Sequential Filter Trees for Ecient D 3D and 4D Orientation Estimation Mats ndersson Johan Wiklund Hans Knutsson omputer Vision Laboratory Linkoping Un

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1 Sequential Filter Trees for Ecient D 3D and 4D Orientation Estimation Mats ndersson Johan Wiklund Hans Knutsson LiTH-ISY-R-7 998

2 Sequential Filter Trees for Ecient D 3D and 4D Orientation Estimation Mats ndersson Johan Wiklund Hans Knutsson omputer Vision Laboratory Linkoping University, Sweden bstract recursive method to condense general multidimensional FIR-lters into a sequence of simple kernels with mainly one dimensional extent has been worked out. onvolver networks adopted for, 3 and 4D signals is presented and the performance is illustrated for spherically separable quadrature lters. The resulting lter responses are mapped to a non biased tensor representation where the local tensor constitutes a robust estimate of both the shape and the orientation (velocity) of the neighbourhood. qualitative evaluation of this General Sequential Filter concept results in no detectable loss in accuracy when compared to conventional FIR (Finite Impulse Response) lters but the computational complexity is reduced several orders in magnitude. For the examples presented in this paper the attained speed-up is, and 3 times for D, 3D and 4D data respectively The magnitude of the attained speed-up implies that complex spatio-temporal analysis can be performed using standard hardware, such asapowerful workstation, in close to real time. Due to the soft implementation of the convolver and the tree structure of the sequential ltering approach the processing is simple to recongure for the outer as well as the inner (vector length) dimensionality of the signal. The implementation was made in VS (pplication Visualization System) using modules written in.

3 ontents Introduction General Sequential Filters 3 Optimization of Filter oecients 4 Tensor Representation and Error Estimation 4 Filtering Structure for D Signals 6 6 Filtering Structure for 3D Signals 9 7 Filtering Structure for 4D Signals 8 Implementation spects 9 cknowledgment Sequential lters for D 6 Sequential Filters for 3D 7 Sequential Filters for 4D 8 D References

4 Introduction In multidimensional signal processing, the main part of the computational power is usually spent on the initial ltering. The lters do for natural reasons need to be of the same `dimensionality' as the signal and for 3D and 4D data, such as time sequences and time sequences of volumes, the number of lter coecients increases dramatically and often limits the practical (interactive) use of the algorithm. In this paper (see also [9]) a new recursive method to optimize general multidimensional lters has been worked out which uses only a small fraction of the number of lter coecients that is required by conventional ltering. The attained speed up implies that multidimensional signal analysis can be performed using standard hardware, such as a powerful workstation, in close to real time. n observation that is very important for the proposed approach is that the performance of a convolver, implemented in standard hardware is directly proportional to the number of kernel coecients, but independent of the spatial coordinates for these coecients. software convolver will also be independent of built in limitations in special purpose hardware and can by simple means be recongured for signals of dierent dimensionality and vector length, see [] for more details. Using N coecients it is always possible to nd a best approximation to a given lter using all coecients at once for a single lter. In many cases, however, a far more ecient way of attaining essentially the same lter is to use the same number of coecients but distribute them over a number, (M), of smaller lters. These lters are then applied in sequence (sequential convolution) to obtain the nal lter response. The proposed method involves the following three steps:. Determine the number, M, of sequential lters that is appropriate for the present application.. Determine the distribution of the N coecients, i.e. determine the coordinates of the coecients for each of the M sequential lters. 3. Optimize the values of the N coecients (distributed over the M lters) so that the combined eect of the lter sequence matches the reference lter as closely as possible. To optimize these steps simultaneously is a complex problem indeed and an overall optimal solution has not been found, (and probably never will be). In order to reduce the complexity of the problem the number of lter components and the distribution of coecient coordinates are performed by intuition and by taking advantage of the symmetry properties of the reference lter. Given the number of lter M and the spatial coordinates of each lter the last step is still cumbersome. recursive method is proposed that in each step optimize each lter component in relation to the remaining M, components. onvergence of the algorithm is fast and typically only 4-6 iterations are needed. This sequential lter optimization method is applied to polar separable quadrature lters for D, 3D and 4D signals and a convolver tree structure and performance is presented.

5 General Sequential Filters The objective ofintroducing sequential ltering is to approximate an ideal lter, F (u), by M lter components, F k (u). In the Fourier domain this is expressed as: F (u) F (u) = MY k= F k (u) () where u is the frequency variable. This operation is meaningful if the lters F k (u) can be implemented using a considerable smaller number of kernel coecients in comparison to a direct (M = ) implementation of F (u). In the spatial domain eq. () corresponds to: f() =f () f () ::: f M () () where are spatial coordinates. The lter function F k (u)ofaconvolution kernel with N k coecients is given by its Fourier transform F k (u) = XN k n= f k ( n ) exp(,i n u) (3) where n dene the coordinates where f k () 6=. Note that so far no mention has been made on how these coordinates are chosen. This decision depend on the number of lter components M as well as the total numberofkernel coecients N. In most cases the symmetry of F (u) reduce the number of possibilities to a manageable level. The coecients of a lter component can for example be concentrated to a line, a circle or just be spread sparsely to cover a large region. 3 Optimization of Filter oecients onsider for the moment that the number of lter components M and the coordinates for the nonzero coecients for each lter are dened. The last step of the sequential lter optimization procedure is consequently to assign a value to each coecient such that such that the dierence between the reference function, F (u), and F (u) is minimized according to some distance measure. This distance is dened as the weighted square of the dierence D = X W (u)jjf (u), F (u)jj =fu : ju i j <g (4) The purpose of the weighting function is to make the importance of a close t proportional to the energy contribution. For most natural images the energy spectra decays as W (u) = juj, where < < 3. In this paper =, for further discussion about the weighting function see [6].

6 ^ F( u) W( u) ε optimizer module F( u) f( ε) Figure : The optimizer module have three inputs, the ideal lter function F (u), the frequency weight W (u) and the spatial coordinates The Optimizer Module onsider for a moment the case where M = in eq. (). The coecients of the convolution kernel that minimizes eq. (4) is computed by taking the partial derivative of D with respect to the N kernel coecients of f (). Setting the partial derivatives to zero results in a set of coupled equations from which the coecients of f () are computed. The computations are straightforward but cumbersome and are for that reason left out here. In g. such a single level lter optimizer is illustrated as a module having three inputs, the ideal lter function for the kernel F (u), the weighting function W (u) and the spatial coordinates for the nonzero coecients. This optimization can unfortunately not be performed simultaneously for two or more lter components which is essential for the use of the sequential lter concept. It is, however, possible to use this method to optimize a single lter component with respect to the present value of the other M, components. recursive use of the lter optimization method can consequently be expected to converge rapidly if the coecient coordinates of the M lter components enable a sucient degree of freedom. Sequential Filter Optimization To extend the kernel optimizer for recursive use require some considerations as the ideal lter function and weight function for a certain lter component depends on the reference function F (u) as well as the current value of the other M, lter components. onsider the optimization of lter component k [;M]. The ideal lter function F (u) is expressed as since in the ideal case F k (u) = F (u) = Q l6=k MY k= F (u) F l(u) () F k (u) (6) according to eq. (). For the weight function W (u) the relation is not as obvious as for the ideal function but for reasons that will be apparent later on the weight 3

7 function for F (u) is dened as W k (u) =juj, Y l6=k F l (u) (7) where juj, relates to the energy spectra of the image. Insertion of eq. () and eq. (7) into the distance measure of eq. (4) results in D = X Q juj,4 ( F l(u)) [ l6=k where F k (u) are dened in eq. (3). Simplifying eq. (8) yields: D = X juj,4 [F (u), Q l6=k F (u), F k (u)] (8) F l(u) MY k= F k (u)] (9) which shows that the recursive optimization procedure will tend to minimize the dierence between the combined eect of the component lters, F k (u), and the ideal lter F (u). Recursive lgorithm for Sequential Filters y using the result in eq. () and eq. (7) it is obvious how asettheoptimizer modules can be connected to perform a recursive sequential lter optimization.the algorithm can be explained in the following steps. Set F (u);f 3 (u);::: ;F M (u) =8u. Dene the nonzero coordinates for f ();f ();:::f M () 3. For k= to M opt[f k ()] : ( F k (u) = F (u)= Q l6=k F l(u) W k (u) = juj, Q l6=k F l(u) 4. Repeat step 3 until convergence (-6 iterations) Given enough degree of freedom in the spatial coordinates the algorithm converges rapidly (typically -6 iterations). In the following parts of the paper this algorithm is applied to optimization of polar separable quadrature lters for D, 3D and 4D with M ranging from, 4. 4 Tensor Representation and Error Estimation efore the actual lter implementation is presented it may be appropriate to discuss how the performance of the resulting lters are estimated. One obvious 4

8 possibility isto use the distance measure of eq. (4) to estimate the weighted distortion within the lter. lthough such measurements are appropriate it is more important to estimate the performance of low-level vision tasks such as estimation of shape and orientation. To enable local orientation estimates the resulting quadrature lters are mapped to a Local Tensor representation [7],[6, ch 6]. Let q k be the magnitudes of a set of polar separable quadrature lters implemented by the proposed sequential method. The local tensor T is attained by projecting the magnitude of the quadrature lter responses q k onto a corresponding set of projection tensors T k Figure : SNR=d. Frame number ; 4; and 3 of the 64-cube test sequence, T = X k T k q k () where the T k are symmetrical second order tensors of the same dimensionality as the signal. The computation of the projection tensor is based on the optimized sequential lter responses and not the ideal lter functions. This process enables consequently a compensation or whitening of the tensor metric due to any bias that are present in the lters or the lter orientations. derivation of the tensor whitening method is found in [8]. The local tensor T constitute continuous geometrical representation of the neighbourhood. The distribution of the eigenvalues of T dene the local shape of the neighbourhood. The most common case for natural images is the rank case. Such a tensor can be written T ' ^e ^e T () where ^e is the eigenvector of T that corresponds to the largest eigenvalue. In D this corresponds to a local neighbourhood consisting of a line or an edge. In 3D, the environment isinterpreted as a planar structure or (for time sequences) as a moving line. In 4D the rank case corresponds to a moving plane. For further discussion on interpretation of the eigenvalues see

9 [6, ch 6]. The eigenvector ^e corresponding to the largest eigenvalue dene the orientation of the neighbourhood. To estimate the orientation error in the tensor representation require a test image containing rank neighbourhoods and the ideal eigenvectors corresponding to this image. In 3D such a test image consists by avolume containing concentric spherical shells. Some frames from this test sequence are displayed in g.. Locally all neighbourhoods are planar and all possible orientations are present. The angular RMS error ' is nally computed as: vu u ' = sin t LX, where: L l= k^x^x T, ^e ^e T k () ^x is a unit vector in the correct orientation, ^e is the eigenvector corresponding to the largest eigenvalue of the estimated tensor T, L is the number of points and ' is the angular RMS error. Filtering Structure for D Signals s mentioned earlier, the examples in this paper are focused on polar separable quadrature lters. The reason for this is the simplicity by which a robust and continuous tensor representation can be attained from this type of lters. This does not imply that this ltering approach is limited to this types of lters. In common for D, 3D and the 4D case is that the radial part of F (u) is of bandpass type with a center frequency of =( p ) : and a bandwidth of octaves. The orientation of each quadrature lter is dened by a vector ^n and the angular function of F (u) is dened as (^u ^n) for (u ^n) > andzero elsewhere. The number of degrees of freedom that are present in a symmetric second order tensor is (d +)d (3) where d is the dimensionality of the signal. In two dimensions the minimal number of lters is consequently three. Using sequential lters where each lter component only contain few coecients it is of out most importance to make advantage of all present symmetry aspects. The lter orienting vectors, ^n, must consequently be chosen with care using sequential lters as opposed to conventional FIR-ltering. The most ecient choice is to direct the lters along the main coordinate axes which takes care of two lters. The second best alternative is to orient the lters symmetrically between the main coordinate axes which also gives two alternatives. From earlier experiences on conventional lters we 6

10 .. Figure 3: Upper part: Resulting Filter function in the FD. Lower part: Deviation from the ideal lter function. know that using four lters increase the robustness in noisy and ambiguous neighbourhoods considerably in D. The the lter orienting vectors for D are therefore dened as: ^n = ( ; ) T ^n == p ( ; ) T ^n 3 = ( ; ) T ^n 4 == p (,; ) T (4) The following problem is to decide how to separate the quadrature lters in the above directions into sequential lter set. Since the lters are two dimensional it seems reasonable that (M=) lter components should be sucient. The choice of the coordinates is dependent of the type of lter that is to be implemented but in this case it is probably most ecient to concentrate the coecients along a line. gain with the symmetry aspect in mind the nonzero coecient of these two lters are distributed along with, and orthogonal to the lter orienting vectors in eq. (4). The sequential ltering structure for sequential lters in D is for this approach expressed as q ()= f y () f x () q ()=f,x;y () f x;y () q 3 ()= f x () f y () q 4 ()= f x;y () f,x;y () () where the indices indicate the distribution of the nonzero coecients of the lter components. The ltering is performed in two steps, the rst lter componentis 7

11 M= (3 coe) M= (648 coe) SNR ng. err Density(%) ng. err Density(%) d : : d 9: : d : :7 8. 6d 3: : d :4 9.9 : d : :66 9. d : 9.6 : :9 9.9 : Table : Performance of the sequential D lters in comparison to a direct implementation. 6 ngular RMS error (deg) Seq GOP SNR d Figure 4: ngular RMS error for dierent SNR. The solid line refers to the sequential (M=) implementation while the dashed line corresponds to a traditional (M=) implementation real valid and is expected to be of low passtype. The second lter component, which is directed in the main direction of the quadrature lter, is expected to be complex valued. The resulting optimized sequential lter set are found below together with the corresponding projection tensors. The spatial extension of the lter components vary from -9 pixels. Note that only four lters need to be computed. The remaining lters are obtained by mirroring according to the indices in eq. (). The reason for f,x;y () and f,x;y () to be tri-diagonal is simply to obtain suf- cient sampling density. The resulting frequency response for q () in eq. () is displayed in the 8

12 upper part of g. 3. The lower part of the same gure show the deviation from the ideal lter F (u). The weighted distance measure of eq. (4) accumulates to D = :. In comparison to a traditional implementation using a single complex lter of size 9 9 the distance measure is twice as high. The weighted distance measure is, however, of limited use when the performance of a lter set is to be judged. To obtain a more relevant comparison the local tensor T is computed from these lter responses using the projection tensors below. This computation where performed for both the sequential lters and the traditional (M = ) implementation for dierent SNR. From the tensor data the angular RMS error was computed according to eq. (). The result the result is listed in table () and exceeds our most optimistic expectations as there are no detectable loss in accuracy while the total number of coecients used in the sequential implementation is 3 as opposed to 648 for a direct implementation. The computational complexity is consequently reduced times in this example. The density in table () is a graded coverage estimate for the tensor representation. The density denes to howwell the local one dimensional neighbourhoods of the test image is reected in the tensor representation. In contrast to the angular RMS error the density only refers to the shape of the neighbourhood and is computed from the two largest eigen values and of the local tensor T. Density = P (, ) P (6) The dierence in density between the sequential and the direct method is in the same magnitude as for the angular RMS error, i.e. insignicant. From the measurements in table () it is concluded that the sequential ltering approach support an ecient and robust tensor representation in D. 6 Filtering Structure for 3D Signals Figure : The icosahedron, one of platonic polyhedra. In 3D, the minimal number of lters required to produce a second order tensor representation is six, eq. (3). The apparent choice using conventional lter- 9

13 ing is to dene the lter orienting vectors as the vertices of a hemi-icosahedron [7, 6]. The icosahedron is a regular diametrically symmetric polyhedron (g. ) which implies that the lters are equally spread. The six lter orienting vectors dened by the icosahedron are ^n = c ( a ; ; b ) T ^n 3 = c ( b ; a ; ) T ^n = c ( ; b ; a ) T ^n = c (,a ; ; b ) T ^n 4 = c ( b ;,a ; ) T ^n 6 = c ( ; b ;,a ) T (7) where: a = b = ( + p ) c = ( + p ),= These orientations are unfortunately not suitable for a sequential implementation because of the symmetry aspects discussed earlier. In agreement with the D case the rst choice of lter orienting vectors for sequential ltering in 3D are along the coordinate axes (3 possibilities) and the second symmetrically between two coordinate axes (6 possibilities). lthough 6 lters are sucient in theory we choose to use nine lters for two reasons. The rst reason is that the orientation error due to the sequential lters not being polar separable is reduced by increasing the number of lters. The second reason is that using the convolver tree structure presented below, the increase in computational complexity by computing 9 lter responses instead of 6 is very small. The 9 lter orienting vectors for 3D are consequently dened as ^n == p ( ;,; ) T ^n == p ( ; ; ) T ^n 3 = ( ; ; ) T ^n 4 == p (,; ; ) T ^n == p ( ; ; ) T ^n 6 = ( ; ; ) T ^n 7 == p ( ; ; ) T ^n 8 == p (,; ; ) T ^n 9 = ( ; ; ) T These lter orientations are not exactly equally spread as in the lters orientations in eq. (7). To eliminate any orientation bias in the local tensor T the projection tensors are computed using the tensor whitening method discussed earlier, [8]. The sequential ltering structure for the 3D case is consequently

14 One dimensional low pass filters One dimensional quadrature filters f y f x q 9 f z f x,y f x,y q 8 f x,y f x,y q 7 f x f z q 6 f y f x,z f x,z q Image sequence f x,z f x,z q 4 f z f y q 3 f x f y,z f y,z q f y,z f y,z q Figure 6: rst level. convolvers organized in 3 levels. Only 3 lters are required on the expressed as q ()=f x () f y;z () f,y;z () q ()=f x () f,y;z () f y;z () q 3 ()=f x () f z () f y () q 4 ()=f y () f x;z () f,x;z () q ()=f y () f,x;z () f x;z () q 6 ()=f y () f x () f z () q 7 ()=f z () f,x;y () f x;y () q 8 ()=f z () f x;y () f,x;y () q 9 ()=f z () f y () f x () (8) Each quadrature lter response are computed in three steps. The rst two components are real valued and are both oriented in directions orthogonal to the lter orienting vectors. The last component is complex valued and directed along the main direction of the ideal lter. In the sequential lter organization above, the initial ltering step is identical for several lters. This observation is used to dene an ecient convolver tree structure of g. 6 consisting of convolvers on 3 levels. Using only 3 convolvers at the top level. The optimized sequential lter components are displayed below. In g. 7 the resulting frequency response for q 7 () is illustrated in the u;u-plane. The angular RMS error of the local tensor were computed for dierent SNR and the result is given in table. The

15 .. Figure 7: Upper part: Frequency response for q 7 () in the u ;u -plane. Lower part: Deviation from the ideal lter function. left column correspond to a lter set consisting of 9 quadrature lters where each quadrature lter response are computed sequentially by three lter components according to g. 6 The computation of nine quadrature lter responses requires 34 multiplications. If on the other hand conventional quadrature lters are used, six complex lters of size require almost 9 multiplications. The computational load is consequently reduced times in this example while the dierence in accuracy is insignicant, see table. SNR M =3 M = 34 coe coe. :76 :7 d 3: 3:3 d 9:3 9:69 Table : ngular RMS error in degrees for a sequential lter with 3 components and the corresponding lter implemented by a single component. 7 Filtering Structure for 4D Signals In 4D, the choice of lter orienting vectors is quite simple. From eq. (3) we conclude the the the minimum number of lters required to produce a tensor representation in 4D is. If the lters are localized symmetrically between

16 two main coordinate axes this results in possibilities, which is sucient. The lter orienting vectors in 4D are dened as ^n == p ( ; ; ; ) T ^n == p ( ; ; ;, ) T ^n 3 == p ( ; ; ; ) T ^n 4 == p ( ;,; ; ) T ^n == p ( ; ; ; ) T ^n 6 == p ( ; ; ;, ) T ^n 7 == p ( ; ; ; ) T ^n 8 == p (,; ; ; ) T (9) ^n 9 == p ( ; ; ; ) T ^n == p ( ; ; ;, ) T ^n == p ( ; ; ; ) T ^n == p (,; ; ; ) T These lter orientations also correspond to the vertices of the 4-cell, one of three regular polyhedra in 4D. The above lter orientations are consequently equally spread in the 4D signal space which simplies to computation of the projection tensors. In agreement with the 3D case the sequential ltering structure for 4D signals is expressed as q () = f x () f y () f,z;w () f z;w () q () = f x () f y () f z;w () f z;,w () q 3 () = f x () f w () f,y;z () f y;z () q 4 () = f x () f w () f y;z () f,y;z () q () = f y () f z () f,x;w () f x;w () q 6 () = f y () f z () f x;w () f x;,w () q 7 () = f y () f w () f,x;z () f x;z () q 8 () = f y () f w () f x;z () f,x;z () q 9 () = f z () f x () f,y;w () f y;w () q ()=f z () f x () f y;w () f y;,w () q ()=f z () f w () f,x;y () f x;y () q ()=f z () f w () f x;y () f,x;y () The ltering is performed in 4 levels where the three rst are real valued and the last lter is complex. The distribution of the nonzero coecients within each lter component is indicated by the indices. Note that the indices of the 3

17 One dimensional low pass filters One dimensional quadrature filters f z fw f x fx,y f x,y f y,w f y,w f x,y f x,y fy, w f y,w q q q q 9 Time sequence of volumes f y f w f z fx,z f x,z f x,w f x,w f x,z f x,z fx, w f x,w q 8 q 7 q 6 q f x f w f y f y,z f y,z f z,w f z,w f y,z f y,z fz, w f z,w q 4 q 3 q q Figure 8: onvolver organization in 4D, 33 convolvers in 4 levels components within each lter are orthogonal and that in the two rst ltering steps the same lter combination occurs several times. I g. 8 a 4D convolver structure in 4 levels is displayed. y careful combination of the results in the sequential ltering structure only 3 and 6 convolversare required at the rst and second level respectively. The optimized lters with center frequency % =: and bandwidth =octaves are found below. t this point it starts to get really interesting to compare the the computational complexity of sequential convolution to traditional implementations. The spatial extent of the optimized lter components vary from -9 pixels. To compute a single quadrature lter response require = 6 real multiplications. Using the convolver tree structure of g. 8 reduces this gure further by about % resulting in 67 multiplications for lters. In conventional FIR-ltering a complex lter of size require almost 6 real valued multiplication to compute quadrature lter responses. The sequential approach is consequently almost 8 times more ecient. 4

18 D: D Image 3D: Time sequence of images or a 3D volume 4D: Time sequence of volumes Figure 9: Sequential convolver organization in D,3D and 4D 8 Implementation spects The general convolver module is implemented in with the graphical user interface in VS (pplication Visualization System), see [] for more details. The tree structure of the convolver organization, g. 9, provide an ecient optimization of the processing performance for a wide variety computer architectures. The soft implementation of the convolver network enable a simple reconguration for dierent types of input data (dierent outer dimensions) e.g D image, 3D spatiotemporal image sequences (3D volumes) and 4D volume sequences. The vector length (i.e. the inner dimension) of the kernel data is likewise simple to adapt from e.g scalars, RG-signals, D vector elds, 3D vector elds, tensor elds etc. 9 cknowledgment The support of the National Swedish oard for Technical Development, NUTEK (project 93-) is greatfully acknowledged.

19 Sequential lters for D These are the sequential lter components used in the experiments. From these four lter components all lters in eq. () can be produced by mirroring according to the indices. The coecients are converted to integers to support an ecient implementation. f x () = f xy () = Re[f x ()] =,87,673, ,7,673,87 63 Im[f x ()] = 67,4, ,,67 Re[f xy ]= Im[f xy ]= Projection tensors in D,46,87,689,869,46 93,44, ,689,44 93,46,869,689,87,46,9,484 73,39, ,44 44,73,, , :7 : T = :,:,: : T 3 = : :7 : : T = : : :,: T 4 =,: : 6

20 Sequential Filters for 3D f x () = f xy () = Re[f x ()] =,,37,6, 4,,6,37, Im[f x ()] = 6,8,7, ,6 Re[f xy ]= Im[f xy ]= ,64,38,487,89,64 644,369, ,487, ,64,89,487,38,64,97,447 49,6, , ,49,444, ,

21 Projection Tensors in 3D These projection tensors where computed using the tensor whitening method ([8]) to compensate for any orientation bias due to uneven spread of the lters. T = T 3 = T = T 7 = T 9 = :69 :4 :33 :33 :4,:84 :38,:84 :4 :33 :69 :33 :4 :4,:33,:33 :4 :69 :38,:84,:84 T = T 4 = T 6 = T 8 = :69 :4,:33,:33 :4 :4,:33 :69,:33 :4,:84,:84 :38 :4 :33 :33 :4 :69 Sequential Filters for 4D Re[f xy ]= f x () = f xy () = , , ,98, ,7, ,98,7 8 38,84,

22 Im[f xy ]= 6 4 Projection Tensors in 4D,963,78,3,, ,3, ,77, T = T 3 = T = T 7 = T 9 = T =,:,: : : : :,: : : : :,: : :,:,: : : : :,: : :,:,: : :,: : : : : : :,:,: T = T 4 = T 6 = T 8 = T = T =,:,: :,:,: :,: :,:,: :,: :,:,:,:,: : :,:,:,: :,:,: :,:,:,: : :,:,: :,:,: 9

23 References [] M. ndersson. ontrollable Multidimensional Filters in Low Level omputer Vision. PhD thesis, Linkoping University, Sweden, S{8 83 Linkoping, Sweden, September 99. Dissertation No 8, ISN 9{787{ 98{4. [] M. T. ndersson and H. Knutsson. ontrollable 3-D Filters for Low Level omputer Vision. In Proceedings of the 8th Scandinavian onference on Image nalysis, Troms, May 993. SI. [3] R. racewell. The Fourier Transform and its pplications. McGraw-Hill, nd edition, 986. [4] D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Processing. Prentice-Hall signal processing series. Prentice-Hall, 984. ISN [] G. H. Granlund. In search of a general picture processing operator. omputer Graphics and Image Processing, 8():{78, 978. [6] G. H. Granlund and H. Knutsson. Signal Processing for omputer Vision. Kluwer cademic Publishers, 99. ISN [7] H. Knutsson. Representing local structure using tensors. In The 6th Scandinavian onference on Image nalysis, pages 44{, Oulu, Finland, June 989. Report LiTH{ISY{I{9, omputer Vision Laboratory, Linkoping University, Sweden, 989. [8] H. Knutsson and M. ndersson. Robust N-Dimensional Orientation Estimation using Quadrature Filters and Tensor Whitening. In Proceedings of IEEE International onference on coustics, Speech, & Signal Processing, delaide, ustralia, pril 994. IEEE. LiTH-ISY-R-798. [9] H. Knutsson, M. ndersson, and J Wiklund. dvanced Filter Design. In Proceedings of the Scandinavian onference on Image analysis, June 999. Submitted. [] J. Wiklund and H. Knutsson. Generalized onvolver. In Proceedings of the 9th Scandinavian onference on Image nalysis, Uppsala, Sweden, June 99. SI.