## NONSUBSAMPLED CONTOURLET TRANSFORM: FILTER DESIGN AND APPLICATIONS IN DENOISING

### Cached

### Download Links

Citations: | 57 - 4 self |

### BibTeX

@MISC{Cunha_nonsubsampledcontourlet,

author = {Arthur L. da Cunha and et al.},

title = {NONSUBSAMPLED CONTOURLET TRANSFORM: FILTER DESIGN AND APPLICATIONS IN DENOISING},

year = {}

}

### OpenURL

### Abstract

In this paper we study the nonsubsampled contourlet transform. We address the corresponding filter design problem using the McClellan transformation. We show how zeroes can be imposed in the filters so that the iterated structure produces regular basis functions. The proposed design framework yields filters that can be implemented efficiently through a lifting factorization. We apply the constructed transform in image noise removal where the results obtained are comparable to the state-of-the art, being superior in some cases.

### Citations

1815 |
Ten Lectures on wavelets
- Daubechies
- 1992
(Show Context)
Citation Context ..., G (1D) 0 (x) each has zeros at x = −1. Then, in order to produce a suitable zero-phase mapping function for the pyramid NSFB we consider the class of maximally-flat filters given by the polynomials =-=[30]-=- ( ) N L−1−N 1 + x ∑ PN,L(x) := 2 l=0 ( ) ( N + l − 1 1 − x l 2 ) l , (11) where N controls the degree of flatness at x = −1 and L controls the degree flatness at x = 1. Following Proposition 2, we ca... |

1070 | The Laplacian pyramid as a compact image code
- Burt, Adelson
- 1983
(Show Context)
Citation Context ...veral representation schemes have recently been proposed [10]–[15]. The contourlet transform [14] is a multidirectional and multiscale transform that is constructed by combining the Laplacian pyramid =-=[16]-=-, [17] with the directional filter bank (DFB) proposed in [18]. The pyramidal filter bank structure of the contourlet transform has very little redundancy, which is important for compression applicati... |

899 |
Multirate Systems and Filter Banks
- Vaidyanathan
- 1993
(Show Context)
Citation Context ...cy plane into 2 3 = 8 wedges. (b) Equivalent multi-channel filter bank structure. The number of channels is L = 2 l , where l is the number of stages in the tree structure. Using multirate identities =-=[18]-=-, the tree-structured DFB can be put into the equivalent form shown in Figure 3 (b), where the downsampling/upsampling matrices Sk for 0 ≤ k ≤ 2l − 1 are given by ⎧ ⎨ diag Sk = ⎩ ( 2l−1 , 2 ) , for 0 ... |

579 |
Uncertainty relation for resolution in space, spatial frequency and orientation optimized by 2D visual cortical filters
- Daugman
- 1985
(Show Context)
Citation Context ...her important feature of a transform is its stability with respect to shifts of the input signal. The importance of the shiftinvariance property in imaging applications dates back at least to Daugman =-=[3]-=- and was also advocated by Simoncelli et al. in [4]. An example that illustrates the importance of shiftinvariance is image denoising by thresholding where the lack of shift-invariance causes pseudo-G... |

489 | Wavelets and Subband Coding
- Vetterli, Kovačević
- 1995
(Show Context)
Citation Context ...ocessing tasks are efficiently carried out in the domain of an invertible linear transformation. For example, image compression and denoising are efficiently done in the wavelet transform domain [1], =-=[2]-=-. An effective transform captures the essence of a given signal or a family of signals with few basis functions. The set of basis functions completely characterizes the transform and this set can be r... |

450 | Shiftable Multi-scale Transforms
- Simoncelli, Freeman, et al.
- 1992
(Show Context)
Citation Context ...ty with respect to shifts of the input signal. The importance of the shiftinvariance property in imaging applications dates back at least to Daugman [3] and was also advocated by Simoncelli et al. in =-=[4]-=-. An example that illustrates the importance of shiftinvariance is image denoising by thresholding where the lack of shift-invariance causes pseudo-Gibbs phenomena around singularities [5]. Thus, most... |

449 |
lifting scheme: a custom-design construction of biorthogonal wavelets
- Sweldens
- 1996
(Show Context)
Citation Context ...-D. If we relax the tightness constraint,DA CUNHA et al.: NSCT: THEORY, DESIGN, AND APPLICATIONS 3093 This can be shown by following the same reasoning as in the critically-sampled case discussed in =-=[30]-=-. The complexity can be reduced further if the lifting steps in the 1-D prototype are monomials and the mapping filter has the form Fig. 6. Lifting structure for the NSFB designed with the mapping app... |

378 | Image denoising using scale mixtures of Gaussians in the wavelet domain
- Portilla, Strela, et al.
- 2003
(Show Context)
Citation Context ...mage denoising by thresholding where the lack of shift-invariance causes pseudo-Gibbs phenomena around singularities [5]. Thus, most state-of-the-art wavelet denoising algorithms (see for example [6]–=-=[8]-=-) use an expansion with less shift sensitivity than the standard maximally decimated wavelet decomposition—the most common being the nonsubsampled wavelet transform (NSWT) computed with the à trous al... |

323 |
The contourlet transform: an efficient directional multiresolution image representation
- Do, Vetterli
(Show Context)
Citation Context ...resentation has to account for the geometrical structure pervasive in natural scenes. In this direction, several representation schemes have recently been proposed [10]–[15]. The contourlet transform =-=[14]-=- is a multidirectional and multiscale transform that is constructed by combining the Laplacian pyramid [16], [17] with the directional filter bank (DFB) proposed in [18]. The pyramidal filter bank str... |

304 | The Curvelet Transform for Image Denoising
- Starck, Candès, et al.
- 2000
(Show Context)
Citation Context ...CT relative to other transforms, we perform hard threshold on the subband coefficients of the various transforms. We choose the threshold for each subband. This has been termed -sigma thresholding in =-=[34]-=-. We set for the finest scale and for the remaining ones. We use five scales of decomposition for both 2 The regularity exponent of a scaling function (t) is the largest number such that 8(!) decays a... |

263 | New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities
- Candès, Donoho
(Show Context)
Citation Context ...xible in that it allows any number of directions in each scale. In particular, it can satisfy the anisotropic scaling law—a key property in establishing the expansion nonlinear approximation behavior =-=[13]-=-, [14]. This property is ensured by doubling the number of directions in the NSDFB expansion at every other scale. The NSCT has redundancy given by , where denotes the number of levels in the NSDFB at... |

249 | Adaptive wavelet thresholding for image denoising and compression
- Chang, Yu, et al.
- 2000
(Show Context)
Citation Context ...is image denoising by thresholding where the lack of shift-invariance causes pseudo-Gibbs phenomena around singularities [5]. Thus, most state-of-the-art wavelet denoising algorithms (see for example =-=[6]-=-–[8]) use an expansion with less shift sensitivity than the standard maximally decimated wavelet decomposition—the most common being the nonsubsampled wavelet transform (NSWT) computed with the à trou... |

190 |
A Wavelet Tour
- Mallat
- 1998
(Show Context)
Citation Context ...ge-processing tasks are efficiently carried out in the domain of an invertible linear transformation. For example, image compression and denoising are efficiently done in the wavelet transform domain =-=[1]-=-, [2]. An effective transform captures the essence of a given signal or a family of signals with few basis functions. The set of basis functions completely characterizes the transform and this set can... |

187 | Spatially adaptive wavelet thresholding with context modeling for image denoising
- Chang, Yu, et al.
- 1998
(Show Context)
Citation Context ...e images and then averaged to stabilize the results. We refer to this method as local adaptive shrinkage (LAS). Effectively, our LAS method is a simplified version of the denoising method proposed in =-=[36]-=- that works in the NSCT or NSWT domain. In the LAS estimator we use four scales for both the NSCT and NSWT. For the NSCT we use 3, 3, 4, 4 directions in the scales from coarser to finer, respectively.... |

159 |
The discrete wavelet transform: Wedding the a trous and Mallat algorithms
- Shensa
- 1992
(Show Context)
Citation Context ... expansion with less shift sensitivity than the standard maximally decimated wavelet decomposition—the most common being the nonsubsampled wavelet transform (NSWT) computed with the à trous algorithm =-=[9]-=-. 1 In addition to shift-invariance, it has been recognized that an efficient image representation has to account for the geometrical structure pervasive in natural scenes. In this direction, several ... |

159 | Sparse geometric image representations with bandelets
- Pennec, Mallat
- 2005
(Show Context)
Citation Context ...metrical structure pervasive in natural scenes. In this direction, several representation schemes have recently been proposed. These include adaptive schemes such as wedgelets [9], [10] and bandelets =-=[11]-=-, and non-adaptive ones such as curvelets [12] and contourlets [13]. The contourlet transform is a multi-directional and multi-scale transform that is constructed by combining the Laplacian pyramid [1... |

152 |
Translation invariant denoising
- Coifman, Donoho
- 1995
(Show Context)
Citation Context ...i et al. in [4]. An example that illustrates the importance of shiftinvariance is image denoising by thresholding where the lack of shift-invariance causes pseudo-Gibbs phenomena around singularities =-=[5]-=-. Thus, most state-of-the-art wavelet denoising algorithms (see for example [6]–[8]) use an expansion with less shift sensitivity than the standard maximally decimated wavelet decomposition—the most c... |

132 |
Wedgelets: Nearly minimax estimation of edges
- Donoho
- 1999
(Show Context)
Citation Context ...cognized that an efficient image representation has to account for the geometrical structure pervasive in natural scenes. In this direction, several representation schemes have recently been proposed =-=[10]-=-–[15]. The contourlet transform [14] is a multidirectional and multiscale transform that is constructed by combining the Laplacian pyramid [16], [17] with the directional filter bank (DFB) proposed in... |

129 | A filter bank for the directional decomposition of images: theory and design
- Bamberger, JT
- 1992
(Show Context)
Citation Context ...[15]. The contourlet transform [14] is a multidirectional and multiscale transform that is constructed by combining the Laplacian pyramid [16], [17] with the directional filter bank (DFB) proposed in =-=[18]-=-. The pyramidal filter bank structure of the contourlet transform has very little redundancy, which is important for compression applications. However, designing good filters for the contourlet transf... |

125 |
Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r
- Kovačević, Vetterli
- 1992
(Show Context)
Citation Context ...FIR filters, this means that . Such condition can only be met with linear phase FIR filters if and are either trivial delays or combination of two delays (for a formal proof, see [22, pp. 337–338] or =-=[23]-=-). Because the NSFB is redundant, an infinite number of inverses exist. Among them, the pseudoinverse is optimal in the least-square sense [1]. Given a frame of analysis filters, the synthesis filters... |

109 |
Fast Algorithms for Digital Signal Processing
- Blahut
- 1985
(Show Context)
Citation Context ...G (1D) 1 (x) such that the Bezout identity is satisfied. In this case it follows that gcd{H (1D) 0 , H (1D) 1 } = 1. The Euclidean algorithm then enables us to factor the filters in the following way =-=[28]-=-, [26], [27]: ⎛ ⎝ H(1D) 0 (x) H (1D) 1 (x) ⎞ ⎠ = ⎛ N∏ ⎝ i=0 ⎞ ⎛ 1 0 ⎠ ⎝ 0 1 P (1D) i (x) 1 1 Q(1D) i (x) ⎞ ⎛ ⎠ ⎝ 1 ⎞ ⎠ . (7) 0 May 19, 2005 DRAFTIEEE TRANSACTIONS ON IMAGE PROCESSING, MAY 2005 12 As ... |

107 | Oversampled filter banks
- Cvetković, Vetterli
- 1998
(Show Context)
Citation Context ... tightest positive constants satisfying (4). Consider the NSFB of Figure 6 (a). The family {h0[· −n], h1[· −n]}n∈Z2 is a frame for ℓ2(Z2 ) if and only if there exist constants 0 < A ≤ B < ∞ such that =-=[19]-=- May 19, 2005 DRAFTIEEE TRANSACTIONS ON IMAGE PROCESSING, MAY 2005 9 Thus, the frame bounds of an NSFB can be computed by A ≤ |H0(e jω )| 2 + |H1(e jω )| 2 } {{ } t(ejω ≤ B. (5) ) A = ess. inf ω∈[−π,... |

81 | Bi-variate Shrinkage with Local Variance Estimation
- Sendur, Selesnick
(Show Context)
Citation Context ... THRESHOLDING, THE NSCT CONSISTENTLY OUTPERFORMS CURVELETS AND THE NSWT. THE NSCT-LAS PERFORMS ON A PAR WITH THE MORE SOPHISTICATED ESTIMATOR BLS-GSM [8] AND IS SUPERIOR TO THE BIVSHRINK ESTIMATOR OF =-=[7]-=- NSCT, contourlet transform (CT), and NSWT. For the NSCT and CT we use 4, 8, 8, 16, 16 directions in the scales from coarser to finer, respectively. Table III (left columns) shows the PSNR results for... |

75 | Frame-theoretic analysis of oversampled filter banks
- Bölcskei, Hlawatsch, et al.
- 1998
(Show Context)
Citation Context ...fect reconstruction NSFB system, both analysis and synthesis filters form a frame. If we denote the analysis and synthesis frame bounds by and respectively, the frames will be close to tight provided =-=[24]-=- In the event that , the frame is said to be tight. The frame bounds are the tightest positive constants satisfying (3). Consider the NSFB of Fig. 5(a). The family is a frame for if and only if there ... |

68 | Approximation properties of multivariate wavelets
- Jia
- 1998
(Show Context)
Citation Context ...oximation of the ideal frequency response of Fig. 5, in addition to imposing regularity of the scaling function. We point out that for the approximation of smooth images, point zeros at would suffice =-=[31]-=-. However, our experience shows that point zeros alone do not guarantee a “reasonable” frequency response of the pyramid filters. The following proposition characterizes the mapping function that gene... |

65 |
A new class of two-channel biorthogonal filter banks and wavelet bases
- Phoong, Kim, et al.
- 1995
(Show Context)
Citation Context ...an be linear phase. An effective and simple way to design 2-D filters is the mapping approach first proposed by McClellan [25] in the context of digital filters and then used by several authors [23], =-=[26]-=-–[28] in the context of filter banks. In such an approach, the 2-D filters are obtained from 1-D ones. In the context of NSFBs, a set of perfect reconstruction 2-D filters is obtained in the following... |

63 | Directional multiscale modeling of images using the contourlet transform
- Po, Do
- 2006
(Show Context)
Citation Context ...se variance at scale and direction . It is shown in [6] that shrinkage estimation with , and assuming generalized Gaussian distributed yields a risk within 5% of the optimal Bayes risk. As studied in =-=[35]-=-, contourlet coefficients are well modelled by generalized Gaussian distributions. The signal variances are estimated locally using the neighboring coefficients contained in a square window within eac... |

50 | Framing pyramids
- Do, Vetterli
- 2003
(Show Context)
Citation Context ...representation schemes have recently been proposed [10]–[15]. The contourlet transform [14] is a multidirectional and multiscale transform that is constructed by combining the Laplacian pyramid [16], =-=[17]-=- with the directional filter bank (DFB) proposed in [18]. The pyramidal filter bank structure of the contourlet transform has very little redundancy, which is important for compression applications. H... |

43 | Wavelet-domain approximation and compression of piecewise smooth images
- Wakin, Romberg, et al.
(Show Context)
Citation Context ...account for the geometrical structure pervasive in natural scenes. In this direction, several representation schemes have recently been proposed. These include adaptive schemes such as wedgelets [9], =-=[10]-=- and bandelets [11], and non-adaptive ones such as curvelets [12] and contourlets [13]. The contourlet transform is a multi-directional and multi-scale transform that is constructed by combining the L... |

42 |
Flexible design of multidimensional perfect reconstruction FIR 2-band filters using transformations of variables
- Tay, Kingsbury
- 1993
(Show Context)
Citation Context ...linear phase. An effective and simple way to design 2-D filters is the mapping approach first proposed by McClellan [21] in the context of digital filters and then used by several authors [22], [23], =-=[24]-=-, [25] in the context May 19, 2005 DRAFTIEEE TRANSACTIONS ON IMAGE PROCESSING, MAY 2005 11 of filter banks. In such approach, the 2-D filters are obtained from 1-D ones. In the context of NSFB’s, a s... |

37 |
The design of two-dimensional digital filters by transformation
- McClellan
- 1973
(Show Context)
Citation Context ...omes more flexible. In addition, as we alluded to earlier, non-tight filters can be linear phase. An effective and simple way to design 2-D filters is the mapping approach first proposed by McClellan =-=[25]-=- in the context of digital filters and then used by several authors [23], [26]–[28] in the context of filter banks. In such an approach, the 2-D filters are obtained from 1-D ones. In the context of N... |

21 |
Image Processing and Data Analysis
- Starck, Murtagh, et al.
- 1998
(Show Context)
Citation Context ...ular, one bandpass image is produced at each stage resulting in redundancy. By contrast, the NSWT produces three directional images at each stage, resulting in redundancy. The 2-D pyramid proposed in =-=[19]-=- pp. 21 is obtained with a similar structure. Specifically, the NSFB of [19] is built from low-pass filter . One then sets , and the corresponding synthesis filters . A similar decomposition can be ob... |

15 |
Nonlinear UnSharp Masking Methods for Image Contrast Enhancement
- Ramponi, Strobel, et al.
- 1996
(Show Context)
Citation Context ...sed algorithm with those by the NSWT. In the experiments, we choose and . To evaluate the enhancement performance objectively, the detailed variance (DV) and background variance (BV) were proposed in =-=[38]-=-. The DV and BV values represent the variance of foreground and background pixels, respectively. A good enhancement methods should increase the DV of the original image but not the BV. We use the BV a... |

12 |
Multi-scale color image enhancement
- Velde
- 1999
(Show Context)
Citation Context ...l of image enhancement is to amplify weak edges and to suppress noise. To this end, we modify the NSCT coefficients according to the category of each pixel by a nonlinear mapping function (similar to =-=[37]-=-) (19) where the input is the original coefficient, and is the amplifying gain. This function keeps the coefficients of strong edges, amplifies the coefficients of weak edges, and zeros the noise coef... |

9 | On two-channel filter banks with directional vanishing moments
- Cunha, Do
- 2007
(Show Context)
Citation Context ... linear phase. An effective and simple way to design 2-D filters is the mapping approach first proposed by McClellan [25] in the context of digital filters and then used by several authors [23], [26]–=-=[28]-=- in the context of filter banks. In such an approach, the 2-D filters are obtained from 1-D ones. In the context of NSFBs, a set of perfect reconstruction 2-D filters is obtained in the following way.... |

6 |
Maximally flat half-band diamond-shaped FIR filters using the Bernstein polynomial
- Cooklev, Yoshida, et al.
- 1993
(Show Context)
Citation Context ...lifting factorization of the prototype filters is given by (12) where the polynomials give the class of maximally-flat half-band filters with diamond support. A closed-form expression for is given in =-=[33]-=-. Table II displays the first six mapping functions for the resulting maximally-flat fan filter bank. D. Design Examples The design through mapping is based on a set of 1-D polynomials that satisfies ... |

4 |
A low complexity overcomplete directional image pyramid
- Rosiles, Smith
- 2003
(Show Context)
Citation Context ...zed that an efficient image representation has to account for the geometrical structure pervasive in natural scenes. In this direction, several representation schemes have recently been proposed [10]–=-=[15]-=-. The contourlet transform [14] is a multidirectional and multiscale transform that is constructed by combining the Laplacian pyramid [16], [17] with the directional filter bank (DFB) proposed in [18]... |

3 |
Digital ladder networks
- Mitra, Sherwood
- 1973
(Show Context)
Citation Context ...tion is a 2-D polynomial in . We thus denote it by , where it is implicit that . A. Implementation Through Lifting Filters designed with the mapping approach can be efficiently factored into a ladder =-=[29]-=- or lifting [30] structure that simplifies computations. To see this, assume without loss of generality that the degree of the high-pass prototype polynomial is smaller than that of . Suppose also tha... |

2 |
Coronary angiogram image enhancement using decimation-free directional filter banks
- Khan, Khan, et al.
- 2004
(Show Context)
Citation Context ... directional wedges. A shift-invariant directional expansion is obtained with a nonsubsampled DFB (NSDFB). The NSDFB is constructed by eliminating the downsamplers and upsamplers in the DFB (see also =-=[20]-=-). This is done by switching off the downsamplers/upsamplers in each two-channel filter bank in the DFB tree structure and upsampling the filters accordingly. This results in a tree composed of two-ch... |

1 |
Oversampled filler banks
- Cvetkovic, Vetterli
- 1998
(Show Context)
Citation Context ...rame is said to be tight. The frame bounds are the tightest positive constants satisfying (3). Consider the NSFB of Fig. 5(a). The family is a frame for if and only if there exist constants such that =-=[21]-=- Thus, the frame bounds of an NSFB can be computed by where , and denote the essential infimum and essential supremum respectively. From (4), we see that the frame is tight whenever is almost everywhe... |