Poisson noise removal from images using the fast discrete Curvelet transform
17 Mar 2011-pp 1-5
TL;DR: The results show that the VST combined with the FDCT is a promising candidate for Poisson denoising, and a simple approach to achieve this is presented.
Abstract: We propose a strategy to combine the variance stabilizing transform (VST), used for Poisson image denoising, with the fast discrete Curvelet transform (FDCT). The VST transforms the Poisson image to approximately Gaussian distributed, and the subsequent denoising can be performed in the Gaussian domain. However, the performance of the VST degrades when the original image intensity is very low. On the other hand, the FDCT can sparsely represent the intrinsic features of images having discontinuities along smooth curves. Therefore, it is suitable for denoising applications. Combining the VST with the FDCT leads to good Poisson image denoising algorithms, even for low intensity images. We present a simple approach to achieve this and demonstrate some simulation results. The results show that the VST combined with the FDCT is a promising candidate for Poisson denoising.
Citations
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Journal Article•
TL;DR: The results show that the VST combined with the EZW is an optimistic technique for better Poisson denoising, even for low intensity images.
Abstract: The variance stabilizing transform (VST) is a frequently used denoising algorithm which is applicable in many areas like medical, defence and commercial applications. As the image data transmission issues are raising day by day we are adopting so many new methods as the combinations of VST algorithm. The embedded zero tree wavelet algorithm (EZW) is basically used for compression techniques from the past years. As its accuracy level is high we are going to embed this technique to current VST algorithm. Therefore, it is suitable for denoising applications. Combining VST with the EZW leads to good Poisson image denoising algorithms, even for low intensity images. We present a simple approach to achieve this and demonstrate some simulation results. The results show that the VST combined with the EZW is an optimistic technique for better Poisson denoising.
01 Oct 2012
TL;DR: The Transformada Curvelet can also be used for filtrado of mapas de fase as discussed by the authors, e.g., for the elimination of ruido in a mapa of fase.
Abstract: Se presenta una nueva metodoloǵıa de filtrado de mapas de fase utilizando la Transformada Curvelet. El desenvolvimiento de fase es requerido para construir un mapa de fase continuo a partir de un mapa de fase envuelto en áreas como la interferometŕıa óptica, la Interferometŕıa de Radar de Apertura Sintética y la Resonancia Magnética. Sin embargo, al momento de la adquisición de la imagen la presencia de ruido es inevitable. El proceso de la eliminación de ruido se puede convertir en una tarea dif́ıcil de realizar y se requiere de un filtrado para tratar de eliminar la mayor cantidad de información no deseada. La principal ventaja de nuestra propuesta utilizando la transformada Curvelet es que no se requiere una previa estimación de la orientación del mapa de fase; como es el caso de las técnicas de filtrado direccional recientemente reportadas. La técnica presentada, la cual se encuentra aún en desarrollo presenta buenas expectativas en su desempeño para la eliminación de ruido y con un tiempo de procesamiento razonable.
References
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TL;DR: In this paper, a different approach to problems of multiple significance testing is presented, which calls for controlling the expected proportion of falsely rejected hypotheses -the false discovery rate, which is equivalent to the FWER when all hypotheses are true but is smaller otherwise.
Abstract: SUMMARY The common approach to the multiplicity problem calls for controlling the familywise error rate (FWER). This approach, though, has faults, and we point out a few. A different approach to problems of multiple significance testing is presented. It calls for controlling the expected proportion of falsely rejected hypotheses -the false discovery rate. This error rate is equivalent to the FWER when all hypotheses are true but is smaller otherwise. Therefore, in problems where the control of the false discovery rate rather than that of the FWER is desired, there is potential for a gain in power. A simple sequential Bonferronitype procedure is proved to control the false discovery rate for independent test statistics, and a simulation study shows that the gain in power is substantial. The use of the new procedure and the appropriateness of the criterion are illustrated with examples.
83,420 citations
"Poisson noise removal from images u..." refers methods in this paper
...Then we perform hard thresholding to detect the significant FDCT coefficients for a prespecified false detection rate (FDR) [10]....
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...A few important specifications of our experiments are: five scales and {8, 16, 16, 32, 32} directions (from coarse to fine) for the MSMD transforms except the first generation curvelet, for which four scales are used; the FDCT implementation based on frequency wrapping; images of size 256×256 and intensity in the range [0.9, 20]; and FDR = 10−3 and number of iterations in HSD = 5 [3]....
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TL;DR: This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform in two and three dimensions, based on unequally spaced fast Fourier transforms, while the second is based on the wrapping of specially selected Fourier samples.
Abstract: This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform in two and three dimensions. The first digital transformation is based on unequally spaced fast Fourier transforms, while the second is based on the wrapping of specially selected Fourier samples. The two implementations essentially differ by the choice of spatial grid used to translate curvelets at each scale and angle. Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an orientation parameter, and a spatial location parameter. And both implementations are fast in the sense that they run in O(n^2 log n) flops for n by n Cartesian arrays; in addition, they are also invertible, with rapid inversion algorithms of about the same complexity. Our digital transformations improve upon earlier implementations—based upon the first generation of curvelets—in the sense that they are conceptually simpler, faster, and far less redundant. The software CurveLab, which implements both transforms presented in this paper, is available at http://www.curvelet.org.
2,603 citations
"Poisson noise removal from images u..." refers methods in this paper
...Two separate digital implementation of the second generation curvelets were proposed in [7]....
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...In this paper, we propose a strategy to combine the MS-VST with the second generation curvelet transform, specifically the fast discrete curvelet transform (FDCT) [7]....
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...Similarly, when compared to the NSCT, the FDCT is less redundant and has more frequency resolution and better directionality properties [7]....
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TL;DR: This paper proposes a design framework based on the mapping approach, that allows for a fast implementation based on a lifting or ladder structure, and only uses one-dimensional filtering in some cases.
Abstract: In this paper, we develop the nonsubsampled contourlet transform (NSCT) and study its applications. The construction proposed in this paper is based on a nonsubsampled pyramid structure and nonsubsampled directional filter banks. The result is a flexible multiscale, multidirection, and shift-invariant image decomposition that can be efficiently implemented via the a trous algorithm. At the core of the proposed scheme is the nonseparable two-channel nonsubsampled filter bank (NSFB). We exploit the less stringent design condition of the NSFB to design filters that lead to a NSCT with better frequency selectivity and regularity when compared to the contourlet transform. We propose a design framework based on the mapping approach, that allows for a fast implementation based on a lifting or ladder structure, and only uses one-dimensional filtering in some cases. In addition, our design ensures that the corresponding frame elements are regular, symmetric, and the frame is close to a tight one. We assess the performance of the NSCT in image denoising and enhancement applications. In both applications the NSCT compares favorably to other existing methods in the literature
1,900 citations
"Poisson noise removal from images u..." refers background or methods in this paper
...In fact, the NMISE is very close to that of the NSCT and better than that of the SNSCT....
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...L Donoho, and the NSCT Toolbox prepared by A. L. Cunha, for conducting the simulations given in this paper....
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...4d), which is a semi nonsubsampled version of the NSCT [5], [6]....
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...The MS-VST+NSCT (Fig....
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...Recently, we have proposed [5] an MS-VST Poisson image denoising algorithm, based on the nonsubsampled contourlet transform (NSCT) [6]....
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01 Jan 2000
TL;DR: The basic issues of efficient m-term approximation, the construction of efficient adaptive representation, theConstruction of the curvelet frame, and a crude analysis of the performance of curvelet schemes are explained.
Abstract: : It is widely believed that to efficiently represent an otherwise smooth object with discontinuities along edges, one must use an adaptive representation that in some sense 'tracks' the shape of the discontinuity set. This folk-belief - some would say folk-theorem - is incorrect. At the very least, the possible quantitative advantage of such adaptation is vastly smaller than commonly believed. We have recently constructed a tight frame of curvelets which provides stable, efficient, and near-optimal representation of otherwise smooth objects having discontinuities along smooth curves. By applying naive thresholding to the curvelet transform of such an object, one can form m-term approximations with rate of L(sup 2) approximation rivaling the rate obtainable by complex adaptive schemes which attempt to track' the discontinuity set. In this article we explain the basic issues of efficient m-term approximation, the construction of efficient adaptive representation, the construction of the curvelet frame, and a crude analysis of the performance of curvelet schemes.
1,633 citations
"Poisson noise removal from images u..." refers background or methods in this paper
...The curvelet transforms, first generation [4] as well as...
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...In [3], MS-VSTs were developed for the wavelet, ridgelet and first generation curvelet transforms [4], and the curvelet was shown to yield the best performance....
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TL;DR: This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C2 edges.
Abstract: This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C 2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle-shaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2 -j , each element has an envelope that is aligned along a ridge of length 2 -j/2 and width 2 -j . We prove that curvelets provide an essentially optimal representation of typical objects f that are C 2 except for discontinuities along piecewise C 2 curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the n-term partial reconstruction f C n obtained by selecting the n largest terms in the curvelet series obeys ∥f - f C n ∥ 2 L2 ≤ C . n -2 . (log n) 3 , n → ∞. This rate of convergence holds uniformly over a class of functions that are C 2 except for discontinuities along piecewise C 2 curves and is essentially optimal. In comparison, the squared error of n-term wavelet approximations only converges as n -1 as n → ∞, which is considerably worse than the optimal behavior.
1,567 citations
"Poisson noise removal from images u..." refers background in this paper
...second generation [8], provide a near-optimal sparse representation for images having discontinuities along C(2) (twice differentiable) curves....
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