## A new family of nonredundant transforms using hybrid wavelets and directional filter banks (2007)

Citations: | 10 - 1 self |

### BibTeX

@MISC{Eslami07anew,

author = {Ramin Eslami and Hayder Radha},

title = {A new family of nonredundant transforms using hybrid wavelets and directional filter banks},

year = {2007}

}

### OpenURL

### Abstract

We propose a new family of nonredundant geometrical image transforms that are based on wavelets and directional filter banks. We convert the wavelet basis functions in the finest scales to a flexible and rich set of directional basis elements by employing directional filter banks, where we form a nonredundant transform family, which exhibits both directional and nondirectional basis functions. We demonstrate the potential of the proposed transforms using nonlinear approximation. In addition, we employ the proposed family in two key image processing applications, image coding and denoising, and show its efficiency for these applications.

### Citations

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Citation Context ...nuities. The curvelet transform has been proposed in the continuous domain, and, therefore, implementing it in the discrete domain is challenging. The contourlet transform employs a Laplacian pyramid =-=[6]-=- to extract edges of an image and applies directional filter banks (DFB) [4] to all bandpass outputs of the pyramid with decreasing number of directions when moving to the coarser pyramid subbands. Th... |

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Citation Context ...native to handle the problem adaptively. One can also enjoy this feature of wavelets when extending it for the proposed transforms. Other leading approaches such as curvelets [8], [9] and contourlets =-=[15]-=- use a similar idea of combining large transform coefficients around discontinuities. The curvelet transform has been proposed in the continuous domain, and, therefore, implementing it in the discrete... |

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Citation Context ...ansform family is constructed using hybrid wavelets and directional filter banks (HWD); thus, we refer to them as the HWD transforms. Other nonredundant geometrical image transforms include bandelets =-=[26]-=-, CRISP-contourlets [27], directionlets [39], nonredundant complex wavelets [22], and multiresolution direction filter banks [29]. A primary difference between our proposed transform family and the ot... |

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Citation Context ...dependencies are considered through the parent–children relationships existing among the wavelet coefficients. Note that although there are other superior schemes such as space-frequency quantization =-=[43]-=- and WSFQ [42], since the scope of this paper is not image coding, we just provide our preliminary results using SPIHT coding algorithm. To take advantage of the SPIHT scanning algorithm for the HWD t... |

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Citation Context ...and, therefore, implementing it in the discrete domain is challenging. The contourlet transform employs a Laplacian pyramid [6] to extract edges of an image and applies directional filter banks (DFB) =-=[4]-=- to all bandpass outputs of the pyramid with decreasing number of directions when moving to the coarser pyramid subbands. The DFB stage attempts to decorrelate the dependencies found over the edges in... |

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Citation Context ...y and in VDFB and HDFB, where ` ' and ` ' ) provide a directional basis for . Note that if we utilize orthogonal fan filters in the DFBs, 2 the basis functions are orthogonal in the corresponding DFB =-=[14]-=-. 2 By DFBs, we sometimes mean both full-tree and half-tree DFBs depending on the context. A. Construction We propose to extend the directionality of the WT by employing the DFBs to the highpass chann... |

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Citation Context ...ey application for the proposed transforms is image coding. Although the NLA decay rate of wavelets for images is suboptimal, one can benefit from tree-based coding schemes to improve this decay rate =-=[11]-=-. The SPIHT algorithm is an efficient tree-based wavelet coding scheme [35]. In this scheme, interscale dependencies are considered through the parent–children relationships existing among the wavelet... |

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Citation Context ...ehave the same relationship as the one in the WT, and for the levels , for each subband , , and we can use a similar parent–children relationship as the one considered for the contourlet coefficients =-=[33]-=-. The problem appears when we attempt to define the children of coefficients lying at level . By applying DFBs to level , we almost remove the interscale dependencies that existed between wavelet leve... |

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Citation Context ... image Barbara, our TIHWD (BS) denoising scheme provides better results whereas for other images it shows comparable performance (within 0.25 dB). Our results are also comparable to those reported in =-=[13]-=- for nonsubsampled contourlet transform using local adaptive shrinkage (NSCT-LAS). Fig. 18 demonstrates a visual example for this part for the Barbara image and noise level of . It clearly shows the s... |

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Citation Context ...enoising, nonlinear approximation (NLA), wavelet transform (WT). I. INTRODUCTION RECENTLY, there have been several studies showing that separable 2-D wavelets fail to represent images optimally [28], =-=[40]-=-. It is well known that the wavelet transform provides efficient approximation of 1-D piecewise smooth signals; nevertheless, since natural images possess 1-D singularities in the form of regular edge... |

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Citation Context ...avelets and directional filter banks (HWD); thus, we refer to them as the HWD transforms. Other nonredundant geometrical image transforms include bandelets [26], CRISP-contourlets [27], directionlets =-=[39]-=-, nonredundant complex wavelets [22], and multiresolution direction filter banks [29]. A primary difference between our proposed transform family and the other nonredundant transforms mentioned above ... |

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Citation Context ...re considered through the parent–children relationships existing among the wavelet coefficients. Note that although there are other superior schemes such as space-frequency quantization [43] and WSFQ =-=[42]-=-, since the scope of this paper is not image coding, we just provide our preliminary results using SPIHT coding algorithm. To take advantage of the SPIHT scanning algorithm for the HWD transform coeff... |

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Citation Context ...frequency wrapping (CuTFW) with curvelets at the finest scale [7] (where we found that it gives better denoising results than curvelets via unequispaced FFT approach), translation-invariant WT (TIWT) =-=[12]-=-, dual-tree complex wavelet transform (DTCWT) [23], and translation-invariant CT (TICT) [17] using hard thresholding for the sake of comparison. Except for the Barbara image that we used HWD-F with , ... |

16 | Crisp-contourlets: A critically sampled directional multiresolution image representation
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Citation Context ...ucted using hybrid wavelets and directional filter banks (HWD); thus, we refer to them as the HWD transforms. Other nonredundant geometrical image transforms include bandelets [26], CRISP-contourlets =-=[27]-=-, directionlets [39], nonredundant complex wavelets [22], and multiresolution direction filter banks [29]. A primary difference between our proposed transform family and the other nonredundant transfo... |

16 |
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Citation Context ...er and Smith introduced directional filter banks (DFB) using quincunx and parallelogram filter banks [2], [4]. An improved version of the DFB using tree-structured filter banks was developed recently =-=[31]-=-. In an -level DFB, the frequency space is divided into wedge-shaped subbands [see Fig. 2(a)]. The overall sampling matrices for channels of such a DFB is [31] for for 1 From hereafter, we mention 2-D... |

14 |
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Citation Context ...see Fig. 1(c)]. As a result, wavelets have poor directionality. C. Directional Filter Banks Bamberger and Smith introduced directional filter banks (DFB) using quincunx and parallelogram filter banks =-=[2]-=-, [4]. An improved version of the DFB using tree-structured filter banks was developed recently [31]. In an -level DFB, the frequency space is divided into wedge-shaped subbands [see Fig. 2(a)]. The o... |

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13 |
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Citation Context ...family.ESLAMI AND RADHA: NEW FAMILY OF NONREDUNDANT TRANSFORMS 1159 Fig. 9. Left: Analysis bank of the triplet QFB. Right: The synthesis bank. Fig. 10. Top: Fan filter pair using triplet filter bank =-=[1]-=-. Bottom: The fan filter pair using double-halfband filter bank [32]. C. Efficient Realization While the WT canbe implemented efficientlyusing1-D filters, the DFBs in the HWD need to be treated carefu... |

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Citation Context ...FBs in this scheme. A major drawback of employing DFB is the pseudo-Gibbs phenomena artifacts introduced when some of the transform coefficients are set to zero during NLA, coding [20], and denoising =-=[17]-=-. Since in the DFB we need to use long filters for better directional resolution and since the basis functions are directional, it turns out that the issue of ringing artifacts will be severer for the... |

7 |
mapping-based complex wavelet transforms
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(Show Context)
Citation Context ... (HWD); thus, we refer to them as the HWD transforms. Other nonredundant geometrical image transforms include bandelets [26], CRISP-contourlets [27], directionlets [39], nonredundant complex wavelets =-=[22]-=-, and multiresolution direction filter banks [29]. A primary difference between our proposed transform family and the other nonredundant transforms mentioned above is the following. While HWD is nonad... |

6 |
tight frames of curvelets and optimal representations of objects with C2 singularities
- New
(Show Context)
Citation Context ... packets is an alternative to handle the problem adaptively. One can also enjoy this feature of wavelets when extending it for the proposed transforms. Other leading approaches such as curvelets [8], =-=[9]-=- and contourlets [15] use a similar idea of combining large transform coefficients around discontinuities. The curvelet transform has been proposed in the continuous domain, and, therefore, implementi... |

5 |
Subband video coding based on velocity filters
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(Show Context)
Citation Context ...in key image processing applications such as coding. We should also note that there have been a few attempts in the past to increase the directionality of wavelets using checkerboard filter bank [3], =-=[10]-=-. Although these transforms provide nonadaptive directional extension of wavelets, they are limited to a small number of directions and do not have flexibility when compared with our proposed scheme. ... |

3 | Regular hybrid wavelets and directional filter banks: Extensions and applications
- Eslami, Radha
- 2006
(Show Context)
Citation Context ...cales. 3Note that if we use linear-phase filters, we can benefit from symmetric extension yielding less border artifact. 4We formerly called HWD-F and HWD-H as HWD type 3 and HWD type 2, respectively =-=[19]-=-. Fig. 3. Schematic plot of the HWD-H transform using l =3 directional levels. Similar to the DFB, the major direction represented by each directional subband in HWD is perpendicular to the major axis... |

1 |
subband decompositions and coders for image and video compression
- “New
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(Show Context)
Citation Context ...oyed in key image processing applications such as coding. We should also note that there have been a few attempts in the past to increase the directionality of wavelets using checkerboard filter bank =-=[3]-=-, [10]. Although these transforms provide nonadaptive directional extension of wavelets, they are limited to a small number of directions and do not have flexibility when compared with our proposed sc... |

1 |
image transforms using hybrid wavelets and directional filter banks: Analysis and design
- “New
(Show Context)
Citation Context ... use quincunx sampling at the first level, the shape of subbands and in the spatial domain is diamond. We can change the shape into a rectangle using a unimodular matrix and shifting, as explained in =-=[18]-=-. III. HYBRID WAVELETS AND DFB Fig. 2. (a) Frequency response of a (full-tree) DFB decomposed in three levels. (b) An example of the vertical directional filter bank (VDFB) using three levels. (c) An ... |

1 |
contourlet coding using an SPIHT-like algorithm
- “Wavelet-based
- 2004
(Show Context)
Citation Context ... proper use of the DFBs in this scheme. A major drawback of employing DFB is the pseudo-Gibbs phenomena artifacts introduced when some of the transform coefficients are set to zero during NLA, coding =-=[20]-=-, and denoising [17]. Since in the DFB we need to use long filters for better directional resolution and since the basis functions are directional, it turns out that the issue of ringing artifacts wil... |

1 |
contourlet transform and its application to image coding
- “Wavelet-based
(Show Context)
Citation Context ...s WT. Do and Vetterli attempted to address this issue by applying the DFB to the Laplacian pyramid in which the highpass channels are free from frequency scrambling [14], [15]. In previous work [20], =-=[21]-=-, we applied DFBs to all the highpass channels of WT, which resulted in introducing many artifacts in the smooth regions during NLA and coding. In this work, we address the problem as described below.... |