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14
The curvelet transform for image denoising
 IEEE TRANS. IMAGE PROCESS
, 2002
"... We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform [2] and the curvelet transform [6], [5]. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A cen ..."
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Cited by 284 (38 self)
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We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform [2] and the curvelet transform [6], [5]. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourierdomain computation of an approximate digital Radon transform. We introduce a very simple interpolation in Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudopolar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of à trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with “state of the art ” techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including treebased Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than waveletbased reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement.
Multiscale Entropy Filtering
, 1999
"... We present in this paper a new method for filtering an image, based on a new definition of its entropy. A large number of examples illustrate the results. Comparisons are performed with other waveletbased methods. ..."
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Cited by 15 (8 self)
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We present in this paper a new method for filtering an image, based on a new definition of its entropy. A large number of examples illustrate the results. Comparisons are performed with other waveletbased methods.
WaveletBased Combined Signal Filtering and Prediction
"... Abstract — We survey a number of applications of the wavelet transform in time series prediction. We show how multiresolution prediction can capture shortrange and longterm dependencies with only a few parameters to be estimated. We then develop a new multiresolution methodology for combined noise ..."
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Cited by 7 (0 self)
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Abstract — We survey a number of applications of the wavelet transform in time series prediction. We show how multiresolution prediction can capture shortrange and longterm dependencies with only a few parameters to be estimated. We then develop a new multiresolution methodology for combined noise filtering and prediction, based on an approach which is similar to the Kalman filter. Based on considerable experimental assessment, we demonstrate the powerfulness of this methodology. Index Terms — Wavelet transform, filtering, forecasting, resolution, scale, autoregression, time series, model,
Astronomical Image and Signal Processing  Looking at Noise, Information, and Scale
, 2001
"... is a perfect tool for astronomical images, typically composed of objects of different size, such as stars, galaxies, interstellar clouds, clusters of galaxies, etc. It is clear that if a multiscale representation is capable of representing information which serves to decrease the entropy, it is al ..."
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Cited by 6 (1 self)
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is a perfect tool for astronomical images, typically composed of objects of different size, such as stars, galaxies, interstellar clouds, clusters of galaxies, etc. It is clear that if a multiscale representation is capable of representing information which serves to decrease the entropy, it is also useful for detection of the different components contained in the image. For instance, it has been shown that an image can be completely (or almost completely) reconstructed from its multiscale edges [31], i.e., the edges detected at each scale. So the WT is useful for both kinds of information measurement, the deterministic kind, which indicates what is in the image and where, and the statistical one, which gives a quantity relative to the amount of information. Another aspect of astronomical images, but also of many other classes of image, is the presence of noise. The observed signal can be decomposed into two components: the useful signal (or signal of interest) and the noise. It is
Multiscale Entropy
, 2000
"... Multiscale entropy is based on the wavelet transform and noise modeling. It is a means of measuring information in a data set. It has been recently developed and has been applied successfully to signal and image filtering. We describe in this paper how it can be used for deconvolution, background fl ..."
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Cited by 5 (5 self)
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Multiscale entropy is based on the wavelet transform and noise modeling. It is a means of measuring information in a data set. It has been recently developed and has been applied successfully to signal and image filtering. We describe in this paper how it can be used for deconvolution, background fluctuation analysis, and astronomical image content analysis. A range of examples illustrates the results. Index Terms Wavelet transform, filtering, deconvolution, image restoration, Bayesian estimation, entropy Number of pages: 34 Number of figures: 11 2 List of Figures 1 Lena image (left) and the same data distributed differently (right). These two images have the same entropy, using any of the standard entropy definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 Corrected wavelet coefficient versus the wavelet coefficient with different ff values (from the top curve to the bottom one, ff is respectively equal to 0,0.1,0.5, 1, 2, 5,10). . . . . ...
Multiscale Entropy for Semantic Description of Images and Signals
, 2000
"... Multiscale entropy is based on the wavelet transform and noise modeling. It is a means of measuring information in a data set, which takes into account important properties of the data which are related to content. We describe in this paper how it can be used for signal and image filtering and decon ..."
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Cited by 5 (3 self)
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Multiscale entropy is based on the wavelet transform and noise modeling. It is a means of measuring information in a data set, which takes into account important properties of the data which are related to content. We describe in this paper how it can be used for signal and image filtering and deconvolution. We then proceed to the use of multiscale entropy for description of image content. We pursue two directions of enquiry: determining whether signal is present in the image or not, possibly at or below the image's noise level; and how multiscale entropy is very well correlated with the image's content in the case of astronomical stellar elds. Knowing that multiscale entropy represents well the content of the image, we finally use it to de ne the optimal compression rate of the image. In all cases, a range of examples illustrate these new results.
Image Processing through Multiscale Analysis and Measurement Noise Modeling
 Statistics and Computing
, 2000
"... We describe a range of powerful multiscale analysis methods. We also focus on the pivotal issue of measurement noise in the physical sciences. From multiscale analysis and noise modeling, we develop a comprehensive methodology for data analysis of 2D images, 1D signals (or spectra), and point pat ..."
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Cited by 1 (0 self)
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We describe a range of powerful multiscale analysis methods. We also focus on the pivotal issue of measurement noise in the physical sciences. From multiscale analysis and noise modeling, we develop a comprehensive methodology for data analysis of 2D images, 1D signals (or spectra), and point pattern data. Noise modeling is based on the following: (i) multiscale transforms, including wavelet transforms; (ii) a data structure termed the multiresolution support; and (iii) multiple scale significance testing. The latter two aspects serve to characterize signal with respect to noise. The data analysis objectives we deal with include noise filtering and scale decomposition for visualization or feature detection. 1 Introduction We will describe a recentlyproposed methodology for data analysis in astronomy, medicine, Earth observation and other fields. We will survey results obtained. This methodology takes account of measurement noise in our input data. Noise models include Gaussi...
Image Filtering by Combining Multiple Vision Models
, 1999
"... We compare dierent strategies for data ltering in wavelet space. Filtering strategy concerns both the wavelet transform algorithm used, and the processing carried out on the wavelet coecients. We present and analyze the results obtained from a set of around two hundred ltered images. Then we show th ..."
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Cited by 1 (0 self)
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We compare dierent strategies for data ltering in wavelet space. Filtering strategy concerns both the wavelet transform algorithm used, and the processing carried out on the wavelet coecients. We present and analyze the results obtained from a set of around two hundred ltered images. Then we show that the combination of dierent ltering methods improves the quality of the ltering. 1 1 Introduction Observed data Y in the physical sciences are generally corrupted by noise, which is often additive and which follows in many cases a Gaussian distribution, a Poisson distribution, or a combination of both. Many methods have been developed during this century in order to remove the corrupted part of the signal. Each class of methods consists of considering a vision model, and using this a priori knowledge to make some assumptions about the noise in order to remove it. For instance, Fourier based ltering methods (e.g. Wiener ltering) apodizes certain frequencies where the signaltono...
Distributed Visual Information Management in Astronomy
, 2002
"... ometry (for example, accumulated flux data) oreffecti0HU detect miect objects. Havit brit descri.) the field'ssciOflx)0 needs, we can now look at how astronomers are expli88)0 usiR resolutix and scale toassifi data (idata tabular, and other) handliO) These new DISTRIBUTED VISUAL INFORMATION MANAGE ..."
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Cited by 1 (0 self)
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ometry (for example, accumulated flux data) oreffecti0HU detect miect objects. Havit brit descri.) the field'ssciOflx)0 needs, we can now look at how astronomers are expli88)0 usiR resolutix and scale toassifi data (idata tabular, and other) handliO) These new DISTRIBUTED VISUAL INFORMATION MANAGEMENT IN ASTRONOMY Resolution scale is central to largeimage visualization, offering one way to address astronomers' need to access and retrieve data. In addition, multipleresolution information and entropy are closely related to compression rate, all three of which are related to the relevance and importance of information. FIONN MURTAGH Queen'sUnivwH581 , Belfast JEANLUC STARCK French Atomic Energy Commission MIREILLE LOUYS Univ11754 Louis Pasteur 15219615/02/$17.00 2002 IEEE FEATURE A STRONOMY vantagepoiag help astronomers address the field's sciR8)0fl needs. We first look at how resolutifi and scale arei)xOOHx)0flfi iO8 sci)iHx ici compressi0fl Compressifl i tim to i) formati
Wavelets and Multiscale Transforms in Massive Data Sets
, 1999
"... With the requirements of scientific and medical image database support in mind, we describe a range of useful technologies for storage, transmission and display. These new technologies are all based on discrete wavelet or related multiscale transforms. Other important issues include noise modeling, ..."
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With the requirements of scientific and medical image database support in mind, we describe a range of useful technologies for storage, transmission and display. These new technologies are all based on discrete wavelet or related multiscale transforms. Other important issues include noise modeling, and the innovative use of entropy for information characterization.