2-D Dual Multiresolution Decomposition Through NUDFB and its Application

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1 2-D Dual Multiresolution Decomposition Through NUDFB and its Application Nannan Ma 1,2, Hongkai Xiong 1,2, Li Song 1,2 1 Dept of Electronic Engineering 2 Shanghai Key Laboratory of Digital Media Processing and Transmissions Shanghai Jiao Tong University, Shanghai , China {puppuppy, xionghongkai, song_li}@sjtueducn Abstract This paper aims to attain sparser representation of a 2-D signal by introducing orientation resolution as a second multiresolution besides multiscale, which is formulated to achieve a dual multiresolution decomposition framework by nonuniform directional frequency decompositions (NUDFB) under arbitrary scales In this scheme, NUDFB is fulfilled by changing the topology structure of a non-symmetric binary tree (NSBT) Through this nonuniform division, we can get arbitrary orientation resolution r at a direction of under a target scale Every two-channel filter bank on each node of this NSBT is designed to be a paraunitary perfect reconstruction filter bank, so NUDFB is an orthogonal filter bank This dual multiresolution decomposition will definitely have bright prospect in its application, such as texture analysis, image processing or video coding A potential application is presented by applying NUDFB in wavelet domain I INTRODUCTION It is known that sparse representation is sought to maximally capture features of interest of a signal with maximally decimated coefficients This representation is performed by a signal projection onto another space expanded by a complete orthogonal basis, whose efficiency is embodied by energy convergence to a few of large coefficients which are inner products of signal and basis For 1-D signal, time or position at which frequency jumping happens always is of interest However, position and frequency can not both be specified to arbitrary precision in terms of Heisenberg uncertainty principle Multiscale is dedicated to limit the deviation to a tolerable extent For 2-D signals, eg images, it has been validated that human eyes are extraordinarily sensitive to contours or edges with high orientation selectivity [16] It is expectant for 2-D signals that both frequency-jumping position and orientation are features of interest, and the orientation along which frequency-jumping amplitude achieves the maximum is called frequency-jumping orientation or the orientation vertical to a contour Although 2-D directional filter bank [1], curvelet [2], and contourlet [3] have taken into account orientation, these decompositions treat 2-D signals with no difference by projecting them onto uniformly divided orientation subspaces In fact, most of natural signals are characteristic of nonuniform orientation When limited orientation resolution is available, these decompositions are not optimal or most efficient Traditionally, orientation resolution is also dependent on scale [3] However, textures and edges are variable with various orientation resolutions across scales and within these scales This paper introduces orientation resolution as an isolated variable from scale, achieving a dual multiresolution decomposition In order to achieve an orientation multiresolution under a certain scale, nonuniform directional frequency decompositions (NUDFB) is imperative Despite extensive methods to design 1-D nonuniform filter banks [4-8], the advance on 2-D nonuniform filter banks has been hindered by complicated issues in design process For example, universal anti-aliasing filter bank for arbitrary downsampling matrices, some of which maybe irrational, may be not accessible A non-symmetric binary tree (NSBT) structured filter bank is proposed to fulfil NUDFB, because of the following advantages: (1) minimum branches or channels at each node to reduce the design complexity, especially for 2-D nonseparable filter banks; (2) more flexible to choose an appropriate frequency division; (3) convenient to elaborate the binary tree structure, which is important to know the decomposition structure if a nonuniform decomposition is used Although biorthogonal filter bank used in contourlet is less constrained in perfect reconstruction, orthogonal filter bank is chosen in this context owing to its attractive properties in subband coding applications [14] For 2-D filters design in a filter bank, there are mainly two methods: to design a 2-D filter directly, and to get the target 2-D filter from a 1-D prototype filter [9] The latter is employed to simplify both the design procedure and the implementation process to reduce the implementation complexity to other than The 2-D nonuniform filter bank with NUDFB structure is of maximal decimation and paraunitary perfect reconstruction The remainder of the paper is organized as follows: Section II formulates the proposed dual multiresolution decomposition framework based on NSBT In Section III, a paraunitary perfect reconstruction condition is provided through a polyphase identical form of filter bank, in terms of 2-D nonseparable filters from a 1-D prototype The NUDFB division is developed in section IV, with regard to the topology structure of binary tree and orientation resolution Potential application is depicted in section V Section VI concludes this paper and gives the future work II DUAL MULTIRESOLUTION REPRESENTATION Suppose we substitute NUDFB for UDFB in wavelet-based contourlet decomposition [11] As the directional filter bank is /08/$ IEEE 509 MMSP 2008

2 designed to capture the high frequency components, NUDFB is performed only after DC components are removed A Space multiresolution (multiscale) Multiscale as the first multiresolution in this scheme can be defined as space multiresolution Its concept is introduced by dividing original space into complete orthogonal subspaces [15], (1) where denotes the lowpass subspace and is excluded in the next directional decomposition analysis Subscript of denotes the scale of this highpass subspace In discrete 2-D separable dyadic wavelet decomposition, one of three highpass filters uniquely defines an orthogonal function via a two-scale equation below: Suppose then the family is an orthonormal basis of for all, where family is an orthogonal basis of which are generated by LH, HL, and HH subbands separately B Orientation multiresolution Applying NUDFB within, we obtain the orientation multiresolution as second multiresolution Using multirate identities, NSBT with any topology structure can be substituted for an identically parallel structure with subbands Suppose these subbands are of the same orientation resolution, can be divided into subspaces, which we can call orientation subspaces From another perspective, is expanded by an orthonormal basis corresponding to each orientation subspace, where scale and position are all fixed [3]: (2) Proposition 1 These subspaces are orthogonal with (3) where denotes the separating operator Corollary 1 From (1), it is easy to infer: (4) we note that is an orthonormal basis for Proposition 2 In corollary 1, for every partition of nonnegative integers into sets of the form,, is an orthonormal basis for At the same time, the sequence composed of integers is extended into a longer sequence with the form of Notice If is set zero for a fixed integer, there is no partition Corollary 2 Denote is the first-level partition scheme for nonnegative integers, and these integers are partition into sets with the form of From proposition 2, we have as an othonormal basis of, and the generating sequence is By carrying on the same partition scheme on, it is easy to prove an orthonormal basis of It is easy to be generalized to an which -level extended case in is also proven to be an orthonormal basis of Definition A NSBT basis of is any orthonormal basis selected from elements family, where and are arbitrary nonnegative integers Proposition 3 Suppose wavelet space is divided into orientation subspaces maybe with different orientation resolutions under a certain NSBT is separated into multiresolution subspaces with different orientation resolutions :, (5) where is the lowest orientation among these subspaces, and is the highest Proof It is an improvement of orientation resolution from to with the process shown in proposition 2 Suppose is an orthogonal basis of before partition In the case of (5), from to, sequence is partitioned, forming a new extended sequence, where If some of the s are set zero for selected integers s, 510

3 where denoting the set of these integers and the ordinal number of these integers, orientation resolution for these subspaces is retained On the other hand, if is not zero for with denoting the set of these integers, these subspaces are extended to more subspaces with higher resolution Subbands with higher resolution are obtained by carrying out operation on, where, ie: As we have shown in corollary 1, is an orthonormal basis of, where is fixed For subbands within, orientation resolution is extended from to according to (6) Accordingly, the resulting basis is also orthonormal for C basis with dual multiresolution Within a multirate system, an arbitrary NSBT with arbitrary subbands can be grouped into two categories: basically horizontal and vertical ones These subbands have identically analysis and synthesis filters with overall sampling matrices which have the diagonal form: (6), (7) corresponding to horizontal and vertical subbands For a NSBT, which is determined by orientation resolution describes the level which the subband belongs to; cardinal number of this subband Let us define then the family is the, (8) is an orthonormal basis of an orientation subspace for, where response of the synthesis filter is obtained by translating the impulse over the sampling lattice [3] Concept of dual multiresolution is embodied by and, respectively, which depict space and orientation resolution of its corresponding subspace III FILTER BANK DESIGN By investigating the filter bank with NSBT structure, we can ascertain it is composed of plenty of two-channel filter banks which are extended from a parent node On this point, there is no difference from the traditional binary tree structure which is involved with a uniform frequency decomposition Therefore, the implicated design procedure is also regarded as a simplified design of a 2-D uniform filter band with two subbands Fig 1 The used filters in the filter bank A 2-D nonseparable filters As shown in Fig 1, 2-D nonseparable filters are chosen [1] For a maximally decimated filter bank, the determinant of the sampling matrix must be two:,,,,, respectively, for,, and, and, and, and to avoid aliasing [10] and achieving maximal decimation B Paraunitary Perfect reconstruction condition Because the structure of filter bank is composed of two-channel filter banks extending from a parent node, it is inferable that if the two-channel filter banks are perfectly reconstructed, the whole filter bank will be definitely perfectly reconstructed From the shape of five filters, relationships between lowpass and highpass filters are (9) where the non-zero coset vector are denoted as for the sampling matrices,,, and By using type 1 polyphase transform for in (9), we can obtain: and and (10) Suppose denotes the polyphase matrix of analysis filter bank defined in z domain, it can be rewritten as: (11) It is easy to prove is paraunitary, ie,, where and is a constant In the case of two channels, we can say is paraunitary pair Similarly, if denotes type 2 polyphase matrix of synthesis filter bank, it is chosen as to satisfy the perfect reconstruction condition: (12) where is an arbitrary constant vector Combining (11) and (12) we can have: 511

4 (13) We finally have the synthesis type 2 polyphase frequency response as: (14) From (14) we can see that is also a paraunitary matrix, so the whole filter bank is orthogonal Like the above analysis, the other four analysis filter banks are also proven to be paraunitary pairs, and they are all orthogonal filter banks whose synthesis filter banks are in the form of (14) C Spatial rotation and rescaling effects Take as an illustration, its spectrum is of a parallel shape Impulse response related to can be written as [9]:, (15) where represents the impulse response of a separable 2-D filter derived from a 1-D filer It is same for,, and to get corresponding impulse responses If denotes an impulse response of an ideal lowpass filter with a form of, the new spatial lattices for 2-D nonspeparable filters respectively for,,, and are shown in Fig 2 We can observe that rotation skewing and rescaling of the coordinates lead to impulse responses with rotated orientations Fig 3 gives impulse responses of these 2-D nonseparable filters which are transformed from the ideal lowpass filter (a) (c) Fig 3 The impulse response for 2-D nonseparable lowpass filters designed from a 1-D ideal lowpass filter (a) (b) (c) (d) D Efficient implementation of this filter bank Generally, the implementation complexity of 2-D filter grows linearly with, where is the filter length However, it can be reduced to by a polyphase form The equivalent form of polyphase decomposition is shown in Fig 4 by decomposing to two separated 1-D lowpass filters, and and are corresponding polyphase components of these two separated filters [9] With the relationship between the polyphase components of lowpass and highpass filter in (11), and the relationship of their corresponding synthesis filters in (12), we can simplify all the filters in the whole filter bank as the form of Fig 6 (b) (d) Fig 4 Suppose, and are polyphase components respectively for and Fig 2 (a) lattice of impulse response for, (b), (c), (d) ( Black dots represent it is non-zero at this location, while the value at the position of white dots is zero) IV ILLUSTRATION AND REPRESENTATION OF NUDFB As shown in Fig 5, the original spectrum is divided into 10 wedge shaped nonuniform directional subbands Subbands with narrower spectrum have more delicate orientation resolution This NUDFB is obtained by a NSBT shown in Fig 6 Part of the spectrum division process is shown in Fig 7 In Fig 6, each subband is labelled with an exclusive binary sequence, such as 000 for subband 1 We call it a trace for a certain subband whose length portrays the affiliated grade, while each bit corresponding to a node in the topology structure symbolizes the branch separated from the last node Vice versa, from the set of all these sequences, we can definitely obtain a NSBT structured NUDFB without overlapping between these subbands Given the minimum and the maximum length among these sequences, we can have the lowest and highest orientation resolution, and the length of these sequences must be different from a NUDFB 512

5 Fig 5 A nonuniform wedge shaped spectrum division Fig 6 The NUDFB with a spectrum division shown in Fig 5 Fig 7 Frequency division process of Fig 6 V WAVELET DOMAIN APPLICATION In this section, NUDFB is applied in wavelet domain When investigating wavelet coefficients around the position of an edge in original image, we can observe that a cluster of significant wavelet coefficients, which look like a visual edge in wavelet space Basically, there are perceptually horizontal edges or details in LH, vertical edges in HL, and diagonal edges in HH It is understandable to get a sparser coefficients representation by applying different spectrum divisions to these subbands Wavelet-based contourlet transform aiming to eliminate the redundancy caused by LP decomposition introduces UDFB in wavelet domain [11] We substitute NUDFB for UDFB, and Fig 9 shows one-level wavelet decomposition in LH, HL and HH with traces of spectrum divisions shown in Fig 8 Table I shows experimental results of two schemes, where both UDFB and NUDFB divide the spectrum into eight subbands Wavelet transform utilizes 9-7 biorthogonal filters [12], and the 2-D nonseparable filters are obtained from Daubechies orthogonal wavelet base with a vanishing moment of 2 [13] It is noticeable that NUDFB performs better than UDFB To show the performance of NUDFB in an intuitive way, Fig 10 shows a comparison between UDFB and NUDFB in LH subband Firstly, a threshold 18 is chosen to select out the significant coefficients in these orientation subbands; Secondly, we upsample these coefficients to rearrange these subbands coefficients together These rearranged coefficients are shown to be much sparser compared with UDFB in Fig 10 In another perception, significant coefficients of NUDFB almost locate at the positions of edges of the original image, while significant coefficients of UDFB capture much more noise In other words, NUDFB retains original image structure much better than UDFB, and it has a much better anti-noise performance VI CONCLUSION In this paper we formulate a dual multiresolution decomposition framework by nonuniform directional frequency decompositions (NUDFB) under arbitrary scales NUDFB is fulfilled by arraying the topology structure of a non-symmetric binary tree (NSBT), as a simplified symmetric extension from a two channel filter bank The paraunitary perfect reconstruction condition is provided through a polyphase identical form of filter bank, in terms of 2-D nonseparable filters from a 1-D prototype This nonuniform division can get arbitrary orientation resolution r at a direction of under a target scale By applying this NUDFB in wavelet domain, a sparse representation of image signal has been improved in comparison with contourlet transform It is imaginable that if the orientations of textures and edges in original image could be instructive to design the corresponding trace for NUDFB that maximally captures these orientation details The implicated coding scheme for image compression is the future concern TABLE I EXPERIMENTAL RESULTS FOR UDFB AND NUDFB SCHEME Coeffs MSE Coeffs MSE number UDFB NUDFB number UDFB NUDFB

6 Fig 9 Fig 8 Binary tree structure in the experiment NUDFB in LH HL and HH in a one-level wavelet domain ACKNOWLEDGMENT The work has been partially supported by the NSFC grants No , No , No and the National High Technology Research and Development Program of China (863 Program) (No 2006AA01Z322) REFERENCES [1] R H Bamberger and M J T Smith, A filter bank for the directional decomposition of images: Theory and design, IEEE Trans Signal Processing, vol 40, no 4, pp , Apr 1992 [2] E J Candes and D L Donoho, Curvelets-A Surprisingly Effective Nonadaptive Representation For Objects with edges, In Curve and Surface Fitting, A Cohen, C Rabut and L L Schumaker, eds, Nashville, TN Vanderbilt Univ Press, 1999 [3] M N Do and M Vetterli, The contourlets transform: an efficient directional multiresolution image representation, IEEE Trans Image Proc, vol 14, no12, December 2005 [4] T Nagai, T Futie, M Ikehara, "Direct Design of Nonuniform Filter Banks, IEEE International Conference on Acoustics, Speech, and Signal Processing 1997, vol 3, pp [5] Kambiz Nayebi, Thomas P Barnwell, and Mark J T Smith Nonuniform filter banks: A reconstruction and design theory IEEE Trans Signal Processing, vol 41, no3, Mar 1993 [6] Tongwen Chen, Nonuniform multirate filter banks: analysis and design with an performance measure, IEEE Trans Signal Processing, vol 45, no3, pp , Mar 1997 [7] P-Q Hoang, and P P Vaidyanathan, Non-uniform multirate filter banks: theory and design, IEEE International Symposium on Circuits and Systems 1989, pp [8] J Li, T Q Nguyen, and S Tantaratana, A simple design method for nonuniform multirate filter banks, Asilomar Conference on Signals, Systems, and Computers 1994, pp [9] Tsuhan Chen and P P Vaidyanathan, multidimensional multirate filters and filter banks derived from one-dimensional filters, IEEE Trans Signal Processing, vol 41, no 5, pp , May 1993 [10] P P Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, Englewood Cliffs, 1993 [11] Eslami R, and Radha H, Wavelet-based contourlet transform and its application to image coding, Proceeding of IEEE International Conference on Image Processing, USA, pp , 2004 [12] A Cohen, I Daubechies, J-C Feauveau, Biothogonal bases of compactly supported wavelets, Commun On Pure and Appl Math,45: , 1992 [13] I Daubechies, Orthonormal bases of compactly supported wavelets, Comm Pure & Appl Math, 41, pp , 1988 [14] S Venkataraman and B C Levy, A comparison of design method for 2-D FIR Orthogonal Perfect Reconstruction Filter Bank, IEEE Trans Circuits and Systems, vol 42, no 8, Aug 1995 [15] SMallat, A theory of multiresolution signal decomposition: The wavelet transform, IEEE Trans, PAMI-11(7), ,1989 [16] Andrews, D P, Perception of contours in the central fovea Nature, Lond 205, (a) UDFB in LH (b) NUDFB in LH Fig 10 Significant coefficients larger than threshold 18 are retained 514