IEEE TRANSACTION ON SIGNAL PROCESSING,VOL .1, NO.1,DECEMBER 2005
1
Multiresolution Direction Filter Banks: Theory, Design and Applications Truong T. Nguyen, Student Member, IEEE, and Sootorn Oraintara, Senior Member, IEEE
Abstract— This paper presents new developments of directional filter banks (DFB).The motivation for the paper is the existence of multiresolution and multidirection orthogonal transform for twodimensional discrete signals. Based on the frequency supports of the ideal transform, a new uniformly, maximally decimated DFB with six highpass directional subbands and two lowpass subands is introduced. The uniform DFB (uDFB) can be implemented by a binary tree structure of two-channel filter banks. The filter bank employed in the tree is shown to be alias-free decimation and permissible. The uDFB is then extended to a non-uniform case (nuDFB), which is still maximally decimated, by combining the two lowpass subbands. The nuDFB yields non-uniform frequency division which composes of one lowpass filter with a decimation factor of one-fourth and six highpass directional filters with a decimation factor of one-eighth. The new DFBs offer alternative image decompositions, which overcome the limited directional selectivity of the separable wavelets and the limited multiresolution of the conventional DFB. The lowpass subband of the nuDFB can be used to obtain a multiresolution representation by simply re-iterating the same nuDFB decomposition. On the other hand, the directional subbands can also be further refined by simply applying a two-channel conventional DFB at each highpass component. A simple design method yielding near orthogonal uniform and non-uniform multidimensional filter banks is presented. Finally, the performances of the newly proposed nuDFB are compared with other conventional transforms in nonlinear approximation, image denoising and texture classification to demonstrate its potential.
I. I NTRODUCTION 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. Manuscript received April, 2004; revised October, 2004. Truong T. Nguyen and Soontorn Oraintara are with the Electrical Engineering Department, University of Texas at Arlington, Arlington, TX 76011 USA (email:
[email protected],
[email protected]).
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 1(b), is the way the low frequency band is divided. 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 In [11], Do et al. proposed a pyramidal DFB (PDFB) in order to implement the contourlet transform, a discrete version of the curvelet transform [12]. 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
c 0000–0000/00$00.002005 IEEE
2
IEEE TRANSACTION ON SIGNAL PROCESSING,VOL .1, NO.1,DECEMBER 2005
HH1
LH1 HH2
H L
3
H H
HH2
HL2
3
HL1
Z1
H H
LH3
3
3
LH2
1
2
S , S 3 7
4
3
H L
HL2
0
HH2 H H
LH3
3
HL1
HH1
LH2 H H
Z2
S , S
Z2
5
6
6
5
HH2
7 HH1
Z1
LH1
4 3
HH1
S , S
2
1
A Note on notation. Uppercase bold face letters and lowercase bold face letters represent 2 × 2 square matrices and 2 × 1 column vectors, respectively. |M| is defined as the determinant of the matrix M. The superscripts T and −T denote the transpose and transpose of the inverse operators, respectively.
0
S , S
(a)
(b) Z2
1
S , S 3
2
7
1 5
0
6
4
Z2
6
4
2
S , S 3
7
Z1
5
Z1 6
0
6
0 5
7 3
2
S , S
5
7 3
1
2
1
S , S
(c)
C. 2D Multirate Systems, Quincunx and Parallelogram Filter Banks If a 2D signal x(n) is downsampled by a decimation matrix M, then the discrete Fourier transform of the downsampled signal y(n) = x(Mn) is [18] 1 Y (ω) = X(M−T ω − 2πM−T k), (1) |M| T k∈N (M )
T
(d)
Fig. 1. Frequency partition of the filter banks: (a) discrete wavelet transform (DWT), (b) conventional DFB, (c) proposed uniform DFB and (d) nonuniform DFB.
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 [13], which is 4/3. In recent work [14], 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. The paper is organized as follows. For the rest of this section related notations of 2D multirate signal processing and twochannel perfect reconstruction are briefly reviewed. Section II discusses the multiresolution and multidirection representation of 2D discrete signals. Different frequency division of the 2D frequency plane with critical sampling properties is also presented. Section III discusses the structure, properties and design issues of the proposed DFB. Limitations of the binary tree structure used in the construction of the maximally decimated DFBs in [14], [15] are discussed. In particular we will show that the sixteen-band DFB in [14] can be easily obtained by simply cascading the proposed eight-band DFB with simple two-channel FBs. An extension to nonuniform decomposition which allows for multiresolution decomposition is presented in Section IV. In Section V, a simple filter design algorithm adapted from [16] for both uniform and non-uniform cases is discussed. Several numerical experiments on image processing application are presented in Section VI to demonstrate the potential of the new DFBs, and Section VII concludes the paper. Preliminary versions of this work have been published in [15], [17].
where N (M ) is the set of all integer lattice points belong to the symmetric parallelepiped (SPD) generated by matrix MT , (SP D(MT )), defined as [18] : SP D(M) = {Mx, x ∈ [−1, 1)2 }. For the case of upsampling by M, the discrete Fourier transform of an upsampled signal x(M−1 n), if n = Mk, y(n) = (2) 0, otherwise, can be given by
Y (ω) = X(MT ω).
(3)
If x(n) is downsampled and then upsampled by the matrix M, from (1) and (3), the Fourier transform of the resulting signal y(n) is 1 Y (ω) = X(ω − 2πM−T k). (4) |M| T k∈N (M )
Two-channel multirate filter banks have been studied thoroughly [18]. In this paper, in the case of 2D filter banks, we classify them according to their sampling lattices. The followings are some special matrices that are used to decimate subband images in the paper: 1 1 2 0 1 0 Q1 = , D0 = , D1 = , −1 1 0 1 0 2 2 2 2 0 Q2 = , D2 = . −2 2 0 2 In the case of the quincunx matrix Q1 , we call the corresponding filter bank a quincunx filter bank (QFB). Note that the lowpass filters in QFB can have diamond-shape or fanshape support. The quincunx lattice is the only non-separable lattice in 2D two-channel filter banks. In multidimensional multirate systems, different subsampling matrices have the same lattice if they only differ by a unimodular matrix. In lattices other than quincunx, a signal is decimated by either row or column for the two-channel case. If the downsample matrix is diagonal (D0 , D1 ), in order to get perfect reconstruction, the shape of filter support must consist of one or
NGUYEN AND ORAINTARA: MULTIRESOLUTION DIRECTION FILTER BANKS: THEORY, DESIGN AND APPLICATIONS
3
several parallelograms. We refer to this type of filter banks as parallelogram filter banks (PFB). For general conditions on 2D PR filter banks and notations of 2D multirate signals used in this paper, see [18](chapter 12).
Let hj1 (n) be the compliment filters whose frequency response are described by √ 2 2, (ω1 , ω2 ) ∈ W1j , j (10) H1 (ω1 , ω2 ) = / W1j . 0, (ω1 , ω2 ) ∈
II. M ULTIRESOLUTION MULTIDIRECTION DIVISION OF 2D
Theorem 1: the filters hj1 (n), (j = 1, . . . , 6) are orthonormal filters with respect to the downsampling matrix Q2 , i.e.
FREQUENCY PLANE
< hj1 (n), hj1 (n − kQ2 ) > = δk , (j = 1, . . . , 6).
Let x(n) denote a 2D signal, then the discrete wavelet transform (DWT) of the signal can be defined as the outputs of a four-channel analysis filter bank: hij (k)x(D2 n − k), (5) xij (n) = k
<
hi1 (n), hj1 (n
Z2 3
4
where i, j ∈ {0, 1} and hij (n) are the impulse responses of the analysis filters.
− kQ2 ) > =
2
4
5
6
3
6
0, 0
W12
Z2
5
7
3
4
Z1
5
2
5
1
7
2
1
7
4 1
W
3
7
4
3
5
2
5
2
7
1
6
0, 0
Z1
6 1
S , S
(a)
W13
S , S
4
1
S , S W11
0, (i, j = 1, . . . , 6; i = j).(12)
S , S 7
(11)
(b)
Fig. 3. The directional filters and its aliasing frequency images shifted by T 1 2πQ−T 2 k, k ∈ N (Q2 ) tile the discrete frequency plane : (a) H1 and (b) H12 .
W15
V1
V1
W16
W1
Proof: We offer a geometrical proof. Note that ⏐ < hj1 (n), hj1 (n − kQ2 ) >= (hj1 (n) ∗ hj1 (−n))Q2 (k). (13)
W11
(a)
(b)
Fig. 2. Frequency division of subspaces generated by (a) wavelet filter banks and (b) directional filter banks.
We can view the signal x(n) as that it belongs to the space of 2D discrete functions l2 (n). The above DWT transform of x(n) is its projections onto spaces spanned by filter coefficients hij (n−D2 k). According to Figure 2(a), let us call the subspace spanned by h00 (n − D2 k) V1 and the subspace spanned by the other three wavelet filters (hij (n−D2 k), hij ∈ {h10 , h01 , h11 }) W1 , then their direct sum is l2 (n) = V0 = W1 ⊕ V1 .
(6)
In the literature, V1 is called the coarse approximation subspace and W1 is called the detail subspace. If the same filter bank is reiterated m times at the first filter h00 , a sequence of nested subspaces is generated as V0 = W1 ⊕ W2 ⊕ . . . ⊕ Vm ,
(7)
Vm ⊂ . . . ⊂ V1 ⊂ V0 .
(8)
with
Now we describe a multidirectional decomposition of the high frequency space W1 . From Figure 2(b), the space W1 can be decomposed into subspaces of functions, with frequency supports in regions W1j , j = 1, . . . , 6, as: W1 = W1j , (9) j=1,...,6
The Fourier transform of (hj1 (n) ∗ hj1 (−n)) is |H1j (ω1 , ω2 )|2 . Consider (hj1 (n) ∗ hj1 (−n)) downsampled and upsampled by the same matrix Q2 , the Fourier transform of resulting signal is the sum of |H1 (ω1 , ω2 )|2 and its shifted images at all aliasing frequencies, divided by the determinant of the matrix Q2 (4), which is 8. As shown in Figure 3(a), the main frequency support and its shifted images tile the whole frequency plane, i.e. ⏐ (14) F (hj1 (n) ∗ hj1 (−n))Q = 1, 2
which proves (11). Similarly, all other filters hj1 , j = 2, 3, . . . , 6 are also orthonormal with respect to Q2 . They are also orthogonal to each other because their frequency supports are not overlapping, so (12) is proven. The orthogonality of hi1 (n) stated in (11),(12) is equivalent to < hi1 (n), hj1 (n − kQ2 ) >= δk,i−j .
(15)
Let h1 (n) represent the impulse response of the ideal brickwall filter with frequency support in region V1 i.e. 2, (ω1 , ω2 ) ∈ V1 , H1 (ω1 , ω2 ) = (16) / V1 . 0, (ω1 , ω2 ) ∈ The impulse response h1 (n) is the 2D sinc function, and it is clear that h1 (n − kD2 ) and hi1 (n) are orthonormal. From the above result, it can be easily shown that if the filters H1 (ω1 , ω2 ) and H1j (ω1 , ω2 ), (j = 1, . . . , 6) are
4
IEEE TRANSACTION ON SIGNAL PROCESSING,VOL .1, NO.1,DECEMBER 2005
employed in analysis and synthesis size of a 2D filter bank with decimation matrices Q2 , D2 , respectively, then the implemented filter bank is perfect reconstruction. It will be shown later in section III that the detail subspace W1 can be divided up to 3.2m1 directional subspaces W1j . The coarse subspace V1 can also be further decomposed to a lower resolution coarse subspace V2 and directional subspaces W2j . Therefore we have shown that there exist a multiresolution and multidirectional decomposition of a 2D discrete function in the ideal case with brick-wall filters, i.e. ⎞ ⎛ (17) l2 (n) = V0 = V1 ⊕ ⎝ W1j ⎠
[19], filter banks with permissible supports allow us to design filters with high stopband attenuation.
j=1,...,6
⎛ = Vm ⊕ ⎝
m i=1
⎛ ⎝
n 3.2 i
j=1
PFB type I
H1
1
H 0
H 0
2
p
H 3
3
H1
3
D0
H 2
D1
1
p
D1
2
p
D1
3
p
D1
0
p
D0
4
p
D0
5
p
D0
7
p
D0
6
3
3
H 4
2
p
D1
PFB type II
H1 1
H 3
p
PFB type II
H 0
p Q1
H 5
3
H 6
3
H 7
3
2
III. T HE UNIFORM DIRECTIONAL FILTER BANK ( U DFB)
For notational convenience, let Hik (z) (Fik (z)) denote the i-th analysis (synthesis) filter at stage k (i = 0, . . . , 2k − 1). According to Figure 4, at the first stage, the image is decomposed into two directional subbands by a fan filter bank (H01 (z) and H11 (z)) with corresponding decimation matrix Q1 , and hence it is a QFB. Note that a fan filter bank typically can be obtained by a diamond-shaped filter bank modulated by π in either horizontal or vertical direction. The remaining stages of the tree employ three different PFBs. We denote the PFB in the second stage as type I, and in the last stage as types II and III. The corresponding upsampling and downsampling matrices of all three PFBs are either D0 or D1 . The supports of these filters satisfy two requirements of maximally decimated 2D filter banks, namely, alias free support (or admissibility) and permissibility [18], [19], [20] (Figure 6). As pointed out in
D0
p Q1
Wij ⎠⎠ .
In order to achieve the frequency division of the eightchannel filter bank in Figure 1(c), we use a tree-structure filter bank, which composes of three stages as shown in Figure 4. At each stage, the image is filtered and decimated by a twochannel PR filter bank. The PR property means that if there are no quantized errors in the filter bank, the reconstructed signal is the same as the input signal. Because the two-channel filter bank at each stage in the tree structure filter bank is PR, the entire filter bank is also PR. The downsampling matrices employed in the tree-structure are Q1 , D0 , and D1 . At the end of stage 3, eight subbands with (approximately) equal numbers of samples are obtained from the decomposition.
3
H 2 p
⎞⎞
The partitioning of the frequency plane in the conventional and the proposed DFB is illustrated in Figures 1(b) and (c), respectively. The motivation for the new frequency division lies in geometrical features of real world images. Typical images usually have smooth regions with edges and/or regions with textures. Directional highpass subbands should be able to extract directional information in an image, while lowpass subbands provide a coarse approximation.
PFB type III
2
p
D1
PFB type I
PFB type III
Fig. 4.
A binary tree structure of the proposed DFB.
Because of the symmetry between the upper and lower branches of the DFB, let us consider only the upper branches’ filters. At the end of stage 1, the frequency bands 0, 1, 2 and 3 (Figure 1(c)) fill the whole discrete frequency plane [−π, π)2 as in Figure 5(a). At stage 2, a two-channel filter bank (H02 (z) and H12 (z)) with supports as shown in Figure 4 separates the frequency bands 0 and 3 from bands 1 and 2. After being decimated by D0 , the frequency partitions of 0, 1, 2 and 3 subbands in two output channels are shown in Figures 5(b) and (c), respectively. The filter bank of two filters H02 (z) and H12 (z) is a PFB of type I. Figure 6(a) shows the frequency supports of a PFB of type I, which can be used to generate H02 (z) and H12 (z) by modulating by π in the vertical direction. The subbands 0 and 3 in Figure 5(b) are separated by H03 (z) and H13 (z), which are denoted as a PFB of type II, together with decimation matrix D1 . Supports of a type II PFB in Figure 6(b) can be used to generate H03 (z) and H13 (z) by horizontally modulating by π. Finally, H23 (z) and H33 (z) (see Figure 4) with D1 , denoted as a PFB of type III, are used to divide subbands 1 and 2 in Figure 5(c). The same discussion apply to stages 2 and 3 of the lower branches by simply rotating analysis and synthesis filters, and changing D0 and D1 . Some observations about the new DFB: •
•
Because of the cascaded structure, if the filters employed in each level are orthogonal, the whole eight-band decomposition is an orthogonal representation of the image. The corresponding standard form of the tree-structure filter bank described above has eight channels with frequency supports in Figure 1(c) and same decimation
NGUYEN AND ORAINTARA: MULTIRESOLUTION DIRECTION FILTER BANKS: THEORY, DESIGN AND APPLICATIONS
Z2 2
Z2 3
3
0
•
1
Z1
Z1
0
(a)
5
In the conventional DFB [5], the number of frequency wedges is fixed, and it is difficult to achieve a multiresolution decomposition since there is no specific lowpass subband that can be further divided for lower resolution, while maintaining the critical sampling. The uDFB, however, can be cascaded with two-channel filter banks with simple passband shape to obtain arbitrary (3×2n ) number of directional subbands at different resolutions.
(b) Z2
A. Alias free support of filter banks (admissibility) 2
1
The Alias free support (AFS) property with respect to a decimation matrix is a necessary property of a PR filter bank. Specifically, if the filters are ideal brickwall, it can be proven that the filter bank is PR if it satisfies the AFS property. The following theorem gives a condition on the filters passband supports that have no aliasing when being decimated by M.
Z1
(c)
Z2
Fig. 5. Frequency division in the upper channels of the proposed DFB tree: (a) the output at stage 1 (subbands 0-3), (b) the output at stage 2 (subbands 0 and 3) and (c) the output at stage 2 (subbands 1 and 2).
S , S
SPD(S M T )
Z1
S 2
S , S (a)
(b)
Fig. 7. Frequency support of subband 1 which satisfies the AFS condition, i.e. the shifted frequency on the 2πM −T grid fit in SPD(πM −T )
Theorem 2: A region of support S in [−π, π)2 is said AFS with respect to a matrix M if and only if the following condition is satisfied: For every point x ∈ SP D(πM−T ), there exists one and only one point y ∈ S and m ∈ Z 2 such that x − y = 2πM−T m. (c) Fig. 6. Frequency support of the two-channel 2D PFBs used in the tree structure of the proposed DFB: (a) type I, (b) type II and (c) type III.
•
•
matrix Q2 . All eight subband channels use the same decimation matrix Q2 , and are different fromthose ofthe conventional 4 0 2 0 DFB which use two matrices and . 0 2 0 4 The output subband signals of the new DFB are rotated by 45 degrees comparing to the input image. It is easy to see that, to obtain the sixteen-channel DFB presented in [14], one only needs to cascade the proposed eight-channel DFB by eight identical fan filter banks. Hence the currently proposed DFB is more flexible and is easier to design especially when the standard structure is used instead of the tree structure.
Proof: The theorem is generalized from the theorem on alias free decimation in [18] for filters whose supports are shifted copies of SPD(πM −T ). The AFS property of an ideal filter having support in region S with respect to decimation matrix M is satisfied when there is no overlapping between the main component of (1) and its aliasing images. Suppose there is one overlapping region between the main term and aliasing terms of (1). Let us define the region S0 = {MT ω|ω ∈ S} and region S1 = {MT ω +2πk|ω ∈ S, k = 0} that are overlapping. There exist ω0 , ω1 ∈ S such that MT ω0 = MT ω1 + 2πk
(18)
Because k = 0 ⇒ ω0 = ω1 . From the theorem, there are ˜ 1 ∈ SP D(πM−T ), corresponding to exist unique ω ˜ 0 and ω ˜0 ω0 and ω1 , respectively, such that ω0 = 2πM−T m0 + ω and ω1 = 2πM−T m1 + ω ˜ 1 . From the uniqueness of mapping
6
IEEE TRANSACTION ON SIGNAL PROCESSING,VOL .1, NO.1,DECEMBER 2005
from SP D(πM−T ) to S and ω0 = ω1 imply ω ˜ 0 = ω ˜ 1 . But continue from (18)
responses of the fan filters [26], [20]. The remaining PFBs of types I and III are designed using the transformation of variables method presented in [24], which is essentially the generalized McCllelan transform [27]. The advantage of this method is that it allows one to design a 2D PR filter bank with different sampling lattices and frequency supports. The design procedure for PFBs of types I and III is as follows:
MT (2πM−T m0 +˜ ω0 ) = MT (2πM−T m1 +˜ ω1 )+2πk. (19) In other words MT (˜ ω0 − ω ˜ 1 ) = 2π(k + m1 − m0 ).
(20)
−T
From definition of SP D(πM ) and the above equation, we have (k+m1 −m0 ) ∈ (−1, 1)2 . But all k, m1 , m0 are belong to the integer lattice, therefore (k + m1 − m0 ) = 0. It makes ˜ 1 , and leads to contradiction. ω ˜0 = ω From the theorem above, one simple test of the alias free decimation for a filter of any shape with respect to the matrix M can be stated as follows: If one can find a partition of the support shape of the filter such that there exist mk ∈ Z 2 , so that when the element kth of the partition being shifted by 2πM−T mk , then the shifted collection of the partition fit in SP D(πM−T ). Note that mk can be different for each element of the partition. Figure 7 shows an example of the test to show that the directional subband 1 in the proposed DFB satisfying the AFS condition. B. Permissibility of filter banks The permissibility condition of a filter bank is a necessary condition for PR filter banks to have good frequency characteristics for non-ideal brickwall filters [21]. In practice, the filters are approximated and not ideal, but the system can be PR with good frequency characteristics if it satisfies this permissibility property. This condition was further discussed in details for the case of 2D cosine modulated filter banks in [22], [23]. Indeed, it can be shown that a two-channel filter bank satisfies the permissibility property if it is AFS because there is only one image frequency in each subband. Supposed the synthesis filters have the same support as the corresponding analysis ones, which frequency selective PR filter banks always do, the aliasing frequencies are at the border of the frequency support. These aliasing always come in pairs in a two-channel FB, and it is possible to cancel each other. However, the permissibility property of two-channel filter banks employed in the tree structure does not imply that the overall filter bank is permissible. C. Filter design for the tree structure As discussed above, the key to design and implement the frequency division in Figure 1(c) is to use appropriate PFBs in stages 2 and 3 of the tree structure in Figure 4. In case of ideal filters, it is a critically sample, multiresolution, and directional representation of images. The challenge is how to approximate these filter banks and implement them efficiently. There are many methods to design PR QFB [5], [24], [25], which will not be discussed in this paper. The PFB of type II with frequency partition presented in Figure 6(b) is easily obtained by simple modulation and resample the impulse
1) Design the transformation kernel M (z1 , z2 ) that satisfies M (z1 , z2 ) = −M (−z1 , z2 ) for PFB of type I,M (z1 , z2 ) = −M (z1 , −z2 ) for PFB of type III, and ⎧ if (ω1 , ω2 ) is in the support ⎨ 1, in Figures 6(a) and (c), and M (ejω1 , ejω2 ) ≈ ⎩ −1, otherwise. At this step, a simple choice would be taking the inverse Fourier transform of an ideal filter, and then truncating the impulse response by a certain 2D window. 2) Use a 1D prototype Lagrange halfband filter to factorize into two symmetric filters, and apply the generalized McClellan transform with kernel M (z1 , z2 ) obtained from the previous step. This results in the analysis and synthesis filters of one of the two channels, say H0 (z1 , z2 ) and F0 (z1 , z2 ). The other channel can be achieved by following the aliasing cancellation condition [26]. For example, for PFB of type I where the decimation matrix is D0 , the other two filters can be obtained by [26]: H1 (z1 , z2 ) = F0 (−z1 , z2 ), and F1 (z1 , z2 ) = −H0 (−z1 , z2 ). One notes that, although this variable transformation method is simple and flexible, it does not allow imposition of some important filter characteristics such as regularity [28], which is very important for image processing. Furthermore, with this class of filter banks, orthogonality of the basis functions is not possible. D. Filter banks with no DC leakage in the directional subbands One important and desirable property of filter banks for the purpose of image processing is that all highpass filters have zero frequency response at (ω1 , ω2 ) = (0, 0) to ensure that there is no DC energy leaking into these subbands (1, 2 and 3). According to Figure 5(a), it can be seen that, in order to prevent the DC component from leaking into subbands 1 and 2, H12 (1, 1) must be zero. Similarly, from Figure 5(b), if there is no DC leakage into subband 3, H13 (1, 1) must also be zero. These two conditions pose a linear constraint that the kernel M (z1 , z2 ) in the design of the above PFBs of type I and II must satisfy. Specifically, for these two types, M (1, 1) = −1. Again, these constraints must be applied to the lower branches of the tree for the design of subbands 4-7. Upon the success of this design, only subbands 0 and 4 contain the DC energy after decomposition.
NGUYEN AND ORAINTARA: MULTIRESOLUTION DIRECTION FILTER BANKS: THEORY, DESIGN AND APPLICATIONS
1
H 0 z p Q1
Fig. 8.
H1
2
z
One branch of the tree structure to implement DFB.
One design problem with the binary tree is the effect of frequency ‘scrambling’ [4], which means that the aliased high frequency component is next to the lowpass component after decimation. Such a problem occurs in the DFB in [5], and is corrected by using a resampling block in [9]. A similar problem appears in the implementation of the tree-structured DFB in [14], [15]. In order to illustrate this problem, let us consider the top branch in the second level of the tree structure in Fig. 4 as in Fig. 8. Using a noble identity [18], it is clear that the equivalent decimation filter is given by: H(z)
(1)
(2)
= H0 (z)H1 (zQ ),
H(ω1 , ω2 ) =
(1) (2) H0 (ω1 , ω2 )H1 (ω1
7
shown in Figure 9(b). The other six directional subbands remain the same. The frequency partitioning of the nuDFB is presented in Figure 1(d). The nuDFB can also be seen as the implementation of the multiresolution and multidirection signal representation discussed in section II. This nuDFB has a number of advantages. It is similar to the DWT in Figure 1(a) in that they have the same lowpass frequency support (−π/2, π/2)2 . Instead of decomposing the image details into horizontal, vertical and diagonal subbands, the nuDFB uses six directional subbands. It provides more flexibility in the decomposition: (i) in order to obtain a (directional) multiresolution of an image, one can reiterate the entire filter bank on the lowpass channel, and (ii) in order to double the directional resolution, one can cascade a two-channel filter bank with fan-shaped passband at the output of each of the highpass channels.
V. A DESIGN METHOD FOR THE U DFB AND NU DFB − ω2 , ω1 + ω2 ).(21)
If this branch is a highpass filter with one zero at DC (one (2) vanishing moment), it is required that H1 (0, 0) = 0 since (1) (1) H0 (0, 0) is in the transition band of H0 (ω1 , ω2 ). This implies that H(±π, ±π) = 0, which severely affect the passband shape of the highpass filter. Furthermore, some high frequency components are leaking into the lowpass channel after decomposition. In order to avoid the above problem, the uDFB is proposed to be optimized directly as a uniform eight-channel filter bank as shown in Figure 9(a). This framework has a number of advantages: (i) The filter banks can be designed with high degrees of regularity, and (ii) the size of the impulse response can be more compact than that from the binary tree structure since the entire eight channel filters can be implemented directly. IV. M ULTIRESOLUTION DIRECTIONAL DECOMPOSITION USING NONUNIFORM DFB( NU DFB) A natural extension to a multiresolution image decomposition using a filter bank is to reiterate the same filter bank on the lowpass coefficients (coarse approximation). For a 1D octave-band multiresolution, the passband support (bandwidth) of the lowpass filter is one-half of the entire frequency space, and hence is one-fourth for the 2D case. The above uDFB has two low frequency subbands (0 and 4), which should be combined to obtain a decimation factor of one-fourth. One simple solution is to use a two-channel PR filter bank (transmultiplexer) to combine the two lowpass signals. This approach increases computational complexity of the design and implementation since additional filters are used. Moreover, the sizes of the overall support of the filter’s impulse responses are less compact. A better approach is to use a nonuniform filter bank with one lowpass component with the decimation matrix D2 of which the decimation factor is one-fourth as
Designing non-separable multidimensional filter banks is a challenging problem. A common approach is transformation of variable [27], [24], by which the 2D filter bank is designed based on a 1D PR filter bank. The method is difficult to apply in the case of more than two channels. Up to now, there is no systematic method to design 2D PR filter banks with constraints in regularity and frequency characteristic (stopband attenuation, passband ripples, etc). In this paper we use an optimization design method to design filters in the filter banks without using the tree structure. Unlike other filter bank optimization design methods [29], the reconstruction error of this method is calculated in time domain, which is a bilinear function of the analysis and synthesis filter coefficients. The design procedure then tries to reduce the reconstruction error at each step. The method allows one to design a near PR orthogonal or biorthogonal filter bank only by solving linear equations iteratively, and it is flexible enough to design filter banks for both uDFB and nuDFB cases. The original approach is introduced in [30] for 1D PR filter banks design, and the formulation for 2D filter banks is presented in [16]. For notational convenience, let S = {1, 2, 3, 5, 6, 7} be the set of the indices of the directional subbands in nuDFB case, and M = 8 is the number of subbands in uDFB case. The design problem can be formulated as an optimization problem with an objective function
Φ =
M −1
fj (−n)hj (n) − 1
n
j=0
+
2
M −1
j=0 m=0
+
M −1
j,k=0,j=k m
n
n
2 fj (mQ2 − n)hj (n) 2
fj (mQ2 − n)hk (n)
, or (22)
8
IEEE TRANSACTION ON SIGNAL PROCESSING,VOL .1, NO.1,DECEMBER 2005
H
1
(z )
H
2
( z)
H
3
(z )
H
0
( z)
H
4
( z)
H
5
(z )
H
6
( z)
H
7
(z)
x ( n0 , n1 )
↓
Q2
1
↓
Q2
2
↓
Q2
3
↓
Q2
0
↓
Q2
4
↓
Q2
5
↓
Q2
6
↓
Q2
7
0
( z)
H
1
(z )
H
2
( z)
H
3
(z )
H
5
(z )
H
6
( z)
H
7
(z)
x ( n0 , n1 )
(a) Fig. 9.
H
↓
D2
↓
Q2
↓
Q2
↓
Q2
↓
Q2
↓
Q2
↓
Q2
(b)
(a) Uniform eight channels DFB and (b) non-uniform seven channels DFB.
Φ =
j∈S
+
2
n
j∈S m=0
+
2 fj (mQ2 − n)hj (n)
n
j,k∈S,j=k m
+
n
j∈S m
In addition, for the case when the analysis and synthesis filters have equal degree of regularity, i.e. the number of zeros at DC of the highpass filters (vanishing moments), can be imposed with the following linear constraints [28], [31]:
fj (−n)hj (n) − 1
2 fj (mQ2 − n)hk (n) 2
fj (mD2 − n)h0 (n)
.
(23)
∂ r1 +r2 fj (z1 , z2 ) =0 ∂z r1 ∂z2 r2 z1 ,z2 =0 r1 +r2 1 ∂ hj (z1 , z2 ) =0 , ∂z1 r1 ∂z2 r2 z1 ,z2 =0
(25)
n
where (22) and (23) are, respectively, for the cases of uDFB and nuDFB. The objective function mainly consists of two parts constituting the PR or biorthogonal property. The first term corresponds to the normalization condition, and it attains zero only when inner product of the two filters < fj (−n), hj (n) > is equal to 1. The rest correspond to the orthogonal conditions, which means that all shifted copies of the synthesis filters on the subsampling lattice are orthogonal to the analysis filters. It is clear from (22) and (23) that when hj (fj ) are fixed, Φ is a quadratic function of fj (hj ), which can be minimized by solving a system of linear equations ∂Φ ∂Φ =0 =0 . (24) ∂fj ∂hj
with r1 , r2 being non-negative integers and r1 + r2 < R. The above conditions in (25) serve as extra constraints in the optimization, and can be simultaneously solved with (24) using Lagrange multiplier. The design procedure can be carried out by iteratively solving (25) and (24) while hj (fj ) are fixed. The obtained fj (hj ) are then fixed and used to optimize for hj (fj ). During the iteration, orthogonality of the filter bank can be obtained by imposing that each synthesis filter is the time reverse version of the corresponding analysis filter, i.e. fj (n) = hj (−n). In order for the filters to converge to an orthogonal solution, the average of the analysis and synthesis
NGUYEN AND ORAINTARA: MULTIRESOLUTION DIRECTION FILTER BANKS: THEORY, DESIGN AND APPLICATIONS
filters is used to update the calculated filter coefficients, i.e. (n−1)
(n−1)
(−n) + fj (n) = 2 (n−1) (n−1) (−n) + hj (n) fj (n) hj (n) = . 2 hj
(n) fj (n)
9
VI. N UMERICAL EXPERIMENTS A. Nonlinear approximation
Z2
(26)
S , S
Z1
The design procedure can be summarized as follows: 0 (0)
1) Initialize the analysis filters hj with appropriate passband supports. (0) 2) Solve for the synthesis filters fˆj by solving the above (0) linear equations (25) and (24). Update fj by setting (0)
(0)
h
(−n)+fˆ
(0)
(n)
j . fj (n) = j 2 (0) (1) ˆ 3) Fix fj and solve for hj using the linear equa(1) (1) tions (25) and (24). Update hj by setting hj (n) =
ˆ (1) (n)+f (0) (−n) h j j . 2
(i)
(i)
4) Repeat the same above process for fj and hj the cost function is lower than a defined level.
S , S (a)
until
(b) Fig. 11. Example of the Barbara image decomposed into 3 levels using DWT at the lower levels and nuDFB at the two higher levels: (a) frequency support of each subband at three levels, (b) corresponding output subband coefficients . Fig. 10. Frequency responses of a design example of orthogonal uDFB. Size of each filter is 15 × 15, SNR of typical reconstructed images are about 80 dB.
Remarks: Although the design method is simple and straightforward, however it has some limitations. First, the method is not robust and the convergent rate is strongly dependent on the initialization filters. Second, the cost functions do not impose any frequency response characteristics (except from the regularity conditions) and hence the resulting filters are also dependent on the initialization filters. Finally, the algorithm is not guaranteed to converge to a PR solution at the optimality, i.e. Φ = 0. These problems are in contrast with 1D filter bank design [30], and it is due to the number of constraints on filter coefficients by the biorthogonal condition being much higher in 2D case.
In the first experiment, approximation performances of the DWT and the nuDFB are compared. The Barbara image of size 512 × 512 is decomposed in to 3 levels as in Figure 11. The same image is also decomposed by the DWT using the ‘9-7’ wavelet filters. The image is then reconstructed using the same number of highest magnitude coefficients. The expectation is that the high magnitude coefficients in the directional subbands of the nuDFB filter banks are corresponding to geometrical features of images (edges and textures). Therefore it will quickly approximate the image with fewer large coefficients than other type of filter banks. The reconstructed Barbara images using 6144 coefficients (or 2.3 percent of total number of coefficients) of DWT and nuDFB are compared in Figure 12. The nuDFB yields slightly higher signal-to-noise ratio (SNR) than that using the DWT. Moreover, it can be seen that the reconstruction from the nuDFB has better perceptual quality in
10
IEEE TRANSACTION ON SIGNAL PROCESSING,VOL .1, NO.1,DECEMBER 2005
(a)
(b)
(c)
Fig. 12. Nonlinear approximation of the Barbara image reconstructed from 6144 coefficients with three levels of decomposition: (a) original image, (b) using DWT, SNR = 18.28dB and (c) using nuDFB, SNR = 18.40dB.
the sense that it preserves edges (strips) while aliasing appears in the reconstruction using DWT.
B. Image denoising The approximation power of the new image basis is also verified in the denoising application. As shown in the previous experiment, high coefficients in highpass subbands corresponding to prominent image features. When denoising image by thresholding coefficients, the noisy coefficients will be set to zero, whereas high coefficients will be kept. Because the nuDFB decomposition is better suited to represent image edges and textures, it is expected to offer better denoising image performance, compared to other non-geometrical basis, such as wavelet. For the DWT type filter banks, the ‘9-7’ wavelet filters are used in the experiment. The threshold is set to be the same for both DWT and nuDFB, which is three times of the noise variance. For comparison, we also denoise the image using the contourlet transform (PDFB) with eight directional subbands, which is not a maximally decimated (redundant) transform. Because the PDFB is not orthogonal, the thresholds are selected through a simulation process to estimate the variance of each subband coefficients when input is white noise. According to the results in Figure 13, the nuDFB yields best reconstruction in term of SNR over the DWT and PDFB. The quality of the reconstructed image by the nuDFB is also better than that by the PDFB filter bank, which has clear directional errors in smooth regions. These errors are due to the tree structure of the PDFB.
TABLE I B RODATZ T EXTURE USED IN THE EXPERIMENT
D1 D2 D4 D8 D11
D18 D19 D20 D21 D35
Texture ID D39 D53 D41 D56 D46 D57 D47 D77 D51 D78
D80 D83 D94 D96 D104
D105 D111
variances (energy) of subsampled signals at subband outputs are used to form a vector. For each training sample i (i = 1, . . . , 60) of texture j (j = 1, . . . , 27), a feature vector is formed: (27) fi,j = (v1 , . . . , v6 ) where vj is the energy of the subband output yj of size, i.e. M ×N 1 2 vj = |yj (m, n)| (28) M × N m,n After obtaining the training feature vectors for each texture, a standard method is used to calculate the statistical model. The feature vector is assumed to be a random variable of a multivariate Gaussian distribution, characterized by a mean vector and a covariance matrix (mk , Ck ). The distance of the testing sample feature vector fj to each texture model is measured by the Bayes distance with equal prior: dk (fj ) = (fj − mk )T C−1 k (fj − mk ) + log(det(Ck )), (29) and the sample is classified as the texture with the minimum distance (30) sj = arg min dk (fj ). k
C. Texture classification The classification performance of the nuDFB with a class of textures with strong directional information is tested in the last experiment. Twenty seven texture images of size 640×640 are chosen from the Brodatz album [32] as listed in Table I. For each image, sixty training and ten testing samples of size 256×256 from two different regions are obtained. The
The above process is run on every sample of testing textures and the average is taken to get the rate of correct classification. The same simulation is also run on a reduced feature vector to get the correct classification rate as a function of number of features. For comparison, the same experiment is performed on the conventional DFB and DWT. For DWT simulation, the ‘9-
NGUYEN AND ORAINTARA: MULTIRESOLUTION DIRECTION FILTER BANKS: THEORY, DESIGN AND APPLICATIONS
Original Image
Noisy Image (SNR = 17.99 dB)
Denoise using Wavelets (SNR = 24.03 dB)
Denoise using Contourlets (SNR = 24.33 dB)
11
Denoise using nuDFB (SNR = 24.42 dB)
Fig. 13. Denoising peppers image, threshold is 3ρ of noise variance, (a) original image, (b) noisy image and images denoised by (c) wavelet, (d) contourlet, (e) nuDFB
100
by nuDFB has consistently higher classification rate than the other two filter banks with any number of features. The correct classification rates are 46%, 80.8%, 94.3% with one, two and three features. The corresponding classification rates for the wavelet-type filter bank are 39.2%, 72.5%, 87.3%. The results confirm that nuDFB offers a good characterization of textures with directional information.
90
Correct Classification rate
80
70
60
VII. C ONCLUSION 50
40 nuDFB DFB Wavelet 30
1
2
3 4 Number of features
5
6
Fig. 14. Classification rate: Comparison of nuDFB, DFB, and DWT on the set of texture images in Table I
7’ wavelet filters is selected, and the quincunx filter bank used in the conventional DFB is the McClellan transform of the ‘9-7’ filter bank. In order to get six highpass subbands in DWT, a fan filter bank is applied to the three highpass subbands of DWT. A comparison of the correct classification rates is in Figure 14(a). The results show that the classification
Two new types of DFB, which we called uDFB and nuDFB, were introduced and related design and implementation issues were discussed. The two DFBs can be used as a multidirection multiresolution decomposition of 2D signal. The difficulties arise in designing tree-structure DFB with good frequency characteristic and regularity constraints, leading us to propose a direct design of DFBs. This approach allows us to design good directional filters in highpass bands, at the requirement of higher computational cost. The design method gives the first multiresolution and multidirectional 2D PR filter bank while keeping the maximally decimated property. The directional resolution can be increased easily using simple twochannel filter banks. A simple design technique yielding near orthogonal DFB is included, and the resulting filter designs are tested in several image processing applications. The numerical
12
IEEE TRANSACTION ON SIGNAL PROCESSING,VOL .1, NO.1,DECEMBER 2005
experiments in this work show the great potential of these new types of DFBs.
[23] H. Yan and M. Ikehara, “Two-dimensional perfect reconstruction fir filter banks with triangular supports,” IEEE Transaction on Signal Processing, vol. 47, no. 4, pp. 1003–1009, Apr 1999. [24] D.B.H.Tay and N. G. Kingsbury, “Flexible design of multidimensional perfect reconstruction fir 2-band filters using transformations of variables,” IEEE Transaction on Image Processing, vol. 2, no. 4, pp. 466– 480, Oct 1993. [25] S.-M. Phoong, C. Kim, P. Vaidyanathan, and R. Ansari, “A new class of two-channel biorthogonal filter banks and wavelet bases,” IEEE Transactions on Signal Processing, vol. 43, no. 3, pp. 649–665, Mar. 1995. [26] Y. P. Lin and P. P.Vaidyanathan, “Theory and design of two-dimensional filter banks: a review,” Multidimensional Systems and Signal Processing, vol. 7, pp. 263–330, 1996. [27] J. S.Lim, Two-Dimensional Signal and Image Processing. Englewood Cliffs, NJ: Prentice Hall, 1990. [28] M. Vetterli and J. Kovacevic, Wavelets and Subband Coding. Englewood Cliffs, NJ: Prentice Hall, 1995. [29] W.-S. Lu, A. Antoniou, and H. Xu, “A direct method for the design of 2-d nonseparable diamond-shaped filter banks,” IEEE Transaction on Circuits and Systems II, vol. 45, no. 8, pp. 1146–1150, Aug 1998. [30] K. Nayebi, T. P. Barnwell, and M. J. Smith, “Time-domain filter bank analysis: a new design theory,” IEEE Transactions on Signal Processing, vol. 40, no. 6, pp. 1412 – 1429, June 1992. [31] S. Oraintara, T. D. Tran, P. N. Heller, and T. Q. Nguyen, “Lattice structure for regular paraunitary linear-phase filterbanks and m-band orthogonal symmetric wavelets,” IEEE Transactions on Signal Processing, vol. 49, no. 11, pp. 2659 – 2672, 2001. [32] P. Brodatz, A Photographic Album for Artists and Designers. Newyork: Dover, 1966.
R EFERENCES [1] G. Strang and T. Q. Nguyen, Wavelets and Filter Banks. Wellesley Cambridge, 1996. [2] E. J. Cand`es and D. L. Donoho, “Ridgelets: a key to higher-dimensional intermittency?” Phil. Trans. R. Soc. Lond. A., pp. 2495–2509, 1999. [3] M. Vetterli, “Wavelets, approximation and compression,” IEEE Signal Processing Magazine, vol. 18, no. 5, pp. 59–73, 2001. [4] M. N. Do, “Directional multiresolution image representations,” Ph.D. dissertation, Swiss Federal Institute of Technology, Lausanne, Nov. 2001. [Online]. Available: http://www.ifp.uiuc.edu/ minhdo/publications/thesis.pdf [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. [11] M. N. Do and M. Vetterli, “Contourlets,” in Beyond Wavelets, G. V. Welland, Ed. Amsterdam, The Netherlands: Academic Press, 2003, ch. 4, pp. 83–105. [12] E. J. Cand`es and D. L. Donoho, “Curvelets - a surprisingly effective nonadaptive representation for objects with edges,” in Curves and Surfaces, L. L. S. et al., Ed. Nashville, TN: Vanderbilt University Press, 1999. [13] P. J. Burt and E. H. Adelson, “The laplacian pyramid as a compact image code,” IEEE Transaction on Communication, vol. 31, no. 4, pp. 532–540, Apr. 1983. [14] Y. Lu and M. N. Do, “Crisp-contourlets: a critically sampled directional multiresolution image representation,” in Proc. of SPIE conference on Wavelet Applications in Signal and Image Processing X, San Diego, USA, Aug. 2003. [15] T. T. Nguyen and S. Oraintara, “A directional decomposition: theory, design and implementation,” in International Symposium on Circuits and Systems, Vancouver, Canada, May 2004. [16] M.Ikehara and T. Q. Nguyen, “On 2d perfect reconstruction linear phase filter banks,” in Thirty-First Asilomar Conference on Signals, Systems and Computers, 1997, pp. 721–725. [17] T. T. Nguyen and S. Oraintara, “A multiresolution direction filter banks for image applications,” in International Conference on Acoustics, Speech, and Signal Processing, Montreal, Canada, May 2004. [18] P. Vaidyanathan, Multirate Systems and Filter Banks. PrenticeHall,Englewood Cliffs, NJ, 1993. [19] T. Chen and P. Vaidyanathan, “Considerations in multidimensional filter bank design,” in Proceedings of IEEE International Symposium on Circuits and Systems, May 1993, pp. 643–645. [20] S. Oraintara and T. Q. Nguyen, “Multidimensional 2-channel filter banks,” in The Thirty-First Asilomar Conference on Signals, Systems and Computers, vol. 2, Nov. 1997, pp. 1269–1273. [21] T. Chen and P. Vaidyanathan, “Recent developments in multidimensional multirate systems,” IEEE Transactions on Circuits and Systems for Video Technology, vol. 3, no. 2, pp. 116 –137, Apr. 1993. [22] Y. P. Lin and P. P.Vaidyanathan, “Theory and design of twoparallelogram filter banks,” IEEE Transaction on Signal Processing, vol. 44, no. 11, pp. 2688–2705, Nov 1996.
Truong T. Nguyen (S’03) was born in Haiphong, Vietnam. He received the B.Eng. and the M.Eng. degrees in Electrical Engneering from Hanoi University of Technology, Hanoi, Vietnam, in 1995 and 2001, respectively. He has worked at Alcatel from 1995 to 2002. He is currently working toward the Ph.D. degree in the Department of Electrical Engineering at University of Texas at Arlington. His research interest is in the area of multirate signal processing.
Soontorn Oraintara (S’97-M’00-SM’04) received the B.E. degree (with first-class honors) from the King Monkuts Institute of Technology Ladkrabang, Bangkok, Thailand, in 1995 and the M.S. and Ph.D. PLACE degrees, both in electrical engineering, respectively, PHOTO from the University of Wisconsin, Madison, in 1996 HERE and Boston University, Boston, MA, in 2000. He joined the Department of Electrical Engineering, University of Texas at Arlington (UTA), as Assistant Professor in July 2000. From May 1998 to April 2000, he was an intern and a consultant at the Advanced Research and Development Group, Ericsson Inc., Research Triangle Park, NC. His current research interests are in the field of digital signal processing: wavelets, filterbanks, and multirate systems and their applications in data compression, signal detection and estimation, communications, image recontruction, and regularization and noise reduction. Dr. Oraintara received the Technology Award from Boston University for his invention on Integer DCT (with Y. J. Chen and T. Q. Nguyen) in 1999. In 2003, he received the College of Engineering Outstanding Young Faculty Member Award from UTA. He represented Thailand in the International Mathematical Olympiad competitions and, respectively, received the Honorable Mention Award in Beijing, China, in 1989 and the bronze medal in Sigtuna, Sweden, in 1990.
