Reprint from IASTED International Conference on Signal and Image Processing (SIP 99). Nassau, Bahamas, October, 1999.

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1 Reprint from IASTED International Conferene on Signal and Image Proessing (SIP 99). Nassau, Bahamas, Otober, Filter Networks Mats Andersson Johan Wiklund Hans Knutsson Computer Vision Laboratory, Department of EE Linköping University, SE Linköping, Sweden phone: fax: matsa@isy.liu.se Abstrat This paper presents a new and effiient approah for optimization and implementation of filter banks e.g. veloity hannels, orientation hannels and sale spaes. The multi layered struture of a filter network enable a powerful deomposition of omplex filters into simple filter omponents and the intermediary results may ontribute to several output nodes. Compared to a diret implementation a filter network uses only a fration of the oeffiients to provide the same result. The optimization proedure is reursive and all filters on eah level are optimized simultaneously. The individual filters of the network, in general, ontain very few non-zero oeffiients, but there are are no restritions on the spatial position of the oeffiients, they may e.g. be onentrated on a line or be sparsely sattered. An effiient implementation of a quadrature filter hierarhy for generi purposes using sparse filter omponents is presented. keywords filter optimization, filter network, sequential onvolution, sparse filters, effiient filtering. 1 Introdution 1 2 n1 n N f 1 f 2 f 3 f M1 f M K nodes L nodes M nodes Figure 1: A general filter network of N levels with M nodes in the final level. Figure 1 depits a general filter network onsisting of N levels and with M output nodes. The nodes onstitute summation points and the filters are loated on the ars onneting two onseutive layers. In the general ase all nodes on layer (n, 1) are onneted to eah node in layer n. The definition of the filter network an be divided into struture and internal properties. The struture of the network are the properties that an be read out from a sketh like in fig. 1 i.e. the number of levels and the number of nodes on eah level. The internal properties omprise the number of oeffiients for eah filter in the network and their spatial position (oordinates). Moreover we need to define the ideal filter funtion f m (u) and weight funtions W m (u) for the final layer, where u are Fourier domain oordinates and m =[1:::M] the number of output nodes. The task is to ompute the filter oeffiients in the network suh that the weighted differene between the resulting and the ideal filter funtions are minimized. min 2 = W m (u) f m (u), f m (u) 2 (1) The Fourier weighting funtion W (u) provide an appropriate Fourier spae metri. The metri will determine the importane of a lose fit for different spatial frequenies, see [7]. 2 Fourier spae metri The hoie of W (u) should if possible be done in the light of the expeted spetra for the signal and the noise. If no a priori information is available eq. 2 is proposed. w(u) =, 0:5 1:5 (2) This form of W (u) is due to two observations. For natural images there is, in general, no reason to expet a nonisotropi spetrum, i.e. the expeted spetrum will only depend on the Fourier domain radius, = kuk. Seondly, there does not seem to be a large differene in terms of spetrum when imaging the real world at very different sales, say a mirosope image vs a satellite image. Consequently the spetrum is approximately sale invariant,, / (a),. The range of the parameter may be argued from the spetrum of randomly sattered lines and edges, see [7]. The onstant relates to the expeted level of broadband noise and aliasing. Note that optimizing

2 Reprint from IASTED International Conferene on Signal and Image Proessing (SIP 99). Nassau, Bahamas, Otober, without using a weighting funtion is equivalent to setting w(u) = 1. 3 Filter network optimization Ideally it would for a given network struture be desirable to optimize the number of oeffiients in eah ar and the position (oordinate) for eah oeffiient for the entire net simultaneously. This is, however, an extremely omplex problem and a method for finding an overall optimal solution has not been found. (and it s doubtful if it ever will be). For this reason the definition of the internal properties of the network have to be made based on experiene. If the internal properties (the non zero oordinates of the filters) is defined it is possible to optimize all filters on the same level simultaneously with respet to the output filters and the urrent state of the network. This proedure is repeated for another layer and so on reating a sequential optimizer loop over the network struture. The onvergene of this approah annot be guaranteed but initial tests prove that for realisti hoies of filter struture and oeffiient distribution the network onverges to a stable solution whih in terms of omputational omplexity outperforms a onventional implementation by orders of magnitude. 3.1 Minimizing the error The network is initialized to provide a non zero transfer funtion for eah ar. The initialization is not ritial but may require some onsideration. Usually a qualified guess or setting the transfer funtions to unity in the FD is suffiient. Pure randomizing is not always reommended as it may introdue unwanted symmetry effets. If a filter in the network is expeted to be e.g. real and even there is no reason not to initialize the filter by suh a funtion. For eah kernel (ar) in the network, a Fourier transform matrix B nkl is omputed suh that the Fourier transform of the filter onneting node k on layer (n, 1) to node l on layer n is omputed as B nkl nkl. The olumn vetor, nkl, ontains the nonzero oeffiients of the filter in this ar of the network. The rows of B nkl ontain the orresponding Fourier basis funtions. In order to minimize eq. 1 selet a layer, n, for the urrent optimization step. For the moment n is assumed to be one of the intermediate layers i.e. n = [2 :::N, 1]. The number of nodes in layer (n, 1) andn is K and L respetively, see fig. 1. Now ompute the transfer funtions, G k (u) k = [1 :::K] from the top node to eah of the nodes in layer (n, 1). To simplify the subsequent notation, G k (u) is reshaped as a diagonal matrix where the transfer funtion is loated in the main diagonal and all offdiagonal values are zero. For the lower part of the network the transfer funtions from eah node in layer n to eah of the final nodes are omputed as: H lm (u) l = [1:::L] m = [1:::M ] (3) For the same reason as above H lm (u) is reshaped into diagonal form. For onveniene, the frequeny oordinate u, is dropped from here on. As a final step towards expressing eq. 1 in terms of the kernel oeffiients of layer n we define: B m =(G1 H1m B n11; G2 H1m B n21; ::: G k H lm B nkl ; ::: ;G K H LM B nkl ) (4) where k =[1:::K] l =[1:::L] m =[1:::M] Equation 1 an now be expressed as: min 2 = k W m ( B m n, f m ) k 2 (5) where n is a vetor ontaining all filters onneting layer (n, 1) and layer n. n =( n11; n21;::: nkl ;:::; nkl ) T (6) Compute the partial 2 n ] n ] Setting the partial derivative equal to zero results in where (7) n = A,1 h (8) A = Bm T W 2 m B m h = Bm T W 2 m f m (9) Note that the omplexity involved in solving this linear equation system is only dependent on the number of nonzero oeffiients in the filters onneting layer (n, 1) and n. The kernels nkl of layer n are updated and the proedure is repeated for another layer until onvergene. For the initial layer the frequeny transfer funtion G degenerates to an identity matrix. The frequeny transfer funtion H will in the same way be be an identity operator in the optimization of the final layer. 3.2 Fourier spae sampling The introdution of the Fourier transform matries, B nkl, in the previous setion implies a sampling of the ontinuous Fourier spae. In priniple the higher the sampling density the loser the sampled ase solution will be to the ontinuous ase. In pratise using 2-3 times as many points, for eah dimension, as the spatial size in pixels (voxels et.) has proven to be adequate. Note that the number of samples does not hange the size of the basi problem, i.e. the size of the matrix A in eq. 9. However, further inreasing the sample density will, as a rule, have an insignifiant effet on the solution. As the Fourier spae representation is repetitive samples are only needed in the interval,= < u = where the inter-sample distane is given by.

3 Reprint from IASTED International Conferene on Signal and Image Proessing (SIP 99). Nassau, Bahamas, Otober, A note on reursive optimization As the optimizer onsiders one level of the network at eah optimization step it may be worthwhile to onsider this restrition in the initialization of the network and when deiding the order in whih to optimize the layers of the network. In general it is reommended to start with a very simple network ontaining only the most neessary nodes and a small number of kernel oeffiients. When developing a network it is equally important to remove ars and oeffiients that are not being used satisfatorily as to add new degrees of freedom. If omplex valued filters are present in the network these filters should, if possible, be plaed lose to the output nodes to minimize the number of omplex multipliations. A real valued solution for a layer may be enfored by solving n =(Re[A]),1 Re[h] (10) whih is obtained by f 1 f 2 f 3 q 1 q 2 q 2 =0 n ] As an example the network in fig. 3 onverges after iterations. f 4 q 4 4 Examples Filters do for natural reasons need a region of support of the same dimensionality as the signal. A areful ombination of simple kernels in a filter network will in relation to a diret implementation be more effiient for signals with high outer dimensionality (3D and 4D data). The filter net example below is, however, limited to 2D to simplify visualization of the result. Note that the omplexity of the optimization is only dependent on the total number of oeffiients within the network and not the orresponding oordinates (i.e. the dimensionality of the resulting filters). The network optimizer is developed and implemented in Matlab A network for logarithmi quadrature hannel deomposition of the FD Figure 2: The proposed logarithmi deomposition of the Fourier domain. Figure 3: Filtering struture for one level of quadrature hannel network. This filtering struture was originally intended for effiient omputation of quadrature filter responses in different orientations and sales for phase based optial flow. The network is however equally useful for a number of purposes suh as e.g orientation estimation, enhanement, subband oding, loal spetrum analysis et. A shemati illustration of the quadrature hannels is given in fig. 2. The network produe a logarithmi deomposition of the FD i.e. the relative bandwidth of the quadrature hannels is onstant. Figure 3 illustrates the struture for one level of suh a network. If this network is ompared to the general network of fig. 1 a number of onditions are not fulfilled. The output nodes are not loalized in the final layer and the ars are not restrited to onnet two subsequent layers. It is, however, straightforward to redraw the network of fig. 3 to meet the onditions in setion 1 by introduing a number of dummy nodes, (in fat this was the way the network was optimized). The large blak dots in fig. 3 indiate the nodes (summation points) while the filtering is performed in the ars. The net ontain eight very simple filters, four real valued LP-filters [f1;:::f4] and four omplex valued quadrature filters [q1;:::q4]. Thenumberofoeffiientsandtheirorresponding oordinates (i.e. filter orientation) is illustrated by the dots on eah ar. The square ontaining grey shaded

4 Reprint from IASTED International Conferene on Signal and Image Proessing (SIP 99). Nassau, Bahamas, Otober, is getting heaper an interesting alternative is to maintain the original size of the signal through the omplete network and use sparse filters. Using sparse filters the distane between the (non zero) filter oeffiients is inreased (by one otave) for eah new network that is added. Using sparse filters with a sample distane of imply that the spetrum will be repetitive with a period of 2=. Both subsampling and sparse filters are straight forward to use with the network optimizer. In this example the sparse filter method is used to support a diret omparison between the sales without interpolation. 4.3 Initial filtering Figure 4: Initial filtering, two passbands separated by an half otave. areas illustrate the energy ontribution in the FD at eah node. Assume for the moment that the spetrum of the signal at the top node in fig. 3 only has energy ontributions inside the shaded diamond, i.e. band limited to along the diagonal diretions. The filter f1 band limits the signal further in the horizontal diretion whih is illustrated at the next node. The differene before and after f1 is an even BP-hannel that is fed to a quadrature filter q1 reating quadrature response in the horizontal diretion. The vertial quadrature filter response is omputed in the same way using f2 and q2 ontaining 5 oeffiients eah. At this stage in the network LP-hain there are no energy ontribution for jjujj >= p 2 and it is onsequently feasible to plae the remaining filters in the diagonal diretion. This implies a sampling distane of = p 2 as opposed to =1for the horizontal and vertial hannel. Now ompare the spetrum of the LP-hannel in the last output node and the spetrum at the input node. Both these nodes share the same diamond shaped spetrum but the bandwidth of the output node is redued by one otave. Consequently another network with the same struture an be attahed to the last node to obtain the next four quadrature hannels and so on to obtain the deomposition of fig Sparse filters A sequential use of the network in fig. 3 require some onsiderations to keep the number of oeffiients onstant for all levels. There are essentially two possibilities to aomplish this. The first is to apply a subsampling by one otave at the last node before the signal is fed to the input node of the next network. Subsampling is memory effiient but require interpolation if several hannels are used simultaneously e.g. to interpolate a broadband filter. As memory Figure 5: The omplete network. Before the network in fig. 3 an be applied some initial filtering is neessary. The sampling of the spatial domain imply a periodi FD. Assuming a sampling distane, = 1, the orners of fig. 2 are loated at [; ] and [;,]. Due to the periodi properties of the FD these four points annot be separated. Consequently it is not feasible to infer a diretion for these frequenies and the initial filtering step is, for that reason, an isotropi LP/HP filter. To the left in fig. 4 a network onsisting of 5 levels using 37 real valued oeffiients is proposed. This network produe two passbands separated by half an otave. The frequeny funtion for the two output nodes are displayed to the right. The result from either of these two output nodes an be used as input to the network of fig. 3 or, as in this example, where they are both used in parallel to obtain two Fourier domain deompositions overlapping eah other by an half otave. In fig. 5 this network is skethed. The top part orresponds to the network of fig. 4. The two output nodes are eah onneted to a network of fig. 4. A network with the same struture is then onneted to the LP-nodes for two further levels. 4.4 The omplete network The network of fig. 5 produes 24 quadrature hannels within 4 diretions and 3 otaves and in addition isotropi HP and LP hannels. The entire network requires 469 real

5 Reprint from IASTED International Conferene on Signal and Image Proessing (SIP 99). Nassau, Bahamas, Otober, π π/2 0 π/2 π π π/2 0 π/2 π Figure 6: Iso-surfae plot of resulting filters, three levels and two frequeny bands. valued multipliations per pixel whih orresponds to multipliations for eah output node. In fig. 6 an iso-surfae plot at the 3dB band width of the resulting quadrature filters is displayed inluding the final LP-filter and the initial HP-filter. Note that for ompleteness the quadrature hannels are displayed in both sides of the FD. The left side of fig. 7 ontain some seleted frequey funtions from the first quadrature hannel network. The top figure show the Fourier transform of the horizontal LPfilter, f1, in fig. 3. Below that the Fourier transform of the orresponding quadrature filter, q1. At the bottom left the result at the first horizontal quadrature hannel node. This result orresponds to the produt of the above frequeny funtions an the initial frequeny funtion displayed at the lower right in fig. 4. The right part of fig. 7 show the orresponding frequeny funtions for a filter in the fourth diretion and on the seond network level. Note that the filters are sparse on this level ( = 2 p 2) whih is refleted by the periodi repetition in the Fourier transform of f4 and q4. 5 Conlusion A novel method for optimizing effiient filters using filter networks has been presented. The optimization proedure is reursive and based on the ideal filter funtions at the output nodes and a weighting funtion defining the Fourier spae metri. It was demonstrated how the filter network optimizer an by used to implement a quadrature filter bandpass pyramid for pratial purposes using spatially sparse kernels. For a image this implementation require 25 times less multipliations ompared to onventional FFT. Figure 7: Left: from network level 1 first LP filter and quadrature filter and resulting response at output node 1. Right: Corresponding plots for network level 2 and diretion 4, note that the filter omponents are sparse in the spatial domain whih results in a repetition in the FD. 6 Aknowledgment This work was supported by NUTEK, the Swedish National Board for Industrial and Tehnial Developement and by WITAS, The Wallenberg Laboratory for Tehnial Development and Autonomous Systems whih is greatfully aknowledged. Referenes [1] M. Andersson. Controllable Multidimensional Filters in Low Level Computer Vision. PhD thesis, Linköping University, Sweden, SE Linköping, Sweden, September Dissertation No 282, ISBN [2] R. Braewell. The Fourier Transform and its Appliations. MGraw-Hill, 2nd edition, [3] D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Proessing. Prentie-Hall signal proessing series. Prentie-Hall, ISBN [4] G. H. Granlund and H. Knutsson. Signal Proessing for Computer Vision. Kluwer Aademi Publishers, ISBN [5] H. Knutsson. Filtering and Reonstrution in Image Proessing. PhD thesis, Linköping University, Sweden, Diss. No. 88. [6] H. Knutsson and M. Andersson. Optimization of Sequential Filters. In Proeedings of the SSAB Symposium on Image Analysis, pages 87 90, Linköping, Sweden, Marh SSAB. LiTH-ISY-R URL: [7] H. Knutsson, M. Andersson, and J. Wiklund. Advaned Filter Design. In Proeedings of the Sandinavian Conferene on Image analysis, Greenland, June SCIA. Also as report LiTH-ISY-R [8] J. Wiklund and H. Knutsson. A Generalized Convolver. In Proeedings of the 9th Sandinavian Conferene on Image Analysis, Uppsala, Sweden, June SCIA.

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