## MULTIDIMENSIONAL MULTIRATE SYSTEMS: CHARACTERIZATION, DESIGN, AND APPLICATIONS (2005)

Citations: | 4 - 0 self |

### BibTeX

@MISC{Zhou05multidimensionalmultirate,

author = {Jianping Zhou},

title = {MULTIDIMENSIONAL MULTIRATE SYSTEMS: CHARACTERIZATION, DESIGN, AND APPLICATIONS },

year = {2005}

}

### OpenURL

### Abstract

Multidimensional multirate systems have been used widely in signal processing, communications, and computer vision. Traditional multidimensional multirate systems are tensor products of one-dimensional systems. While these systems are easy to implement and design, they are inadequate to represent multidimensional signals since they cannot capture the geometric structure. Therefore, “true” multidimensional systems are more suited to multidimensional signals, such as images and videos. This thesis focuses on the characterization, design, and applications of “true” multidimensional multirate systems. One key property of multidimensional multirate systems is perfect reconstruction, which guarantees the original input can be perfectly reconstructed from the outputs. The most popular multidimensional multirate systems are multidimensional filter banks, including critically sampled and oversampled ones. Characterizing and designing multidimensional perfect reconstruction filter banks have been challenging tasks. For critically sampled filter banks, previous one-dimensional theory cannot be extended to the multidimensional

### Citations

2640 | A theory for multiresolution signal decomposition: the wavelet representation - Mallat - 1989 |

2295 | A Wavelet Tour of Signal Processing - MALLAT - 1999 |

1915 | lectures on wavelets - Daubechies, Ten - 1992 |

1735 | Orthonormal Bases of Compactly Supported Wavelets
- Daubechies
- 1988
(Show Context)
Citation Context ...that the energy of errors generated by transmission or quantization will not be amplified. Second, under certain conditions, orthogonal filter banks can be used to construct orthonormal wavelet bases =-=[60, 61]-=-. Third, orthogonal filter banks offer certain conveniences; for example, the best M-term approximation is simply done by keeping those M coefficients with largest magnitude. One-dimensional (1-D) ort... |

1090 | The Laplacian pyramid as a compact image code
- Burt, Adelson
- 1983
(Show Context)
Citation Context ... X HN−1 D D GN−1 Figure 1.1: Multidimensional N-channel filter bank: Hi and Gi are MD analysis and synthesis filters, respectively; D is an M × M sampling matrix. 1puter vision and image compression =-=[2]-=-. After that, 1-D multirate systems and filter banks have been well-studied by a number of researchers, in particular in connection with wavelets and multiresolution analysis [3–11]. Vetterli extended... |

992 |
Emergence of simple-cell receptive field properties by learning a sparse code for natural images,” Nature 381
- Olshausen, Field
- 1996
(Show Context)
Citation Context ...nd filter banks [60, 73]. In recent years, geometric image representation has received interest [29, 30, 74–77]. Olshausen and Field showed that directionality plays a key role in human visual system =-=[78]-=-. Bamberger proposed the critically sampled directional filter banks which decompose images into different directional subbands [29]. Based on directional filter banks, Do and Vetterli proposed an eff... |

943 |
Multirate Systems and Filter Banks
- Vaidyanathan
- 1993
(Show Context)
Citation Context ...defined as LAT(D) = {Dk,k ∈ Z M }. (2.1) When the sampling matrix is a diagonal matrix, the sampling lattice is separable. Otherwise, the sampling lattice is nonseparable. The N -set of D is given by =-=[7]-=- N(D) = { integer vector Dt : t ∈ [0, 1) M} , (2.2) and its size is equal to the sampling rate |D|. We extend the 1-D downsampling and upsampling to the MD case using the sampling matrix and sampling ... |

628 | MPEG: A video compression standard for multimedia applications
- Gall
- 1991
(Show Context)
Citation Context ... new image compression standard JPEG 2000 adopts the wavelet filter bank for transform coding [22]. Multirate filter banks have been used in the texture coding part of new video compression standards =-=[23, 24]-=-, and are promising to make a breakthrough in scalable video coding [25]. Traditional MD multirate systems are separable and straightforward extensions from 1-D ones. Transfer functions of a separable... |

499 | Wavelets and subband coding - Vetterli, Kovacevic - 1995 |

462 | Shiftable Multi-scale Transforms
- Simoncelli, Freeman, et al.
- 1992
(Show Context)
Citation Context ...nd upsampling, the contourlet transform is shift-variant. However, shift-invariance is desirable in image analysis applications such as edge detection, contour characterization, and image enhancement =-=[74]-=-. In this chapter, we present the nonsubsampled contourlet transform (NSCT), which is a shift-invariant version of the contourlet transform. The NSCT is built upon iterated nonsubsampled filter banks ... |

387 | Image denoising using scale mixtures of Gaussians in the wavelet domain - Portilla, Strela, et al. - 2003 |

368 |
Using Algebraic Geometry
- Cox, Little, et al.
- 1998
(Show Context)
Citation Context ...role in the matrix inverse problem [10]. However, the Euclidean algorithm fails for multidimensional (MD) filters. Algebraic geometry and Gröbner bases are powerful tools for multivariate polynomials =-=[89, 90]-=- and are widely used in multidimensional signal processing [91–95]. Rajagopal and Potter recently applied algebraic geometry to compute polynomial matrix inverse [96]. However, the filters they consid... |

330 |
The contourlet transform: an efficient directional multiresolution image representation
- Do, Vetterli
(Show Context)
Citation Context ...he MD structure directly, resulting in more freedom and better frequency selectivity; for example, see [26–28]. In addition, nonseparable systems lead to flexible directional decomposition of MD data =-=[29,30]-=-. In recent years, “true” MD multirate systems have received more and more interest [7,8,31–34]. Figure 1.2 illustrates the difference between traditional separable systems and new nonseparable system... |

324 |
Image compression through wavelet transform coding
- DeVore, Jawerth, et al.
- 1992
(Show Context)
Citation Context ...on representation that captures the discontinuity locally in the signal. Moreover, wavelets provide better nonlinear approximation, which implies better performance in data compression and processing =-=[72, 73]-=-. The decay rate of wavelet coefficients is O(N −1 ) for piecewise smooth signals, while that of Fourier coefficients is O(N −1/2 ). Vanishing moments play a key role in the design of wavelets and fil... |

310 | The curvelet transform for image denoising
- Starck, s, et al.
- 2002
(Show Context)
Citation Context ...andard deviation of the subbands at a specific level. We first estimate the noise variance of the input image with the robust median operator [142] and then compute the noise variance of each subband =-=[128]-=-. The goal 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 149each pixel by a nonlinear mapping function... |

284 |
Subspace methods for the blind identification of multichannel FIR filters
- Moulines, Duhamel, et al.
- 1995
(Show Context)
Citation Context ...f sensors and computing units. The theory and applications of multichannel deconvolution have grown rapidly, such as general deconvolution theory [110–112], channel equalization for multiple antennas =-=[113]-=-, multichannel image deconvolution [105,114–117], and polarimetric calibration of radars [96]. Figure 6.1 shows the multichannel deconvolution setup, where the original signal is filtered by multiple ... |

190 |
A.K.: Digital image restoration
- Banham, Katsaggelos
- 1997
(Show Context)
Citation Context ...oth nonsubsampled and oversampled FIR filter banks. 122CHAPTER 6 MULTIDIMENSIONAL MULTICHANNEL FIR DECONVOLUTION 6.1 Introduction The traditional single-channel deconvolution problem is well-studied =-=[108, 109]-=-. In general, this problem is ill-posed since the convolution output does not contain information at frequencies corresponding to the zeros of the convolution filter. Over the last decade, multichanne... |

190 | Spatially adaptive wavelet thresholding with context modeling for image denoising
- Chang, Yu, et al.
- 2000
(Show Context)
Citation Context ... is a parameter ranging from 1 to 5, and σ is the noise standard deviation of the subbands at a specific level. We first estimate the noise variance of the input image with the robust median operator =-=[142]-=- and then compute the noise variance of each subband [128]. The goal of image enhancement is to amplify weak edges and to suppress noise. To this end, we modify the NSCT coefficients according to the ... |

180 | Image and Video Compression Standards: Algorithms and Architecture - Bhaskaran, Konstantinides - 1995 |

172 | Subband Coding of Images - Woods, O’Neil - 1986 |

162 | Painless nonorthogonal expansions - Daubechies, Grossmann, et al. - 1986 |

153 | Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency - Şendur, Selesnick - 2002 |

133 | A filter bank for directional decomposition of images: theory and design
- Bamberger, Smith
- 1992
(Show Context)
Citation Context ...n image representation called the contourlet transform [30,134,135]. The contourlet transform employs Laplacian pyramids [2, 136] to achieve multiresolution decomposition and directional filter banks =-=[29, 57, 137]-=- to achieve directional decomposition. Owing to the geometric information, the contourlet transform achieves better results than the discrete wavelet transform in image analysis applications such as d... |

123 |
Non-separable bidimensional wavelet Bases, Rev
- Cohen, Daubechies
- 1993
(Show Context)
Citation Context ...that the energy of errors generated by transmission or quantization will not be amplified. Second, under certain conditions, orthogonal filter banks can be used to construct orthonormal wavelet bases =-=[60, 61]-=-. Third, orthogonal filter banks offer certain conveniences; for example, the best M-term approximation is simply done by keeping those M coefficients with largest magnitude. One-dimensional (1-D) ort... |

110 | Oversampled filter banks
- Cvetković, Vetterli
- 1998
(Show Context)
Citation Context ...ruction oversampled FIR filter banks (that is, all filters are FIR), which are much easier to implement and thus more popular. One-dimensional (1-D) oversampled filter banks have been investigated in =-=[10, 11, 87, 88]-=-. For 1-D oversampled FIR filter banks, the Euclidean algorithm plays a key role in the matrix inverse problem [10]. However, the Euclidean algorithm fails for multidimensional (MD) filters. Algebraic... |

110 |
Fast Algorithms for Digital Signal Processing
- Blahut
- 1987
(Show Context)
Citation Context ...lem [101]. If the greatest common divisor (GCD) of {H1,...,HN} is 1, then the Bezout identity problem has a solution. We can use the Euclidean algorithm to find the GCD and also a set of {G1,...,GN} =-=[102]-=-. However, the univariate GCD criterion fails for multivariate polynomials. To illustrate this, we give an example. Example 5.1 Let H1(z1,z2) = 1 − z1 and H2(z1,z2) = 1 − z2. The GCD of H1(z1,z2) and ... |

98 | Contourlets: A directional multiresolution image representation
- Do, Vetterli
- 2002
(Show Context)
Citation Context ...SNR = 32.4 dB. 140CHAPTER 7 NONSUBSAMPLED CONTOURLET TRANSFORM 7.1 Introduction Do and Vetterli proposed an efficient directional multiresolution image representation called the contourlet transform =-=[30,134,135]-=-. The contourlet transform employs Laplacian pyramids [2, 136] to achieve multiresolution decomposition and directional filter banks [29, 57, 137] to achieve directional decomposition. Owing to the ge... |

95 | Mulfiresolufion analysis, Haar bases, and self-similar filings of lin - Gr0chenig, Madych - 1992 |

93 |
Multirate Digital Filters, Filter Banks, Polyphase Networks, and Applications
- Vaidyanathan
- 1990
(Show Context)
Citation Context ...ilter banks. In a critically sampled filter bank, the number of total output samples is equal to that of the input samples. Critically sampled filter banks are also known as biorthogonal filter banks =-=[38,39,41,42]-=-. The orthogonal (or loseless) filter bank is a special class of biorthogonal filter banks that is tight frame [43]. In contrast, in an oversampled filter banks, the number of total output samples is ... |

82 | Filter banks allowing perfect reconstruction - Vetterli - 1986 |

77 |
Multidimensional sub-band coding: some theory and algorithms
- Vetterli
- 1984
(Show Context)
Citation Context ...d by a number of researchers, in particular in connection with wavelets and multiresolution analysis [3–11]. Vetterli extended 1-D multirate systems and filter banks to the multidimensional (MD) case =-=[12]-=-. Since then, MD multirate systems have been widely used for MD data, such as image compression [13–15], video compression [16–19], and computer vision [20, 21]. Multidimensional filter banks have ach... |

76 |
A theory of multirate filter banks
- Vetterli
- 1987
(Show Context)
Citation Context ...ilter banks. In a critically sampled filter bank, the number of total output samples is equal to that of the input samples. Critically sampled filter banks are also known as biorthogonal filter banks =-=[38,39,41,42]-=-. The orthogonal (or loseless) filter bank is a special class of biorthogonal filter banks that is tight frame [43]. In contrast, in an oversampled filter banks, the number of total output samples is ... |

76 | Brushlets: a tool for directional image analysis and image compression - Meyer, Coifman - 1997 |

69 | Weighted Median Filters: A Tutorial - Yin, Yang, et al. - 1996 |

63 |
Linear Multivariable Systems
- WOLOVICH
- 1974
(Show Context)
Citation Context ...ect reconstruction condition for an oversampled filter bank is equivalent to a matrix inverse problem in the polyphase domain. Perfect reconstruction oversampled IIR filter banks have been studied in =-=[85, 86]-=- in the context of control theory. In this chapter, we are interested in perfect reconstruction oversampled FIR filter banks (that is, all filters are FIR), which are much easier to implement and thus... |

63 | Directional multiscale modeling of images using the contourlet transform, to appear in
- Po, Do
- 2006
(Show Context)
Citation Context ...sition. Owing to the geometric information, the contourlet transform achieves better results than the discrete wavelet transform in image analysis applications such as denoising and texture retrieval =-=[138]-=-. Due to downsampling and upsampling, the contourlet transform is shift-variant. However, shift-invariance is desirable in image analysis applications such as edge detection, contour characterization,... |

62 | The analysis and design of multidimensional FIR perfect reconstruction filter banks with arbitrary sampling lattices - Viscito, Allebach - 1991 |

60 |
Wavelets and recursive filter banks
- Herley, Vetterli
- 1993
(Show Context)
Citation Context ...hogonal filter bank is completely determined by its N − 1 synthesis filters and a phase factor in the last synthesis filter. Although this result was shown for 1-D two-channel orthogonal filter banks =-=[50]-=- and MD two-channel orthogonal filter banks [33], to the best of our knowledge, this is the first time it is proved for general orthogonal filter banks of any dimension and any number of channels. Mor... |

60 |
The sampling and reconstruction of time-varying imagery with application in video systems
- Dubois
- 1985
(Show Context)
Citation Context ...se representation Multidimensional sampling plays a key role in MD multirate systems. Compared to the 1-D sampling, MD sampling is more complex since it involves sampling matrix and sampling lattices =-=[31, 51, 52]-=-. A sampling matrix D is an M × M integer matrix and its sampling rate is equal to the absolute value of its determinant, denoted by |D|. A sampling lattice of a sampling matrix D is defined as LAT(D)... |

60 | The nonsubsampled contourlet transform: Theory, design, and applications
- Cunha, Zhou, et al.
- 2006
(Show Context)
Citation Context ...e number of directions is increased with frequency. 7.3 Application: Image Enhancement In this section, we apply the NSCT to image enhancement. For the applications of the NSCT to image denoising see =-=[131, 133]-=-, and for applications to coutour characterization see [139]. 7.3.1 Image enhancement algorithm Image enhancement is widely used in medical and biological imaging to improve the image quality. The pur... |

58 | Three dimensional subband coding of video - Karlsson, Vetterli - 1988 |

58 |
Quadrature mirror filter banks, M-band extensions and perfect reconstruction techniques
- Vaidyanathan
- 1987
(Show Context)
Citation Context ...ilter banks. In a critically sampled filter bank, the number of total output samples is equal to that of the input samples. Critically sampled filter banks are also known as biorthogonal filter banks =-=[38,39,41,42]-=-. The orthogonal (or loseless) filter bank is a special class of biorthogonal filter banks that is tight frame [43]. In contrast, in an oversampled filter banks, the number of total output samples is ... |

58 | Lossy Compression of
- Chan, Vetterli
- 1995
(Show Context)
Citation Context ...on representation that captures the discontinuity locally in the signal. Moreover, wavelets provide better nonlinear approximation, which implies better performance in data compression and processing =-=[72, 73]-=-. The decay rate of wavelet coefficients is O(N −1 ) for piecewise smooth signals, while that of Fourier coefficients is O(N −1/2 ). Vanishing moments play a key role in the design of wavelets and fil... |

57 |
Advances in scalable video coding
- Ohm
- 2005
(Show Context)
Citation Context ... transform coding [22]. Multirate filter banks have been used in the texture coding part of new video compression standards [23, 24], and are promising to make a breakthrough in scalable video coding =-=[25]-=-. Traditional MD multirate systems are separable and straightforward extensions from 1-D ones. Transfer functions of a separable system are products of multiple 1-D filters. Therefore, tensor products... |

51 | Framing Pyramids
- Do, Vetterli
- 2003
(Show Context)
Citation Context ...1 Introduction Do and Vetterli proposed an efficient directional multiresolution image representation called the contourlet transform [30,134,135]. The contourlet transform employs Laplacian pyramids =-=[2, 136]-=- to achieve multiresolution decomposition and directional filter banks [29, 57, 137] to achieve directional decomposition. Owing to the geometric information, the contourlet transform achieves better ... |

49 | Filter bank frame expansions with erasures - Dragotti, Goyal, et al. - 2002 |

48 | Advanced modern algebra - Rotman - 2002 |

46 |
Computer Algebra
- Davenport, Y, et al.
(Show Context)
Citation Context ... and numerical stability The main difficulty with Algorithm 5.2 is computing Gröbner bases. The computational complexity of Gröbner bases has been studied in the literature, for example in [119] and (=-=[120]-=- Chapter 3). Detailed complexity analysis is beyond the scope of this chapter. Experimental results show that computing Gröbner bases is very fast (less than 2 seconds) in practical deconvolution prob... |

45 |
Flexible design of multidimensional perfect reconstruction FIR 2-band filters using transformations of variables
- Tay, Kingsbury
- 1993
(Show Context)
Citation Context ...er the last decade, the theory and applications of filter banks have grown rapidly [8, 46, 52, 56–58]. Among them, orthogonal filter banks received particular attention due to their useful properties =-=[31, 43, 45, 59]-=-. First, orthogonality implies energy preservation, which guarantees that the energy of errors generated by transmission or quantization will not be amplified. Second, under certain conditions, orthog... |

45 |
Perfect blind restoration of images blurred by multiple filters: Theory and efficient algorithms
- Harikumar, Bresler
- 1999
(Show Context)
Citation Context ...act deconvolution problem where both convolution and deconvolution filters are finite impulse response (FIR), and the reconstruction signal equals the original signal in the absence of additive noise =-=[104, 105, 118]-=-. Such FIR exact deconvolution is more computationally efficient than traditional least-square solutions. Moreover, FIR deconvolution filters limit the noise propThis chapter includes research conduct... |