Multiresolution analysis using wavelet, ridgelet, and curvelet transforms for medical image segmentation

Copyright @ 2011 Shadi AlZubi et al. This article has been made available through the Brunel Open Access Publishing Fund.The experimental study presented in this paper is aimed at the development of an automatic image segmentation system for classifying region of interest (ROI) in medical images which are obtained from different medical scanners such as PET, CT, or MRI. Multiresolution analysis (MRA) using wavelet, ridgelet, and curvelet transforms has been used in the proposed segmentation system. It is particularly a challenging task to classify cancers in human organs in scanners output using shape or gray-level information; organs shape changes throw different slices in medical stack and the gray-level intensity overlap in soft tissues. Curvelet transform is a new extension of wavelet and ridgelet transforms which aims to deal with interesting phenomena occurring along curves. Curvelet transforms has been tested on medical data sets, and results are compared with those obtained from the other transforms. Tests indicate that using curvelet significantly improves the classification of abnormal tissues in the scans and reduce the surrounding noise

3D multiresolution statistical approaches for accelerated medical image and volume segmentation

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Medical volume segmentation got the attraction of many researchers; therefore, many techniques have been implemented in terms of medical imaging including segmentations and other imaging processes. This research focuses on an implementation of segmentation system which uses several techniques together or on their own to segment medical volumes, the system takes a stack of 2D slices or a full 3D volumes acquired from medical scanners as a data input. Two main approaches have been implemented in this research for segmenting medical volume which are multi-resolution analysis and statistical modeling. Multi-resolution analysis has been mainly employed in this research for extracting the features. Higher dimensions of discontinuity (line or curve singularity) have been extracted in medical images using a modified multi-resolution analysis transforms such as ridgelet and curvelet transforms. The second implemented approach in this thesis is the use of statistical modeling in medical image segmentation; Hidden Markov models have been enhanced here to segment medical slices automatically, accurately, reliably and with lossless results. But the problem with using Markov models here is the computational time which is too long. This has been addressed by using feature reduction techniques which has also been implemented in this thesis. Some feature reduction and dimensionality reduction techniques have been used to accelerate the slowest block in the proposed system. This includes Principle Components Analysis, Gaussian Pyramids and other methods. The feature reduction techniques have been employed efficiently with the 3D volume segmentation techniques such as 3D wavelet and 3D Hidden Markov models. The system has been tested and validated using several procedures starting at a comparison with the predefined results, crossing the specialists’ validations, and ending by validating the system using a survey filled by the end users explaining the techniques and the results. This concludes that Markovian models segmentation results has overcome all other techniques in most patients’ cases. Curvelet transform has been also proved promising segmentation results; the end users rate it better than Markovian models due to the long time required with Hidden Markov models

Face recognition in 2D and 2.5D using ridgelets and photometric stereo

A new technique for face recognition - Ridgefaces - is presented. The method combines the well-known Fisherface method with the ridgelet transform and high-speed Photometric Stereo (PS). The paper first derives ridgelet projections for 2D/2.5D face images before the Fisherface approach is used to reduce the dimensionality and increase the spread of the resulting feature vectors. The ridgelet transform is attractive because it is efficient at extracting highly discriminating low-frequency directional features. Best recognition is obtained when Ridgefaces is performed on surface normals acquired from PS, although good results are also found using standard 2D images and PS-derived albedo maps. © 2012 Elsevier Ltd. All rights reserved

Finite Pseudo-Differential Operators, Localization Operators for Curvelet and Ridgelet Transforms

Pseudo-differential operators can be built from the Fourier transform. However, besides the difficult problems in proving convergence and L^2-boundedness, the problem of finding eigenvalues is notoriously difficult. Finite analogs of pseudo-differential operators are desirable and indeed are constructed in this dissertation. Energized by the success of the Fourier transform and wavelet transforms, the last two decades saw the rapid developments of new tools in time-frequency analysis, such as ridgelet transforms and curvelet transforms, to deal with higher dimensional signals. Both curvelet transforms and ridgelet transforms give the time/position-frequency representations of signals that involve the interactions of translation, rotation and dilation, and they can be ideally used to represent signals and images with discontinuities lying on a curve such as images with edges. Given the resolution of the identity formulas for these two transforms, localization operators on them are constructed. The later part of this dissertation is to investigate the L^2-boundedness of the localization operators for curvelet transforms and ridgelet transforms, as well as their trace properties

Directional edge and texture representations for image processing

An efficient representation for natural images is of fundamental importance in image processing and analysis. The commonly used separable transforms such as wavelets axe not best suited for images due to their inability to exploit directional regularities such as edges and oriented textural patterns; while most of the recently proposed directional schemes cannot represent these two types of features in a unified transform. This thesis focuses on the development of directional representations for images which can capture both edges and textures in a multiresolution manner. The thesis first considers the problem of extracting linear features with the multiresolution Fourier transform (MFT). Based on a previous MFT-based linear feature model, the work extends the extraction method into the situation when the image is corrupted by noise. The problem is tackled by the combination of a "Signal+Noise" frequency model, a refinement stage and a robust classification scheme. As a result, the MFT is able to perform linear feature analysis on noisy images on which previous methods failed. A new set of transforms called the multiscale polar cosine transforms (MPCT) are also proposed in order to represent textures. The MPCT can be regarded as real-valued MFT with similar basis functions of oriented sinusoids. It is shown that the transform can represent textural patches more efficiently than the conventional Fourier basis. With a directional best cosine basis, the MPCT packet (MPCPT) is shown to be an efficient representation for edges and textures, despite its high computational burden. The problem of representing edges and textures in a fixed transform with less complexity is then considered. This is achieved by applying a Gaussian frequency filter, which matches the disperson of the magnitude spectrum, on the local MFT coefficients. This is particularly effective in denoising natural images, due to its ability to preserve both types of feature. Further improvements can be made by employing the information given by the linear feature extraction process in the filter's configuration. The denoising results compare favourably against other state-of-the-art directional representations

A discrete approach to monogenic analysis through Radon transform

8 pagesInternational audienceMonogenic analysis is gaining interest in the image processing community as a true signal processing tool for 2D signals. Unfortunately, it is only defined in the continuous case. We address this issue by proposing an innovative scheme that uses a discrete Radon transform based on discrete geometry. Radon domain signal processing and monogenic analysis is studied and performance is shown to be equivalent to the usual FFT-based algorithms. The advantage is that extensions to filterbanks and to higher dimensions are facilitated, thanks to the perfect invertibility and computational simplicity of the used Radon algorithm

Quantum Ridgelet Transform: Winning Lottery Ticket of Neural Networks with Quantum Computation

A significant challenge in the field of quantum machine learning (QML) is to establish applications of quantum computation to accelerate common tasks in machine learning such as those for neural networks. Ridgelet transform has been a fundamental mathematical tool in the theoretical studies of neural networks, but the practical applicability of ridgelet transform to conducting learning tasks was limited since its numerical implementation by conventional classical computation requires an exponential runtime $\exp(O(D))$ as data dimension $D$ increases. To address this problem, we develop a quantum ridgelet transform (QRT), which implements the ridgelet transform of a quantum state within a linear runtime $O(D)$ of quantum computation. As an application, we also show that one can use QRT as a fundamental subroutine for QML to efficiently find a sparse trainable subnetwork of large shallow wide neural networks without conducting large-scale optimization of the original network. This application discovers an efficient way in this regime to demonstrate the lottery ticket hypothesis on finding such a sparse trainable neural network. These results open an avenue of QML for accelerating learning tasks with commonly used classical neural networks.Comment: 27 pages, 4 figure

Analysis of the Spatial Distribution of Galaxies by Multiscale Methods

Galaxies are arranged in interconnected walls and filaments forming a cosmic web encompassing huge, nearly empty, regions between the structures. Many statistical methods have been proposed in the past in order to describe the galaxy distribution and discriminate the different cosmological models. We present in this paper results relative to the use of new statistical tools using the 3D isotropic undecimated wavelet transform, the 3D ridgelet transform and the 3D beamlet transform. We show that such multiscale methods produce a new way to measure in a coherent and statistically reliable way the degree of clustering, filamentarity, sheetedness, and voidedness of a datasetComment: 26 pages, 20 figures. Submitted to EURASIP Journal on Applied Signal Processing (special issue on "Applications of Signal Processing in Astrophysics and Cosmology"

Fast Fourier Transform at Nonequispaced Nodes and Applications

The direct computation of the discrete Fourier transform at arbitrary nodes requires O(NM) arithmetical operations, too much for practical purposes. For equally spaced nodes the computation can be done by the well known fast Fourier transform (FFT) in only O(N log N) arithmetical operations. Recently, the fast Fourier transform for nonequispaced nodes (NFFT) was developed for the fast approximative computation of the above sums in only O(N log N + M log 1/e), where e denotes the required accuracy. The principal topics of this thesis are generalizations and applications of the NFFT. This includes the following subjects: - Algorithms for the fast approximative computation of the discrete cosine and sine transform at nonequispaced nodes are developed by applying fast trigonometric transforms instead of FFTs. - An algorithm for the fast Fourier transform on hyperbolic cross points with nonequispaced spatial nodes in 2 and 3 dimensions based on the NFFT and an appropriate partitioning of the hyperbolic cross is proposed. - A unified linear algebraic approach to recent methods for the fast computation of matrix-vector-products with special dense matrices, namely the fast multipole method, fast mosaic-skeleton approximation and H-matrix arithmetic, is given. Moreover, the NFFT-based summation algorithm by Potts and Steidl is further developed and simplified by using algebraic polynomials instead of trigonometric polynomials and the error estimates are improved. - A new algorithm for the characterization of engineering surface topographies with line singularities is proposed. It is based on hard thresholding complex ridgelet coefficients combined with total variation minimization. The discrete ridgelet transform is designed by first using a discrete Radon transform based on the NFFT and then applying a dual-tree complex wavelet transform. - A new robust local scattered data approximation method is introduced. It is an advancement of the moving least squares approximation (MLS) and generalizes an approach of van den Boomgard and van de Weijer to scattered data. In particular, the new method is space and data adaptive

Recovering edges in ill-posed inverse problems: optimality of curvelet frames

We consider a model problem of recovering a function $f(x_1,x_2)$ from noisy Radon data. The function $f$ to be recovered is assumed smooth apart from a discontinuity along a $C^2$ curve, that is, an edge. We use the continuum white-noise model, with noise level $\varepsilon$. Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model mean squared errors (MSEs) that tend to zero with noise level $\varepsilon$ only as $O(\varepsilon^{1/2})$ as $\varepsilon\to 0$. A recent innovation--nonlinear shrinkage in the wavelet domain--visually improves edge sharpness and improves MSE convergence to $O(\varepsilon^{2/3})$. However, as we show here, this rate is not optimal. In fact, essentially optimal performance is obtained by deploying the recently-introduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build "curvelet shrinkage" estimators based on thresholding of the noisy curvelet coefficients. In effect, the estimator detects edges at certain locations and orientations in the Radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain. We prove that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE $O(\varepsilon^{4/5})$ as noise level $\varepsilon\to 0$. This rate of convergence holds uniformly over a class of functions which are $C^2$ except for discontinuities along $C^2$ curves, and (except for log terms) is the minimax rate for that class. Our approach is an instance of a general strategy which should apply in other inverse problems; we sketch a deconvolution example