This paper presents new developments of directional filter banks. A new directional filter bank for image analysis and classification is proposed. This paper introduces an improved structure in order to visualize subband outputs of the directional filter banks, while retaining the attractive properties of the original directional filter banks such as 1-D separable filtering, perfect reconstruction, and maximal decimation. In order to have nondistorted phase information in the subbands for visualization, both FIR and IIR filter prototypes that can be implemented efficiently are provided for linear phase filtering. This paper shows the approach proposed here can be applied to image analysis and classification.
Keywords- Directional Filter Bank, SubBand, Wavelet, 1-D Filtering, FIR, IIR, Decomposition, Image processing.
I.INTRODUCTION
Wavelets and filter banks have been used effectively in many signal-processing applications [1]. An advantage of using wavelet bases instead of Fourier bases is due to approximation power of wavelet series in signal with singularities since it would take a larger number of Fourier coefficients than wavelet coefficients to represent a signal with discontinuities. Typically, one constructs a two-dimensional (2D) wavelet by taking the tensor product of one-dimensional (1D) wavelets. This 2D wavelet is still effective at approximating point singularities (e.g. points in an image), but not for line singularities (e.g. edges in an image). This fact was notified by many researchers [2], [3], [4], and therefore finding a more effective basis for images is currently a very attractive research area.
Directional information in a given image is important in many image-processing applications [1]. It can be extracted by various methods such as a simple fan filter, Radon transform[2], steerable filter bank[3], or directional filter bank[1]. Among these methods, the main advantages of a directional filter bank are the following: It can be implemented by 1-D separable filtering. Directional filter banks can extract 2-D directional information, as in Figure 1 (a), into subbands, as shown in Figure 1 (b) for this is possible using the poly phase structure[4], in Figure 2 (a). This poly phase structure make the implementation of directional filter banks very efficient not only because of its structure by itself [4], but also because it makes 1-D separable filtering possible[1].
(a)
(b)
Figure 1. Frequency map for 8 band directional filter bank. (a) Directional frequencies of input to be decomposed. (b) Decomposed frequency maps of the 8 subbands.
(a)
Figure 2. Polyphase Structure.
II CONVENTIONAL APPROACHES
A. The conventional Directional Filter Bank
Recently, a directional filter bank (DFB), of which subband partitioning is presented in Figure 1(b), has been introduced by Bamberger and Smith [5]. A major property of the DFB is its ability to extract 2D directional information of an image, which is important in image analysis and other applications. It has been used in texture classification [6], image denoising [7], fingerprint image enhancement and recognition [8], etc. The DFB is maximally decimated and perfect reconstruction (PR). This means that the total number of subbands' coefficients is the same as that of the original image, and they can be used to reconstruct the original image without error. It can be implemented by a tree structure consisting of three levels of two-band systems. Each level can be implemented by using separable polyphase filters, which make the structure extremely computationally efficient. The DFB is improved in [9] where visible subbands are constructed. In [10], an octave directional subband decomposition has also been studied. One major drawback of the conventional DFB, as one can see in Figure 3, is the way the low frequency band is divided.
Figure 3. Frequency partition of the filter bank: Conventional DFB,
It is known that, for natural images, most of the energy is concentrated at DC and its neighboring bands. Hence a small perturbation occurring at this region can have a significant impact on the directional information in the subbands. All the subbands in the conventional DFB meet at DC and thus require very sharp frequency partition in order to have accurate estimation. In practice, such ideal brickwall filters are not feasible (at least for the FIR case), and therefore the low frequency component is usually removed before the DFB is applied. Hence the resulting decomposition is no longer maximally decimated, which explains why the DFB has found limited applications.
B. The Pyramidal Directional Filter Bank and Contourlet
Do et al. proposed a pyramidal DFB (PDFB) in order to implement the contourlet transform, a discrete version of the curvelet transform . The proposed structure is a combination of the Laplacian pyramid and the DFB. Their pyramidal DFB has united two advantages of the two structures, which are multi-resolution and multi-direction. It attempts to separate the low frequency component from the rest directional components, and re-iterates with the same DFB in the lowpass band forming a pyramid structure. While the decomposition solves the problem at low frequency, it is, however, still not maximally decimated. The oversampled factor is the same as the Laplacian pyramid , which is 4/3. In recent work s, the authors have generated a maximally decimated directional filter bank by employing appropriate filters at the third level of the conventional DFB. However, the problem of dividing the DC component is still not resolved because there are four lowpass directional subbands at DC. The decomposition has at least sixteen subbands. In addition, the proposed filter bank is implemented by a binary tree structure with large supported impulse responses.
III PROPOSED SYSTEM
This paper presents a new directional filter bank structure that has the following features:
Maximally-decimated, and perfectly-reconstructing filters with 1-D separable filtering, as in the original implementation.
Visualizable directional subbands. Each of the subbands has intuitive directional information.
Tree-structured filter banks. A 2n band directional filter bank can be implemented by cascading basic components, which are the only components that need to be designed.
Linear phase filtering: In order to preserve the phase information of a given input image during filtering, both linear phase FIR and linear phase IIR filtering are considered and implemented.
Unlike typical tree-structured filter banks, where only one 2- band filter bank block needs to be cascaded, tree-structured directional filter banks need to use two: One, as in Figure 4(a), is used for first 2 stages, and the other, as in Figure 3 (c), is for the remaining stages. Both of structures in Figure 4(a) and (c) can be implemented in polyphase form as in Figure 4 (b). For the structure in Figure 4(a), a diamond filter H0(ω), as in Figure 4 (a), is used,
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(b)
(c)
Figure 4: Overall Structure of a new directional filter banks. (a) First block for first 2 stages. (b) Its polyphase Structure. (c) Second block for the rest of stages.
while an hour-glass filter, Hm0(ω), as in Figure 5(b), can be used as in Figure 4(a) if the modulator is omitted. Both cases are conceptually identical. Once the filter approach for the first two stages is decided upon, the remaining stages use the corresponding
(a)
(b)
Figure 5. 5 Kinds of passbands for directional filter banks. (a) For modulated structure. (b) For non-modulated structure.
IV APPROACH CONSIDERATIONS
One of the main purposes for this paper is to introduce an application, where the analysis system of the directional filter is used without the synthesis system. Here, for visualizing subband outputs without phase distortions, linear phase subband filtering is more important than perfect reconstruction. We use a 2nd-order linear-phase IIR filter[7] that not only guarantees perfect reconstruction but also can be implemented faster than by using an FIR filter with similar characteristics. The linear phase IIR filter coefficients used here are ω1 = 4.126081, ω2 = 4.488383, α= 0.711746, and K= 0.295035 for the following transfer function:
H(z) = K x (1 + z-1) I(ω1)I(ω1)/ (1+α2z-2)(1+(1/ α2) x z-2), (1)
Where,
I(ω) = 1 - (2 cos ω) z-1 + z-2 (2)
Another interested choice for the filter is a linear-phase odd length FIR filter. For a 2N +1 tap linear phase FIR filter, in addition to its whole sample delay property, one of its polyphase filters always has the form of a simple delay, which saves computation by a factor of 2, as in Figure 6.
Figure 6. Polyphase structure using a linear-phase odd-length FIR
filter.
At first, there are totally 4 kinds of quincunx matrices that can be used in directional filter banks as follows:
q1 = [ 1 1 ; -1 1 ],
q2 = [ 1 -1 ; 1 1 ],
q3 = [ -1 1 ; 1 1],
q4 = [ 1 1 ; 1 -1 ]
For first two stages, any quincunx matrix can be used. A typical choice is q1. Note that, unlike q3 or q4, q1*q1 = [0 2 ; 2 0], which is a clockwise-rotation matrix of [ 2 0 ; 0 2 ] so that a counterclockwise rotation needs to be additionally added after the first two stages if q1 is chosen. For the remaining stages, in order to make a simple rule for an expanding tree, a pair of quincunx matrices need to be used as follows:
Qi = {q2 if i is 1 or 4 OR q1 if i is 2 or 3}
The definition of resampling matrix is a matrix whose determinant should be 1 with integer matrix entries so that its inverse matrix is also a resampling matrix. Among the infinitely many resampling matrices, in order to replace the parallelogram filters with the diamond filter, the resampling matrices that define the change of variables[8] are the following:
r1 = [ 1 1 ; 0 1 ],
r2 = [ 1 -1 ; 0 1 ],
r3 = [ 1 0 ; 1 1 ],
r4 = [ 1 0 ; -1 1 ]
At any stage after the second stage, Ri = ri.
V RESULTS
The subband outputs for the cameraman image as in Figure 7 are given in Figure 8. As expected, each of subbands has its directional information. For instance, in Figure 8 (b), one of the tripod's legs are directionally visualized with the cameramen's body contour that has similar direction. But, in Figure 8 (d), most of directional information that is captured in Figure 8 (b) is missing, while another tripod's leg is captured. Also notice that the contour of cameraman's right hand that has slightly different direction from the tripod's left leg is captured in (i), unlike in (b), as expected.
Figure 7: Directionally decomposed images. (a) The original cameraman image.
Figure 8: (a) to (h) corresponds to directional decomposition of bands 1 to 8.
VI CONCLUSION
In this paper, the basic idea of directional filter banks are briefly reviewed. a new structure for directional filter banks is proposed based on the idea in [1] and [6]. This approach has a potential in any application where directional edge detection can solve given problems, also in other applications where rotational invariance is important such as common automatic target detection.
VII REFERENCE
[1] R. H. Bamberger and M. J. T. Smith, "A filter bank for the directional decomposition of images: theory and design," IEEE Trans. on Signal Processing, vol. 40, pp. 882-893, April 1992.
[2] A. K. Jain, Fundamentals of Digital Image Processing. Prentice-Hill, 1989.
[3] W. T. Freeman and E. H. Adelson, "The design and use of steerable filters," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, pp. 891-906, September 1991.
[4] P. P. Vaidyanathan, Multirate Systems and Filter Banks. Prentice-Hall, 1993
[5] R. H. Bamberger and M. J. T. Smith, "A filter bank for the directional decomposition of images: theory and design," IEEE Transactions on Signal Processing, vol. 40, no. 7, pp. 882 -893, Apr. 1992.
[6] J. Rosiles and M. J. Smith, "Texture classification with a biorthogonal directional filter bank," in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, 2001, pp. 1549 -1552.
[7] "Image denoising using directional filter banks," in Proceedings of IEEE International Conference on Image Processing, 2000, pp. 292-295.
[8] C. H. Park, J. J. Lee, M. J. Smith, S. il Park, and K.-H. Park, "Directional filter bank-based fingerprint feature extraction and matching," IEEE Transactions on Circuits and Systems for Video Technology, vol. 14, no. 1, pp. 74 - 85, Jan. 2004.
[9] S. I. Park, M. J. Smith, and R. M. Mersereau, "A new directional filter bank for image analysis and classification," in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, Mar. 2000, pp. 1417 -1420.
[10] P. Hong and M. J. Smith, "An octave-band family of non-redundant directional filter banks," in Proceedings. International Conference on Acoustics, Speech, and Signal Processing, Orlando, FL, May 2002, pp. 1165 -1168.
