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Chapter 4 Subband Transforms MIT Media Laboratory Vision and Modeling Technical Report Appears in Subband Coding edited by John Woods Kluwer Academic Press Chapter Subband Transforms y z Eero PSimoncelli and Edward H Adelson Vision Science Group The Media Lab oratoryand y DepartmentofElectrical Engineering and Computer Science z Departmentof Brain and CognitiveScience Massachusetts Institute of Technology Cambridge Massachusetts Linear transforms are the basis for manytechniques used in image pro cessing image analysis and image co ding Subband transforms are a sub class of linear transforms whichoer useful prop erties for these applications In this chapter wediscuss a varietyofsubband decomp ositions and illustrate their use in image co ding Traditionallycoders based on linear transforms are divided into twocategories transform co ders and subband co ders This distinction is due in part to the nature of the computational metho ds used for the twotyp es of representation Transform co ding techniques are usually based on orthogonal linear trans forms The classic example of suchatransform is the discrete Fourier trans form DFT whichdecomp oses a signal into sinusoidal frequency comp o nents Twoother examples are the discrete cosine transform DCT and the KarhunenLo evetransform KLT Conceptually these transforms are com puted bytaking the inner pro duct of the nitelength signal with a set of basis functions This pro duces a set of co ecients which are then passed on to the This work was supp orted bycontracts with the IBM Corp oration agreementdated and DARPA Rome Airforce FC and a grantfrom the NSF IRI The opinions expressed are those of the authors and do not necessarily representthose of the sp onsors SUBBAND IMAGE CODING quantization stage of the co der In practice manyofthesetransforms have ecient implementations as cascades of buttery computations Further more these transforms are usually applied indep endently to nonoverlapping subblo cks of the signal Subband transforms are generally computed by convolving the input sig nal with a set of bandpass lters and decimating the results Eachdecimated subband signal enco des a particular p ortion of the frequency sp ectrum corre sp onding to information o ccurring at a particular spatial scale To reconstruct the signal the subband signals are upsampled ltered and then combined ad ditivelyFor purp oses of co ding subband transforms can b e used to control the relative amounts of error in dierentparts of the frequency sp ectrum Most lter designs for subband co ders attempt to minimize the aliasing resulting from the subsampling pro cess In the spatial domain this aliasing app ears as evidence of the sampling structure in the output image An ideal sub band system incorp orates brickwall bandpass lters whichavoid aliasing altogether Suchlters however pro duce ringing Gibbs phenomenon in the spatial domain whichisperceptually undesirable Although co ders are usually classied in one of these twocategories there is a signicant amountofoverlap b etween the two In fact the latter part of this chapter will fo cus on transforms whichmaybe classied under either cat egoryAsan example consider the blo ckdiscrete cosine transform DCT in which the signal image is divided into nonoverlapping blo cks and eachblock is decomp osed into sinusoidal functions Several of these sinusoidal functions are depicted in gure The basis functions are orthogonal since the DCT is orthogonal and the blo cks are chosen so that they do not overlap Co ders employing the blo ck DCT are typically classied as transform co ders Wemayalso view the blo ck DCT as a subband transform Computing a DCT on nonoverlapping blo cks is equivalenttoconvolving the image with each of the blo ckDCT basis functions and then subsampling bya factor equal to the blo ck spacing The Fourier transform of the basis functions also shown in gure indicates that eachofthe DCT functions is selectivefora particular frequency subband although it is clear that the subband lo calization is rather poor Thus the DCT also qualies as a subband transform Subband Transform Prop erties Given the overlap b etween the categories of transform and subband co ders what criteria should b e used in cho osing a linear transformation for co ding pur poses We will consider a set of prop erties whichare relevanttothe problem Chapter Subband Transforms Figure Several of the p oint DCT basis functions left with their corresp onding Fourier transforms right The Fourier transforms are plot ted on a linear scale over the range from to SUBBAND IMAGE CODING of image co ding Scale and Orientation An explicit representation of scale is widely accepted as b eing imp ortantfor eectiveimage representation Images contain ob jects and features of manydierent sizes whichmaybeviewed over a large range of distances and therefore a transformation should analyze the image simulta neously and indep endently at dierentscales Several authors haveargued that the correct partition in terms of scale is one in whichthe scales are re lated bya xed constantofprop ortionality In the frequency domain this corresp onds to a decomp osition into lo calized subbands with equal widths on a logarithmic scale For twodimensional signals a lo calized region in the frequency plane cor resp onds spatially to a particular scale and orientation Orientation sp ecicity allows the transform to extract higher order oriented structures typically found in images suchasedges and lines Thus it is useful to construct transforma tions whichpartition the input signal into lo calized patches in the frequency domain Spatial lo calization In addition to lo calization in frequencyitisadvantageous for the basis func tions to b e spatially lo calized that is the transform should enco de p ositional information The necessityofspatial lo calization is particularly apparentin machine vision systems where information ab out the lo cation of features in the image is critical This lo calization should not however o ccur abruptly as in the blo ckDCT example given earlier abrupt transitions lead to p o or lo calization in the frequency domain The concept of jointlocalization in the spatial and spatialfrequency do mains maybe contrasted with the twomost common representations used for the analysis of linear systems the sampled or discrete signal and its Fourier transform The rst of these utilizes the standard basis set for discrete signals consisting of impulses lo cated at eachsample lo cation These basis functions are maximally lo calized in space but convey no information ab out scale On the other hand the Fourier basis set is comp osed of even and o dd phase sinu soidal sequences whose usefulness is primarily due to the fact that they are the eigenfunctions of the class of linear shiftinvariantsystems Although they are maximallylocalized in the frequency domain eachone covers the entire spatial extentofthe signal Chapter Subband Transforms It is clear that representation in the space or frequency domains is ex tremely useful for purp oses of system analysis but this do es not imply that impulses or sinusoids are the b est waytoenco de signal information In a numberof recentpapers the imp ortance of this issue is addressed and related to a pap er by Dennis Gab or who showed that the class of linear transformations maybe considered to span a range of jointlocalization with the impulse basis set and the Fourier basis set at the twoextremes He demonstrated that onedimensional signals can b e represented in terms of ba sis functions whicharelocalized b oth in space and frequencyWewill return to Gab ors basis set in section Orthogonality A nal prop ertytobeconsidered is orthogonality The justication usually given for the orthogonality constraintisinterms of decorrelation Given a signal with prescrib ed second order statistics ie a covariance matrix there is an orthogonal transform the KarhunenLo eve transform whichwill decor relate the signal ie diagonalize the covariance matrix In other words the second order correlations of the transform co ecients will b e zero Orthogo nalityisusually not discussed in the context of subband transforms although manysuch transformas are orthogonal The examples in the next section will demonstrate that although orthogonalityisnot strictly necessarya transform that is strongly nonorthogonal maybeundesirable for co ding Linear Transformations on Finite Images The results presented in this chapter are based on analysis in b oth the spatial and the frequency domains and thus rely on two separate notational frame works the standard matrix notation used in linear algebra and the Fourier domain representations commonly used in digital signal pro cessing In this section wedescrib e the twotyp es of notation and makeexplicit the connec tion b etween them For simplicitywewill restrict the discussion to analysis of one dimensional systems although the notation maybe easily extended to multiple dimensions AnalysisSynthesis Filter Bank Formulation We will b e interested in linear transformations on images of a nite size which maybe expressed in terms of convolutions with nite impulse resp onse FIR SUBBAND IMAGE CODING lters The schematic diagram in gure depicts a convolutionbased sys tem known as an analysissynthesis AS lter bank The notation in the diagram is standard for digital signal pro cessing except that for the pur poses of this pap er the b oxes H indicate circular convolution of a nite i input image of size N with a lter with impulse resp onse h nandFourier i transform X jn H h ne i i n Wedonot
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