ACCEPTED FOR PUBLICATION IN IEEE TRANSACTION ON SIGNAL PROCESSING, OCT. 2007

1

The Shiftable Complex Directional Pyramid, Part I: Theoretical Aspects Truong T. Nguyen and Soontorn Oraintara Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX 76019–0016, Phone: 817-272-3482, Fax: 817-272-2253 Email: [email protected], [email protected]

Abstract— It is well-known that the discrete wavelet transform has limitations that make it difficult to apply in image analysis applications: shift sensitivity and poor directional selectivity. This paper presents a complex directional wavelet transform that is energy shift invariant (shiftable) and can provide an image decomposition with arbitrarily high directional resolution at a fixed redundant ratio. The image decomposition is implemented by a filter bank (FB) consists of a multiresolution FB and a pair of directional filter banks (DFB). The two DFBs (primal and dual) are designed in such a way that each pair of corresponding directional filters produced by the primal and dual filter banks has the Hilbert transform phase relation. Therefore, the two subband coefficients can be viewed as the real and imaginary parts of a complex-valued subband image. It is proven that there is no aliasing in the decimated complex-valued signal, which implies that the system is shift-invariant in the energy sense. Furthermore, the two-dimensional FB can be implemented in a separable fashion, which makes the entire structure very computational efficient. Index Terms— Complex wavelet, Dual-tree discrete wavelet transform, Directional filter bank, Directional decomposition, Multidimensional filter bank, Multiresolution representation, Multiresolution DFB, Wavelet

A. Lack of Shift-invariance in Maximally Decimated Filter Banks One of the problems of the DWT is that it is shift-variant, which means that a representation of a signal by wavelet coefficients is highly dependent on its relative position with respect to the downsampling lattice. This problem is the main difficulty when applying DWT in pattern recognition [3]. It is also undesirable in other image processing applications since different results are obtained once the image is shifted. A system with a downsampling operation cannot have the shift-invariant property in the normal sense, i.e. a translation of the input leads to the same translation of the output. However, a reduced form of translation invariance exists, namely, energy shift-invariance or ‘shiftability’ [4], which means that the energy of the output signal is shift invariant. This will happen if aliasing in the decimated signal is negligible. This condition implies that the frequency spectrum of the signal before being decimated is strictly bandlimited inside a region of less than the Nyquist frequency associated with the downsampling ratio. P1 Z

I. I NTRODUCTION

S

Wavelet and Filter Banks (FB) have been a major research topic in signal processing for the last two decades [1]. The discrete wavelet transform (DWT) has been shown to be an optimal representation of one-dimensional (1-D) piecewise smooth signals [2], and found widespread use in many signal and image processing applications. However, there are limitations of the separable DWT that have been pointed out by many researchers [3], [4], [5], [6], namely lack of shift invariance and directionality. In this work we present a shiftable complex directional pyramid transform implemented by a special FB that we called the dual-tree pyramidal directional filter banks (DFB). The first major contribution of the paper is the formulation of the phase condition on a pair of DFB so that directional filters have the Hilbert transform phase relation. The second contribution is the construction of the FB implementing the proposed shiftable transform. We also proved that the transform has many good properties to be used in image analysis, such as shiftable subband, arbitrarily high directionality and low redundancy. The filter design problem and applications of the novel image transform are addressed in a forthcoming paper [7].

P2 Z

P2b Z

P1a Z

P1b Z

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0

3

2S 3

G  n

S

S

p1  n

3

S 3

2S 3

S

q2  n 2

Q1 Z

S

P2a Z 0

G  n

q1  n 1



S

p2  n Q2 Z

S

0

3

(a)

2S 3

S

S

0

S 3

2S 3

S

(b)

Fig. 1. (a) A shift-variant multirate system, and (b) A shift-invariant system in the energy sense.

Consider a filter H1 (z), of which frequency support does not satisfy the Nyquist property when decimated by three, i.e. its bandwidth is greater than 2π/3. Suppose that the input signal is δ(n), and the filter H1 (z) has two frequency components: H1 (ω) = P1a (ω) + P1b (ω) as in Fig. 1(a). Since the decimation ratio is three, the Fourier transform of the subsampled signal q1 (n) is   2 1X ω − 2kπ Q1 (ω) = H1 3 3 k=0      2 1X ω − 2kπ ω − 2kπ = P1a + P1b (1) . 3 3 c 0000–0000/00$00.00 2007 IEEE 3 k=0

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ACCEPTED FOR PUBLICATION IN IEEE TRANSACTION ON SIGNAL PROCESSING, OCT. 2007

By the Parseval’s theorem, the energy of q1 (n) can be given by n=∞ X

Z π 1 |Q1 (ω)|2 dω |q1 (n)| = 2π −π n=−∞     2 Z π 1 ω − 2kπ ω − 2kπ = P1a + P1b dω 18π −π 3 3   2   2 Z π 1 P1a ω − 2kπ + P1b ω − 2kπ + = 18π −π 3 3   ω − 2π ∗ ω + 2π )P1b ( ) dω (2) +2Real P1a ( 3 3 2

In the calculation of the energy of q1 (n) above, the cross term between P1a and
P1b is proportional to Rπ ∗ ω+2π Real P1a ( ω−2π 3 )P1b ( 3 ) dω. If the input signal −π is delayed by one sample, then p1 (n) is also delayed by one sample.  j4π The cross term will be proportional to Rπ ∗ ω+2π Real e− 3 P1a ( ω−2π 3 )P1b ( 3 ) dω, which is differ−π ent from the previous quantity, resulting in a change in the output energy. On the contrary, the filter H2 (z) in Fig. 1(b) has supports restricted in the (±π/3, ±2π/3) regions, and thus there is no aliasing by three. R π when p2 (n) is decimated
Therefore, the cross ∗ ω+2π term −π Real P2a ( ω−2π 3 )P2b ( 3 ) dω becomes zero, and the system is energy shift-invariant. Hence, in order to obtain an energy shift-invariant FB, the passband and transition band of all the filters must be limited within regions satisfying the Nyquist property with respect to the decimation ratio. B. Limited Angular Resolution in Two-dimensional Separable Filter Banks A separable subband decomposition of a two-dimensional (2-D) signal is a direct extension from the 1-D case where a signal is convolved by 1-D filters and subsampled in rows and columns. For a two-channel FB such as that used in the DWT, the corresponding 2-D FB produces four sub-images, which are usually referred to as LL, LH, HL and HH images [1]. The LH and HL images contain features along the horizontal and vertical directions, but the HH image contains diagonal components of both directions. If the directional selectivity of a FB is defined as the ability to extract orientational features into separate images, then the 2-D DWT has very poor directional selectivity. Moreover, the directional resolution of the 2-D DWT can not be increased by having more highpass subbands because every separable 2-D filter must have passband in all four quadrants of the 2-D frequency plane, symmetric with respect to the ω1 and ω2 axes.

are perfectly shift-invariant. On the other hand, a natural approach to increase directionality of the representation is to use nonseparable directional filters. The steerable pyramid [4] yields shift invariance and high directionality by combining these two approaches. The double-density FB in [8] is similar to the dyadic FB for DWT. The only difference is that it has two highpass filters instead of one. Therefore it is an overcomplete FB and can be designed to obtain energy shiftinvariance or shiftable subbands by requiring that each filter satisfies the Nyquist condition. Although the discrete curvelet transform in [9] is proposed in the frequency domain, it can be viewed as a shift-invariant, multiscale, directional transform obtained by oversampling and filtering by directional filters. It should be noted that the dual-tree DWT provides both shiftable subbands and good directional selectivity in a different way [5], [10]. In 1-D dual-tree DWT, each and every subband signal of the FB is maximally decimated and not shift-invariant by itself. However, when one considers a pair of subband outputs at the same branches in the primal and dual trees as the real and imaginary parts of a complex signal, then the new signal is shiftable [5]. Directional selectivity is achieved in the 2-D case by combining the outputs of the FB in such a way that the equivalent complex filters have supports in only one quadrant of the frequency plane [10]. This method of increasing directionality is equivalent to attaching quadrant FBs [11] to the highpass subband of the DWT, and can have only six directional subbands. We refer to [12] for an excellent tutorial and comprehensive references on the dual-tree DWT. The dual-tree DWT is recently generalized from twochannel to M -channel, and applied to seismic image denoising [13]. Other energy shift-invariant decompositions that use the idea of complex subband filters and have high directional selectivity include the brushlet [14], phaselet [15], and the double-density dual-tree filter bank [16]. In [14], a set of local overlapping windows are applied to the frequency plane to create local Fourier bases. These functions are called brushlets, and they have complex values and analytic frequency responses. In [15], the half-sample delay phase condition of the dual-tree DWT is generalized to fractional delay, and the decomposition is called the phaselet transform. The doubledensity FB is recently extended to a double-density dual-tree FB having better performance than the dual-tree DWT [16]. The building blocks of the FB constructed in this paper are a 2-D multiresolution FB and the DFB. Therefore, the proposed transform can be considered as a complex extension of the original contourlet transform [17]. We will elaborate on the connection of the novel decomposition with different contourlet transforms in Section IV-C. D. Notations and Paper Outline

C. Reducing or Eliminating the Effects of Shift-variance and Low Directionality of the DWT Many solutions to the shift-variance and lack of directionality of the DWT have been suggested in the literature. A simple approach for shift-variance is to remove the decimation blocks in the FB, so that there is no aliasing in the output subband signals. In this case, the subbands signals

Here we briefly review the notations and terminology used in 2-D multirate system. For fundamental operations in multidimensional 2-D systems, we refer to [18]. • Uppercase and lowercase bold face letters represent 2 × 2 square matrices and 2 × 1 column vectors, respectively. For example, h(n) is a function defined on the 2-D integer lattice (n1 , n2 )T , and π is (π, π)T . The superscripts

NGUYEN AND ORAINTARA: THE SHIFTABLE COMPLEX DIRECTIONAL PYRAMID, PART I: THEORETICAL ASPECTS

T −T

•

•

, and ∗ denote the transpose, transpose of the inverse and complex conjugate operators, respectively. N (M) is defined as the set of integer vectors of the form Mx where x ∈ [0, 1)2 . |M| represents the determinant of the matrix M. The number of elements in N (M) is equal to |M|. The Fourier transform of a 2-D filter h(n) is defined as X T H(ω) = h(n)e−jω n where ω = (ω1 , ω2 )T . (3) n∈Z 2

•

Matrix exponential notations greatly simplify 2-D multirate system expressions. The notation ω M is defined as   m11 ω1 + m21 ω2 ∆ ωM = , (4) m12 ω1 + m22 ω2 where M=



m11 m21

m12 m22



3

case, the structure consists of two trees of DWT decomposition of the same signal. The filters employed in the two trees are designed in such a way that the aliasing in one branch in the first tree is approximately cancelled by the corresponding branch in the second tree. The dual-tree FB is further explored by Selesnick [22], [23], and it is proven that the wavelet associated with the dual FB is the Hilbert transform of that associated with the primal FB, assuming that the two twochannel FBs in the dual-tree are orthogonal and regular. We briefly review the dual-tree DWT in this section, and derive the phase relation of the two highpass filters in the primal and the dual trees. The analysis in this section is similar to that in [22], but the concentration is on the phase relationship in the dual-tree DWT. H 000a

This notation is equivalent to M T ω. The followings are some special matrices that are used to decimate subband images in the paper:     1 1 2 0 Q= , D2 = . 1 −1 0 2 In this section, we have discussed the shift sensitivity and the lack of directionality problems in the DWT and shown that the problems are inherent in all maximally decimated and separable FBs. Several previously proposed shiftable and directional image decompositions are then briefly discussed and classified into two main categories: overcomplete, nonseparable methods and complex yet separable methods. The main example of complex FBs is the dual-tree DWT which will be reviewed in Section II. It is shown that in order for the filters of the two trees to be related as the real and imaginary parts of a complex filter, the pair of two-channel FBs at the second level of the trees must satisfy the half-sample phase delay condition. This condition is generalized to a 2-D phase function in Section III. A dual-tree FB consisting of two four-channel directional filter banks (DFB) [19] is constructed. Each DFB is constructed by cascading two levels of fan FBs and the 2-D phase function condition is imposed onto the fan FB in the second level of the dual tree. Similar to dual-tree DWT, this dual-tree four-band DFB can be viewed as a realization of a complex DFB. However, it is still not shiftable due to nonideal filters’ responses. A 2-D multiresolution FB is employed in Section IV, and is combined with the dual-tree DFB to create a shiftable pyramidal dual-tree DFB (PDTDFB). The new FB is shown to be approximately energy shift-invariant. The paper is concluded in Section V. As the title suggests, this paper focuses on the theoretical aspects of the complex shiftable pyramid, mainly its construction and properties. The detail implementation and applications are subjects of an accompanied paper [7]. This work was previously presented in part in [20], [21]. II. P HASE R ELATION IN T HE D UAL - TREE DWT In a series of papers [10], [5], Kingsbury proposed a dualtree structure that produces a dyadic complex DWT. In the 1-D

p

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H 01b

p

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p

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Fig. 2. The dual-tree DWT filter bank. The labels under the filter blocks indicate how the two prototype FBs (Hi (z) and HiH (z)), are used in the construction in [5]. .

As mentioned, a dual-tree FB is a combination of two binary trees of two-channel FBs, as depicted in Fig. 2. The trees are fundamentally constructed from four filters [22]. Let H0 (z) and H0H (z) be lowpass filters and H1 (z) and H1H (z) be highpass filters. The lowpass filters are related by the interpolation condition which is given by H0H (ω) = e−jθ(ω) H0 (ω)

(5)

with the half-sample phase delay function θ(ω) defined as ω θ(ω) = , when |ω| < π. (6) 2 In the original construction of the two trees in [10], at the first level, the second FB is a sample delay of the first one, i.e. Hib (ω) = e−jω Hia (ω) = e−jω Hi (ω),

for i = 0, 1.

(7)

(See Fig. 2.) From the second level on, the two FBs (Hi (z) and HiH (z)) are used alternatively between the two trees with possibly a sample delay as indicated in Fig. 2. This construction ensures that, at every level, the overall lowpass filter of the dual tree is always equal to that of the primal tree delayed by half of the equivalent decimation factor. For example, at level M , the overall decimation factor is 2M . If one of the lowpass filters is Hl (z), the other will be M −1 ) z (−2 Hl (z). It is shown in [22] that if the two two-channel FBs with lowpass filters H0 (z) and H0H (z) are regular and orthogonal, then the wavelets of the two FBs are related by

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ACCEPTED FOR PUBLICATION IN IEEE TRANSACTION ON SIGNAL PROCESSING, OCT. 2007

the Hilbert transformation. The relation above implies the following relation between the two highpass filters H1 (z) and H1H (z): H1H (ω) = ejθ(ω−π) H1 (ω). (8) Let us consider the highpass subbands at the second level of the two trees with passband supports in the region [π/4 < |ω| < π/2]: H01 (ω) = H0a (ω)H01a (2ω), H H01 (ω) = H0b (ω)H01b (2ω).

other. It is further assumed that the two filters are zero phase, and the filters used in the primal FB as depicted in Fig. 4(a) are: (f )

H0a (ω) = H00a (ω) = H10a (ω) = H0 (ω),

(11)

(f ) H1 (ω).

(12)

H1a (ω) = H01a (ω) = H11a (ω) =

Fig. 4(b) shows the four equivalent directional filters

(9)

H 00a

H 0a

H 01a

H10a

= = =

pQ

e H0 (ω)H1H (2ω) −j(ω−θ(2ω−π)) e H0 (ω)H1 (2ω) −j(ω−θ(2ω−π)) e H01 (ω).

T Z

2S

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2

Z  T  2Z  S mod 2S S 2S

S

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pQ

H 00b

H 0b

p

D2

H

pQ

H d0

p

D2

p

D2

p

D2

p

D2

p

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pQ

H 01b

H H d1

pQ

Dual DFB H10b

H1b

H H d2

pQ

pQ

H11b

H H d3

pQ

(a)

(b)

Fig. 4. The two-level dual-tree fan FB in: (a) the two-level binary tree structure, and (b) the equivalent filter support of the dual-tree of FBs.

Hdk (z), k = 0, 1, 2, 3, employed in the primal DFB with an overall decimation matrix D2 . These filters can be related to (f ) (f ) the prototype filters H0 (z) and H1 (z) through the tree structure as (f )

(f )

Hd0 (ω) =

H0 (ω)H0 (ω Q ),

Hd1 (ω) =

(f ) (f ) H0 (ω)H1 (ω Q ), (f ) (f ) H1 (ω)H0 (ω Q ), (f ) (f ) H1 (ω)H1 (ω Q ).

Hd3 (ω) =

(13) (14) and

2S

2

The phase functions θ(ω) and ω − θ(2ω − π).

e−jω1 H0 (ω),

H1b (ω) =

(f ) e H1 (ω), (f ) e−jφA (ω) H0 (ω), (f ) e−jφA (ω) H1 (ω), (f ) e−jφB (ω) H0 (ω), (f ) e−jφB (ω) H1 (ω),

H01b (ω) = H10b (ω) =

III. T HE T WO - LEVEL D UAL - TREE FAN F ILTER BANK One may observe that the key to the Hilbert relation between the filters of the dual and primal branches of the dual-tree DWT is the phase delay in (6). The conventional DFB [19] is created by cascading two-channel FBs in a binary tree. This section discusses how to construct a dual-(binary-)tree DFB, whose directional filters have a similar relationship to the Hilbert transform relation in the dual-tree DWT. As an initial step, a dual-tree consisting of two two-level fan FBs is constructed. A four-band DFB is created when one cascades (f ) two levels of the same prototype fan FB. Let H0 (z) and (f ) H1 (z) be two prototype fan filters, complementing each

(f )

H0b (ω) = H00b (ω) =

Fig. 3.

D2

(15) (16)

The filters in each level of the dual tree are chosen as follows:

2 0

H d2

pQ

H11a

Hd2 (ω) = S

p

(10)

The equations above show that the equivalent filter of the dual tree is differed from that of the primal one by the phase function of ω −θ(2ω −π). This function is illustrated in Fig. 3, H and it is clear that the filter H01 (z) is the Hilbert transform of H01 (z). Assuming that h01 (n) is real, let us write H01 (ω) = P (ω)+P ∗ (−ω), where as P (ω) is a 2π-periodic function with P (ω) = 0 when −π ≤ ω < 0, and P (ω) = H01 (ω) when H 0 ≤ ω < π, then H01 (ω) = −jP (ω) + jP ∗ (−ω). Therefore, the dual-tree structure will generate a complex wavelet since the highpass filters can be interpreted as the real and imaginary parts of a complex filter. In other words, there is a complex wavelet associated with each dual-tree if its filters satisfy the conditions discussed above. S

D2

Primal DFB

= H0 (ω)H1 (2ω), −jω

H d1

pQ

H1a

H H01 (ω)

p

pQ

Substituting the two prototype FBs into (9) as labeled in Fig. 2, we have H01 (ω)

H d0

pQ

H11b (ω) =

−jω2

(17) (18) (19) (20) (21) (22)

where φA (ω) and φB (ω) are phase functions, which will be defined later. Theorem 1: Let Ai (z) and Bi (z) be the analysis and the synthesis filters of channel i, (i = 0, 1) of a quincunx perfect reconstruction (PR) FB with a decimation matrix Q. If φ(ω) is a 2π-periodic function satisfying φ(ω) = φ(ω + π), then the four filters having frequency responses e−jφ(ω) A0 (ω), e−jφ(ω) A1 (ω), ejφ(ω) B0 (ω) and ejφ(ω) B1 (ω) render another two-channel PR FB. Proof: Note that φ(ω + π) = φ(ω1 + π, ω2 + π). Since the four filters A0 (z), A1 (z), B0 (z) and B1 (z) create a PR

NGUYEN AND ORAINTARA: THE SHIFTABLE COMPLEX DIRECTIONAL PYRAMID, PART I: THEORETICAL ASPECTS

FB, we have A0 (ω)B0 (ω) + A1 (ω)B1 (ω) = 2, and

(23)

A0 (ω + π)B0 (ω) + A1 (ω + π)B1 (ω) = 0. (24)

5

It can be shown that the function φA (ω Q ) (see Fig. 5(b)) is a (π, 2π)T -periodic function, and φA (ω Q ) = =

Using that φ(ω) = φ(ω + π), the conditions above are equivalent to

φA (ω1 + ω2 , ω1 − ω2 ),  −ω1 + π2 , 0 < ω1 < π, −π < ω2 < π, −ω1 − π2 , −π < ω1 < 0, −π < ω2 < π.

For k = 0 and 1, substituting φA (ω Q ) into equations (25) e A0 (ω)e B0 (ω) + e A1 (ω)e B1 (ω) = 2, and (26), we have  −jπ/2 −jφ(ω+π) jφ(ω) e Hdk (ω), ω1 > 0, e A0 (ω + π)e B0 (ω) + H Hdk (ω) = (30) jπ/2 −jφ(ω+π) jφ(ω) e H ω1 < 0. dk (ω), +e A1 (ω + π)e B1 (ω) = 0, −jφ(ω)

−jφ(ω)

jφ(ω)

jφ(ω)

which completes the proof. A direct implication of Theorem 1 is that the filters defined in (19)-(22) can be the analysis filters of two fan FBs which are PR as long as φm (ω) = φm (ω + π), m ∈ {A, B}. In this case, the corresponding synthesis filters are modulated by ejφm (ω) . Using (17)-(22), the four equivalent filters in the dual tree become: Q

(f ) (f ) e−jω1 e−jφA (ω ) H0 (ω)H0 (ω Q ) Q e−j(ω1 +φA (ω )) Hd0 (ω), Q (f ) (f ) e−jω1 e−jφA (ω ) H0 (ω)H1 (ω Q ) Q e−j(ω1 +φA (ω )) Hd1 (ω), Q (f ) (f ) e−jω2 e−jφB (ω ) H1 (ω)H0 (ω Q ) Q e−j(ω2 +φB (ω )) Hd2 (ω), Q (f ) (f ) e−jω2 e−jφB (ω ) H1 (ω)H1 (ω Q ) Q e−j(ω2 +φB (ω )) Hd3 (ω).

H Hd0 (ω) =

= H Hd1 (ω) =

= H Hd2 (ω)

= =

H Hd3 (ω)

= =

(25) (26) (27) (28)

Fig. 5(a) illustrates the phase function φA (ω) where black and white represent −π/2 and π/2, respectively. I A Z1 , Z2

I A Z1  Z2 , Z1  Z2

2S

Z2 S

S

2S

Z Z1

(0, 0) S

S

2S

Z1 S

S

(0, 0)

It can be shown that the function φB (ω Q ) is (2π, π)T periodic, and can be given by φB (ω Q ) = =

The key idea for the construction of the dual-tree of the four-channel DFBs is to choose φA (ω) and φB (ω) so that the equivalent directional filters of the dual DFB are Hilbert transforms of those of the primal DFB. Let us choose the 2πperiodic phase function φA (ω) as follows:  ω1 − 2 − ω22 + π2 , ω1 + ω2 > 0, (29) φA (ω) = − ω21 − ω22 − π2 , ω1 + ω2 < 0.

Z2

H Therefore, the dual filters Hdk (z) are the Hilbert transforms of the primal filters Hdk (z) with respect to ω1 for k = 0 and 1. Similarly, let φB (ω) be defined as  ω1 − 2 + ω22 + π2 , ω1 − ω2 > 0, φB (ω) = (31) − ω21 + ω22 − π2 , ω1 − ω2 < 0.

φB (ω1 + ω2 , ω1 − ω2 ),  −ω2 + π2 , 0 < ω2 < π, −π < ω1 < π, −ω2 − π2 , −π < ω2 < 0, −π < ω1 < π.

For k = 2 and 3, substituting φB (ω Q ) into (27) and (28) yields  −jπ/2 e Hdk (ω), ω2 > 0, H Hdk (ω) = (32) ejπ/2 Hdk (ω), ω2 < 0. H Hence Hdk (z) are the Hilbert transforms of Hdk (z) with respect to ω2 for k = 2 and 3. S , S 

Z2



S , S

Z2

P  Ȧ

Z2

Z1

Z1

Z1

P  -Ȧ

 S , S

P  -Ȧ

 S , S

 S , S

(a)

(b)

We have shown that the four directional filters of the dual H DFB Hdk (z) are related to those in the primal DFB by Hilbert transform in ω1 or ω2 . Let us consider one pair of H filters Hd3 (z) and Hd3 (z). Assuming that Hd3 (z) has real coefficients and zero phase, it can be considered as a sum of two real functions symmetric through the origin as

where P (ω) =

2S

(a) φA (ω)

Q

(b) φA (ω )

Fig. 5. The phase functions φA (ω) and φA (ωQ ). Black to white shades correspond to value changes from −π/2 to π/2.

(c)

Fig. 6. (a) The ideal support regions of the filter Hd3 (z), (b) the actual support of Hd3 (z) by realizable filters, the dotted regions are the transition band. (c) P (z) as defined in (34).

Hd3 (ω) = P (ω) + P (−ω), S

S , S

P  Ȧ

P  Ȧ



Hd3 (ω), 0 < ω2 , 0, ω2 < 0.

(33) (34)

H The ideal supports of Hd3 (z) and P (z) are shown in Fig. 6(a), where black and white colors represent the passband and H stopband. From (32), Hd3 (ω) can be written as H Hd3 (ω) = −jP (ω) + jP (−ω).

(35)

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ACCEPTED FOR PUBLICATION IN IEEE TRANSACTION ON SIGNAL PROCESSING, OCT. 2007

pair. When the two branches including the synthesis side are considered together, the aliasing components will partially cancel each other in a way similar to the cases of four-channel dual-tree DFB.

Hence

H Hd3 (ω) + jHd3 (ω) . (36) P (ω) = 2 In practice, the ideal support of the directional filter Hd3 (z) in Fig. 6(a) cannot be achieved, and the transition regions between the stopband and the passband, denoted by the dotted areas in Fig. 6(b), must be taken into consideration. By the definition of P (z) in (34), its support is illustrated in Fig. 6(c). Let us denote the subband signal corresponding to the primal (dual) subband 3 of the dual-tree DFB before and after decimaH tion as x3 (n) (xH 3 (n)) and x3d (n) (x3d (n)), respectively. We define the complex subband signal corresponding to subband 3 of the dual-tree DFB before and after decimation as

xC 3 (n) xC 3d (n)

= x3 (n) + jxH 3 (n), H = xH (n) + jx 3d 3d (n).

IV. T HE C OMPLEX D IRECTIONAL P YRAMID A. A Shiftable Pyramid Decomposition

(37) (38)

We have the following relations X3C (ω) = C (ω) = X3d

2X(ω)P (ω), (39) X 1 X3 (D2−T ω − 2πD2−T k).(40) |D2 | T k∈N (D2 )

 2S , 0 §Ȧ Ȧ · X 3Cd ¨ 1 , 2 ¸ © 2 2 ¹

Z2

Z2

Z1

Ȧ · §Ȧ X 3Cd ¨ 1  S , 2 ¸ 2 ¹ © 2

S , S

S , S

 0, 2S

Z1 §Ȧ Ȧ · X 3Cd ¨ 1 , 2  S ¸ © 2 2 ¹

Ȧ §Ȧ · X 3Cd ¨ 1  S , 2  S ¸ 2 © 2 ¹

In order to construct a shiftable multiscale and multidirectional decomposition, a combination of a multiresolution FB with the dual-tree DFB at high frequency is proposed. The combination of the multiresolution FB and the dual-tree DFB has a two-fold objective. First, the multiscale FB provides a multiresolution image decomposition. Secondly, it can also remove the remaining aliasing components in the dual-tree DFB as discussed in the previous section. In order to keep the complexity low, the FB should be efficiently implemented using separable filtering. The proposed multiscale FB consists of an undecimated two-channel FB and an iterated 2-D multiresolution FB having decimation at the lowpass branch. Consider the construction in Fig. 8. At the front end, an undecimated two-channel FB (L0 (z) and R0 (z)) is used to separate the high frequency components near (±π, ·) and (·, ±π), which can potentially cause aliasing in the dual-tree. The highpass filter R0 (z) produces a ‘residual’ image similar to that in the steerable pyramid [4]. It is clear that, for this undecimated FB to be PR, the filters must satisfy |R0 (ω)|2 + |L0 (ω)|2 = 1.

 S , S

 S , S

(a)

(b)

C (ω) in (40).(a) Fig. 7. Frequency supports of the four elements of X3d T T T T k = (0, 0) , (b) k = (1, 0) , (0, 1) , (1, 1) . The overlapping region C (0.5ω , 0.5ω ) and X C (0.5ω − π, 0.5ω ) is encircled. between X3d 1 2 1 2 3d

Following similar arguments on the estimation of the energy C of X3d (ω) as in Section I, the signal xC 3 (n) is shiftable if the C four components of X3d (ω) in (40) are not overlapping. From Fig. 7, we can easily see that they have an overlapping region, which corresponds to the highpass regions in the directional subband 3. In order to have a shiftable image transform, a pyramidal decomposition is used to remove the frequency components in these regions before applying the signal to the dual-tree DFB, which will be discussed in Section IV. A. Extension to the case of 2n Subband Dual-tree DFB The conventional DFB is constructed by using a binary tree of two-channel FBs [19]. These two-channel FBs can be obtained from one prototype fan FB if appropriate resampling blocks, which are up or downsampling blocks having determinant one, are used in the tree [24]. The 2n -channel dual-tree DFB is constructed from the above four-channel dual-tree DFB by cascading two-channel FBs, using the same (f ) (f ) prototype fan FBs (H0 (z) and H1 (z)) and resampling blocks. Therefore, each pair of directional filters in the primal and dual 2n -channel DFBs still form a Hilbert transform

(41)

The output of the wide-band lowpass filter L0 (z) is then fed into the first stage of the multiresolution pyramidal FB where the signal is divided into two parts: the coarse approximation (point L in Fig. 8(a)) and the high frequency component (point H in Fig. 8(a)). This high frequency component is then further decomposed by a dual-tree of DFBs to produce the real and imaginary parts of the 2n complex directional subbands. This decomposition (block P ) is repeated at the low frequency output (point L), forming a pyramid. The filters in blocks P and Q are designed to satisfy the PR and non-aliasing condition 1 |R1 (ω)|2 + |L1 (ω)|2 = 1, (42) 4 π π L1 (ω1 , ω2 ) = 0 when |ω1 | > or |ω2 | > . (43) 2 2 The factor 1/4 on the left side of (42) is needed to ensure the two-channel FB using R1 (±ω) and L1 (±ω) is PR. Let us assume that the input signal of the FB in frequency domain is X(ω). Then the output of the filter L1 (z) is X(ω)L1 (±ω). After being decimated and upsampled by D2 , the signal in frequency domain is 1 (X(ω1 , ω2 )L1 (ω1 , ω2 ) + X(ω1 + π, ω2 )L1 (ω1 + π, ω2 ) + 4 + X(ω1 , ω2 + π)L1 (ω1 , ω2 + π) + +

X(ω1 + π, ω2 + π)L1 (ω1 + π, ω2 + π))

The filter L1 (−z) will keep only the first component and reject the other three aliasing components. Therefore, the

NGUYEN AND ORAINTARA: THE SHIFTABLE COMPLEX DIRECTIONAL PYRAMID, PART I: THEORETICAL ASPECTS

x  n

R 0 Z

R

R

P

n

H

Primal 2 -channel maximally decimated DFB

L0 Z

R 1 Z

7

 Z H i

Fi Z

 H Z H i

FiH Z

Primal 2n -channel

H

maximally decimated DFB

R1  Z

Dual 2n -channel maximally decimated DFB

y  n

R 0  Z

Q

L 0  Z

Dual 2n -channel maximally decimated DFB L1  Z

L1 Z p

L

D2

P

L

Q

n

(a)

D2

(b)

Fig. 8. The structure to implement the shiftable complex directional pyramid : (a) Analysis side, and (b) Synthesis side. Similar P and Q blocks can be reiterated at lower scale to decompose an image into a multiscale representation.

output signal of the FB in the frequency domain is   1 2 2 X(ω1 , ω2 ) |L1 (ω1 , ω2 )| + |R1 (ω1 , ω2 )| 4 In order for the FB to be PR, the term in the brackets must be equal to 1. B. The Pyramidal (PDTDFB)

Dual-tree

Directional

Filter

Bank

The PDTDFB takes the shiftable pyramidal FB discussed above and applies a dual-tree 2n -channel DFB at the highpass output of each level. The first level of the PDTDFB is illustrated in Fig. 8. The blocks P and Q are iterated to provide a multiscale decomposition and synthesis. Based on the construction of the PDTDFB, it will be shown below that the image representation produced by the structure in Fig. 8 is energy shift-invariant. The DFBs employed in the PDTDFB are constructed using the proposed structure in [19]. The primal and dual DFBs of the dual-tree are identical at every level except for the second one where the filters of the two-channel fan FBs in the dual DFB have to satisfy the phase conditions in (29) and (31). By this construction, the resulting directional filters in the primal and dual DFBs satisfy the conditions (30) and (32). Therefore, they form Hilbert transform pairs either in ω1 or ω2 . Let Hi (z) and HiH (z), i ∈ {0, 1, . . . 2n − 1}, be the directional filters of the analysis primal and dual DFBs. Similarly, let ˜ i (z) Fi (z) and FiH (z) be the equivalent synthesis filters. Let H H ˜ and Hi (z) be the equivalent primal and dual analysis filters of the first stage of the PDTDFB, i.e. the transfer functions between the input x(n) to the output of Hi (z) and HiH (z). Similarly, let F˜i (z) and F˜iH (z) be those on the synthesis side (see Fig. 8). The frequency responses of these filters can be expressed as follows: ˜ i (ω) H ˜ HiH (ω) F˜i (ω) F˜ H (ω) i

= L0 (ω)R1 (ω)Hi (ω), = L0 (ω)R1 (ω)HiH (ω),

(44) (45)

= L0 (−ω)R1 (−ω)Fi (ω), and = L0 (−ω)R1 (−ω)FiH (ω).

(46) (47)

From the above expressions, since Hi (z) and HiH (z) ˜ i (z) (Fi (z) and FiH (z)) are Hilbert transform pairs, so are H ˜ H (z) (F˜i (z) and F˜ H (z)). The frequency responses of and H i i each pair of the filters have the Hilbert transform relation

in ω1 in (30) if 0 ≤ i ≤ 2n−1 − 1, and in ω2 in (32) if 2n−1 ≤ i ≤ 2n − 1. It was shown in Section III that if the highpass components are removed, the complex subband signals are shiftable. That discussion is still applicable to the ˜ i (z) and H ˜ H (z). Recall that the frequency components case H i at (±π, ·) and (·, ±π) have already been removed by the wideband lowpass filter L1 (z). Therefore, the overall FB is alias-free, which implies shiftability.

 Z H 0 4 0º » ¬0 2¼

x  n

Z2

F0 Z p ǻ

4 0º » ¬0 2¼

F0H Z

 H Z H 0 p ǻ

4 0º » ¬0 2¼

n ª«

4 0º » ¬0 2¼

1 y n 2

F0C Z p ǻ

4 0º » ¬0 2¼

n ǻ

4 0º » ¬0 2¼

(b)

1

2

3

5

6

7

Z1

yD  n

(a) x  n

3

7

 C Z H 0

S , S

yP  n

n ǻ

xˆ  n

4

 S , S

(c)

Fig. 9. (a) Two directional branches of the PDTDFB considered separately.(b) A complex filters interpretation of the system in (a), and (c) the alias components appear in the multirate system using complex filter in (b).

Let us view the shiftable property of the PDTDFB from a different perspective. Each pair of filters from the two trees are interpreted as the real and imaginary parts of a complex filter. Let us define ˜ i (ω) + j H ˜ H (ω) H ∆ i ˜ C (ω) = , (48) H i 2 F˜i (ω) − j F˜iH (ω) ∆ F˜iC (ω) = . (49) 2 Hence each of these complex filters will have only singlesided passband. Consider an example for n = 3, i.e. there are eight directional subband in each DFB. Fig. 9(a) shows ˜ 0 (z), H ˜ H (z), F˜0 (z) and F˜ H (z). the passband supports of H 0 0 C ˜ 0 (z) and F˜0C (z) have approxThus, the two complex filters H imately zero magnitude responses when ω2 < 0 as illustrated in Fig. 9(b). Assuming that the input x(n) in Fig. 9(b) is real, the output x ˆ(n) will be complex. Since the decimation matrix is diag{4, 2}, there are seven aliasing components (gray regions in Fig. 9(c)) that are in the stopband of the filter F˜0C (z). Assuming that the transition band is sufficiently narrow, the synthesis filter F˜0C (z) will be able to reject all the

8

ACCEPTED FOR PUBLICATION IN IEEE TRANSACTION ON SIGNAL PROCESSING, OCT. 2007

seven aliasing components. Hence, according to Fig. 9(b), ˆ X(ω) ≈

˜ 0C (ω)F˜0C (ω)X(ω). H

(50)

Let yP (n) and yD (n) be the outputs of F˜0 (z) and F˜0H (z) and y(n) = 21 (yP (n) + yD (n)) as indicated in Fig. 9(a). It is easy to show that y(n) = Real(ˆ x(n)). Therefore, the system in Fig. 9(a) produces a single-band complex output which is shiftable. The impulse  response of this system is ˜ C (n) ∗ f˜C (n) . In [20], we approximately equal to Real h 0 0 show that there exist aliasing components in both yP (n) and yD (n), but these components are canceled each other in y(n). Fig. 10 demonstrates aliasing cancellation in the directional filters where yP (n), yD (n) and y(n) are obtained by setting x(n) = δ(n) and the filters are described according to the method described in the next section. One can see that the aliasing displayed as peaks at the transition bands in Figs. 10(a) and (b) are clearly eliminated in Fig. 10(c).

(a)

(b)

(c)

Fig. 10. The frequency content of signals in the system in Fig. 9(a) when the input signal is δ(n): (a) YP (ω) (b) YD (ω) and (c) Y (ω).

Let us now estimate the overcomplete ratio of the decomposition. Since the PDTDFB has two DFBs at each highpass resolution corresponding to a redundant ratio of 2, and the lowpass band of the second multiresolution FB at the output of the filter L1 (z) has 1/4 of the number of pixels compared to the highpass band. Therefore the upper bound of the overcomplete ratio at the output of wide-band lowpass filter 1 + · · · ≈ 83 . L0 (z) can be estimated by 2 + 2 × 41 + 2 × 16 Taking into account the residual subband at the output of filter R0 (z), the overcomplete ratio of the decomposition is 11/3. C. Relationship between the shiftable complex directional pyramid decomposition and the contourlet transform The idea of combining the DFB and a multiresolution decomposition to create a multiresolution multidirectional decomposition is suggested in [26], [27]. In particular, Do and Vetterli build the theory of the contourlet transform from the PDFB in [28], [17]. Therefore, the PDTDFB can be regarded as a shiftable contourlet FB. The relation of the PDTDFB to the contourlet FB in [17] is similar to that of the 1-D dual-tree DWT to the 1-D DWT. The 1-D dual-tree DWT (the PDTDFB) provides a shiftable decompositions similar to the 1-D DWT (the contourlet transform) at the cost of approximately twice overcomplete ratio. Furthermore, both of the 1-D dual-tree DWT and the PDTDFB uses a dual FB to provide complex or one-sided filter supports. Recently, a nonsubsampled contourlet transform is constructed by the same research group [29], [25], [30]. The ideas are to suppress all the decimation block in the original

contourlet FB and to employ the ‘ a` trous’ algorithm [2] to have an efficient implementation. The decomposition is complete shift-invariant and has the overcomplete ratio of N × K, where N is the number of resolution levels and K is the number of directions. Again, the relation between the PDTDFB and the nonsubsampled contourlet FB is similar to that between the dual-tree DWT and the undecimated DWT. Table I summarizes the position of the PDTDFB with respect to different types of the contourlet FBs. V. C ONCLUSION A novel shiftable multiscale multidirectional image transform implemented by the PDTDFB is presented in this work. The PDTDFB is a multiresolution FB, of which each level consists of a two-channel 2-D multiresolution FB and a pair of conventional 2n -channel DFBs. These FBs are designed to satisfy the required criteria so that the equivalent filters of the dual branches are the Hilbert transforms of the ones in the primal branches. Thus, each pair of directional filters in the PDTDFB renders a complex filter with only one region of support in the frequency plane. The image decomposition offered by the PDTDFB has several desirable properties for image analysis applications. It is energy shift-invariant, which makes the representation stable with respect to signal translation. It also has a very low redundant ratio compared to other energy shiftinvariant image decompositions in the literature. The proposed multiscale and multidirectional representation creates favorable conditions for efficient subsequent processing algorithms. Unlike other shiftable directional transforms whose angular resolutions are limited, the number of directional subbands in the PDTDFB can be increased adaptively depending on the processed image without increasing the redundant ratio of the representation. Furthermore, the decomposition provides phase information on the image feature, which can be very useful in several image and video processing tasks, such as motion estimation or edge detection. In an accompanied paper [7], we show that the whole framework can be realized by separable filters and has a complexity of order O(N ), where N is the number of pixels in the processed image. The MATLAB code to implement the PDTDFB transform can be found from the Multirate Signal Processing Lab website at the following address: http://www-ee.uta.edu/msp. R EFERENCES [1] G. Strang and T. Q. Nguyen, Wavelets and Filter Banks. Wellesley Cambridge, 1996. [2] S. Mallat, A Wavelet tour of signal processing, 2nd ed. Academic Press, 1999. [3] G. Strang, “Wavelets and dilation equations: a brief introduction,” SIAM Review, vol. 31, no. 4, pp. 614–627, Dec 1989. [4] E. P. Simoncelli, W. T. Freeman, E. H. Adelson, and D. J. Heeger, “Shiftable multiscale transform,” IEEE Transaction on Information Theory, vol. 38, no. 2, pp. 587–607, Mar 1992. [5] N. G. Kingsbury, “Complex wavelets for shift invariant analysis and filtering of signals,” Journal of Applied and Computational Harmonic Analysis, vol. 10, no. 3, pp. 234–253, May 2001. [6] F. C. A. Fernandes, R. L. C. van Spaendonck, and C. S. Burrus, “A new framework for complex wavelet transforms,” IEEE Transaction on Signal Processing, vol. 51, no. 7, pp. 1825–1837, Jul 2003. [7] T. T. Nguyen and S. Oraintara, “The shiftable complex directional pyramid, part 2: Implementation and applications,” submitted to the IEEE Transaction on Signal Processing, Mar 2007.

NGUYEN AND ORAINTARA: THE SHIFTABLE COMPLEX DIRECTIONAL PYRAMID, PART I: THEORETICAL ASPECTS

9

TABLE I C OMPARISON BETWEEN THE PDTDFB AND THE CONTOURLET FB S . Filter bank PDFB/Contourlet FB [17] PDTDFB Nonsubsampled contourlet FB [25]

Overcomplete ratio 4/3 11/3 N ×K

[8] I. W. Selesnick, “The double density dwt,” in Wavelets in Signal and Image Analysis: From Theory to Practice, A. Petrosian and F. G. Meyer, Eds. Amsterdam, The Netherlands: Kluwer, 2001, ch. 2. [9] J.-L. Starck, E. J. Cand`es, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Transaction on Image Processing, vol. 11, no. 6, pp. 670–684, Jun. 2002. [10] N. G. Kingsbury, “Image processing with complex wavelets,” Phil. Trans. Royal Society London A, vol. 357, no. 1760, pp. 2543–2560, Sept 1999. [11] 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. [12] I. W. Selesnick, R. G. Baraniuk, and N. C. Kingsbury, “The dual-tree complex wavelet transform,” IEEE Signal Processing Magazine, vol. 22, no. 6, pp. 123–151, Nov 2005. [13] C. Chaux, L. Duval, and J.-C. Pesquet, “Image analysis using a dualtree m-band wavelet transform,” IEEE Transaction on Image Processing, vol. 15, no. 8, pp. 2397–2412, May 2006. [14] F. G. Meyer and R. R. Coifman, “Brushlets : Steerable wavelet packets,” in Beyond Wavelets, G. V. Welland, Ed. Amsterdam, The Netherlands: Academic Press, 2003, ch. 3, pp. 61–82. [15] R. A. Gopinath, “The phaselet transform-an integral redundancy nearly shift-invariant wavelet transform,” IEEE Transaction on Signal Processing, vol. 51, no. 7, pp. 1792–1805, Jul 2003. [16] I. W. Selesnick, “The double-density dual-tree dwt,” IEEE Transaction on Signal Processing, vol. 52, no. 5, pp. 1304–1314, May 2004. [17] M. N. Do and M. Vetterli, “The contourlet transform: An efficient directional multiresolution image representation,” IEEE Transactions on Image Processing, vol. 14, pp. 2107– 2116, Dec. 2005. [18] P. Vaidyanathan, Multirate Systems and Filter Banks. PrenticeHall,Englewood Cliffs, NJ, 1993. [19] 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. [20] T. T. Nguyen, “The multiresolution directional filter bank,” Ph.D. dissertation, University of Texas at Arlington, Aug. 2006. [Online]. Available: http://www-ee.uta.edu/msp/truong [21] T. T. Nguyen and S. Oraintara, “A shift-invariant multiscale multidirection image decomposition,” in Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’06), France, May 2006, pp. 153–156. [22] I. W. Selesnick, “Hilbert transform pairs of wavelet bases,” IEEE Signal Processing Letter, vol. 8, no. 6, pp. 170–173, June 2001. [23] ——, “The design of approximate hilbert transform pairs of wavelet bases,” IEEE Transaction on Signal Processing, vol. 50, no. 5, pp. 1144– 1152, May 2002. [24] S. I. Park, M. J. Smith, and R. M. Mersereau, “Improved structure of maximally decimated directional filter banks for spatial image analysis,” IEEE Transactions on Image Processing, vol. 13, no. 11, pp. 1424 – 1431, Nov. 2004. [25] A. L. Cunha, J. Zhou, and M. N. Do, “The nonsubsampled contourlet transform: Theory, design, and applications,” IEEE Transactions on Image Processing, vol. 15, no. 10, pp. 3089–3101, 2006. [26] M. N. Do and M. Vetterli, “Pyramidal directional filter banks and curvelets,” in Proc. of IEEE International Conference on Image Processing (ICIP), Thessaloniki, Greece, Oct. 2001. [27] 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. [28] 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. [29] A. L. Cunha, J. Zhou, and M. N. Do, “Nonsubsampled contourlet transform: filter design and application in image denoising,” in Proc. of International Conference on Image Processing (ICIP 05), Genoa, Italy, Sep. 2005.

Shiftability Shift-variant Shiftable or approximately shift-invariant Complete linear shift-invariant

[30] J. Zhou, A. L. Cunha, and M. N. Do, “Nonsubsampled contourlet transform: construction and application in enhancement,” in Proc. of International Conference on Image Processing (ICIP 05), Genoa, Italy, Sep. 2005.

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, and the Ph.D. degree in Electrical Engineering from the University of Texas at Arlington in 2006. From 1995 to 2002 he worked at Alcatel NSV, Hanoi, Vietnam. From June 2007 to Sep. 2007, he was a Visiting Scientist at Shell Geoscience Solution, Rijswijk, The Netherlands. Currently, he is working as a Research Engineer at the Geophysical Research Center, Paris School of Mines, France. His current research interests are in the field of signal and image processing, computer vision and seismic imaging.

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. degrees, both in electrical engineering, respectively, from the University of Wisconsin, Madison, in 1996 and Boston University, Boston, MA, in 2000. He joined the Department of Electrical Engineering, University of Texas at Arlington (UTA), as an Assistant Professor in July 2000 where he has become an Associate Professor in September 2006. 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 1990 and the bronze medal in Sigtuna, Sweden, in 1991.