H(-w + K) H(w + 7r)l X(w). (5) Note that the aliasing cancellation is exact, independent of the choice of the function /+(U). We should emphasize, however, that it is the overall system aliasing that cancels- the individual sub-bands do contain aliasing. The design problem is now reducedto finding a filter with DTFT H(w)that satisfies the constraint ................. .................................x.x.:.:.:.:.:............. .................. ~[H(w)H(-w)+ H(-w r)H(w + *)I= 1 ....................................... + ........................... ....................................... or > (H(w)I2+ IH(w + *)I2 = 2. (6) Ingeneral, lowpass solutions are desirable since the system then becomes a band-splitting system. Several authors have studied the design and implementation of these filters [25]- [29]. Once filters have been designed so that the overall sys- Fig. 4. Octave band splitting produced by a four-level pyr- tem response is unity, the filter bank may be cascaded to amid Lcascade of a two-band A/S system. The top picture rep- form multiple-band systems.This may bedone in a uniform resents the splitting of the two-band AIS system. Each suc- manner as in [Ill, or in a non-uniform or “pyramid” fashion cessive picture shows the effect of reapplying the system to the lowpass sequence (indicate in grey) of the previous pic- [12].
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