Showing papers on "Multiresolution analysis published in 2005"

The contourlet transform: an efficient directional multiresolution image representation

Abstract: The limitations of commonly used separable extensions of one-dimensional transforms, such as the Fourier and wavelet transforms, in capturing the geometry of image edges are well known. In this paper, we pursue a "true" two-dimensional transform that can capture the intrinsic geometrical structure that is key in visual information. The main challenge in exploring geometry in images comes from the discrete nature of the data. Thus, unlike other approaches, such as curvelets, that first develop a transform in the continuous domain and then discretize for sampled data, our approach starts with a discrete-domain construction and then studies its convergence to an expansion in the continuous domain. Specifically, we construct a discrete-domain multiresolution and multidirection expansion using nonseparable filter banks, in much the same way that wavelets were derived from filter banks. This construction results in a flexible multiresolution, local, and directional image expansion using contour segments, and, thus, it is named the contourlet transform. The discrete contourlet transform has a fast iterated filter bank algorithm that requires an order N operations for N-pixel images. Furthermore, we establish a precise link between the developed filter bank and the associated continuous-domain contourlet expansion via a directional multiresolution analysis framework. We show that with parabolic scaling and sufficient directional vanishing moments, contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves. Finally, we show some numerical experiments demonstrating the potential of contourlets in several image processing applications.

Sparse geometric image representations with bandelets

Abstract: This paper introduces a new class of bases, called bandelet bases, which decompose the image along multiscale vectors that are elongated in the direction of a geometric flow. This geometric flow indicates directions in which the image gray levels have regular variations. The image decomposition in a bandelet basis is implemented with a fast subband-filtering algorithm. Bandelet bases lead to optimal approximation rates for geometrically regular images. For image compression and noise removal applications, the geometric flow is optimized with fast algorithms so that the resulting bandelet basis produces minimum distortion. Comparisons are made with wavelet image compression and noise-removal algorithms.

A comparative analysis of image fusion methods

Abstract: There are many image fusion methods that can be used to produce high-resolution multispectral images from a high-resolution panchromatic image and low-resolution multispectral images Starting from the physical principle of image formation, this paper presents a comprehensive framework, the general image fusion (GIF) method, which makes it possible to categorize, compare, and evaluate the existing image fusion methods Using the GIF method, it is shown that the pixel values of the high-resolution multispectral images are determined by the corresponding pixel values of the low-resolution panchromatic image, the approximation of the high-resolution panchromatic image at the low-resolution level Many of the existing image fusion methods, including, but not limited to, intensity-hue-saturation, Brovey transform, principal component analysis, high-pass filtering, high-pass modulation, the a/spl grave/ trous algorithm-based wavelet transform, and multiresolution analysis-based intensity modulation (MRAIM), are evaluated and found to be particular cases of the GIF method The performance of each image fusion method is theoretically analyzed based on how the corresponding low-resolution panchromatic image is computed and how the modulation coefficients are set An experiment based on IKONOS images shows that there is consistency between the theoretical analysis and the experimental results and that the MRAIM method synthesizes the images closest to those the corresponding multisensors would observe at the high-resolution level

Sparse multidimensional representation using shearlets

Abstract: In this paper we describe a new class of multidimensional representation systems, called shearlets. They are obtained by applying the actions of dilation, shear transformation and translation to a fixed function, and exhibit the geometric and mathematical properties, e.g., directionality, elongated shapes, scales, oscillations, recently advocated by many authors for sparse image processing applications. These systems can be studied within the framework of a generalized multiresolution analysis. This approach leads to a recursive algorithm for the implementation of these systems, that generalizes the classical cascade algorithm.

Advances in multiresolution for geometric modelling

Abstract: Compression.- Recent Advances in Compression of 3D Meshes.- Shape Compression using Spherical Geometry Images.- Data Structures.- A Survey on Data Structures for Level-of-Detail Models.- An Algorithm for Decomposing Multi-dimensional Non-manifold Objects into Nearly Manifold Components.- Encoding Level-of-Detail Tetrahedral Meshes.- Multi-Scale Geographic Maps.- Modelling.- Constrained Multiresolution Geometric Modelling.- Multi-scale and Adaptive CS-RBFs for Shape Reconstruction from Clouds of Points.- Parameterization.- Surface Parameterization: a Tutorial and Survey.- Variations on Angle Based Flattening.- Subdivision.- Recent Progress in Subdivision: a Survey.- Optimising 3D Triangulations: Improving the Initial Triangulation for the Butterfly Subdivision Scheme.- Simple Computation of the Eigencomponents of a Subdivision Matrix in the Fourier Domain.- Subdivision as a Sequence of Sampled Cp Surfaces.- Reverse Subdivision.- $$\sqrt 5 $$ -subdivision.- Geometrically Controlled 4-Point Interpolatory Schemes.- Thinning.- Adaptive Thinning for Terrain Modelling and Image Compression.- Simplification of Topologically Complex Assemblies.- Topology Preserving Thinning of Vector Fields on Triangular Meshes.- Wavelets.- Periodic and Spline Multiresolution Analysis and the Lifting Scheme.- Nonstationary Sibling Wavelet Frames on Bounded Intervals: the Duality Relation.- Haar Wavelets on Spherical Triangulations.

A novel algorithm for ship detection in SAR imagery based on the wavelet transform

Abstract: Carrying out an effective control of fishing activities is essential to guarantee a sustainable exploitation of sea resources. Nevertheless, as the regulated areas are extended, they are difficult and time consuming to monitor by means of traditional reconnaissance methods such as planes and patrol vessels. On the contrary, satellite-based synthetic aperture radar (SAR) provides a powerful surveillance capability allowing the observation of broad expanses, independently from weather effects and from the day and night cycle. Unfortunately, the automatic interpretation of SAR images is often complicated, even though undetected targets are sometimes visible by eye. Attending to these particular circumstances, a novel approach for ship detection is proposed based on the analysis of SAR images by means of the discrete wavelet transform. The exposed method takes advantage of the difference of statistical behavior among the ships and the surrounding sea, interpreting the information through the wavelet coefficients in order to provide a more reliable detection. The analysis of the detection performance over both simulated and real images confirms the robustness of the proposed algorithm.

A bound optimization approach to wavelet-based image deconvolution

Abstract: We address the problem of image deconvolution under I/sub p/ norm (and other) penalties expressed in the wavelet domain. We propose an algorithm based on the bound optimization approach; this approach allows deriving EM-type algorithms without using the concept of missing/hidden data. The algorithm has provable monotonicity both with orthogonal or redundant wavelet transforms. We also derive bounds on the l/sub p/ norm penalties to obtain closed form update equations for any p /spl isin/ [0, 2]. Experimental results show that the proposed method achieves state-of-the-art performance.

Wavelet noise

Abstract: Noise functions are an essential building block for writing procedural shaders in 3D computer graphics. The original noise function introduced by Ken Perlin is still the most popular because it is simple and fast, and many spectacular images have been made with it. Nevertheless, it is prone to problems with aliasing and detail loss. In this paper we analyze these problems and show that they are particularly severe when 3D noise is used to texture a 2D surface. We use the theory of wavelets to create a new class of simple and fast noise functions that avoid these problems.

3-D deformable image registration: a topology preservation scheme based on hierarchical deformation models and interval analysis optimization

Abstract: This paper deals with topology preservation in three-dimensional (3-D) deformable image registration. This work is a nontrivial extension of , which addresses the case of two-dimensional (2-D) topology preserving mappings. In both cases, the deformation map is modeled as a hierarchical displacement field, decomposed on a multiresolution B-spline basis. Topology preservation is enforced by controlling the Jacobian of the transformation. Finding the optimal displacement parameters amounts to solving a constrained optimization problem: The residual energy between the target image and the deformed source image is minimized under constraints on the Jacobian. Unlike the 2-D case, in which simple linear constraints are derived, the 3-D B-spline-based deformable mapping yields a difficult (until now, unsolved) optimization problem. In this paper, we tackle the problem by resorting to interval analysis optimization techniques. Care is taken to keep the computational burden as low as possible. Results on multipatient 3-D MRI registration illustrate the ability of the method to preserve topology on the continuous image domain.

Isotropic polyharmonic B-splines: scaling functions and wavelets

Abstract: In this paper, we use polyharmonic B-splines to build multidimensional wavelet bases. These functions are nonseparable, multidimensional basis functions that are localized versions of radial basis functions. We show that Rabut's elementary polyharmonic B-splines do not converge to a Gaussian as the order parameter increases, as opposed to their separable B-spline counterparts. Therefore, we introduce a more isotropic localization operator that guarantees this convergence, resulting into the isotropic polyharmonic B-splines. Next, we focus on the two-dimensional quincunx subsampling scheme. This configuration is of particular interest for image processing because it yields a finer scale progression than the standard dyadic approach. However, up until now, the design of appropriate filters for the quincunx scheme has mainly been done using the McClellan transform. In our approach, we start from the scaling functions, which are the polyharmonic B-splines and, as such, explicitly known, and we derive a family of polyharmonic spline wavelets corresponding to different flavors of the semi-orthogonal wavelet transform; e.g., orthonormal, B-spline, and dual. The filters are automatically specified by the scaling relations satisfied by these functions. We prove that the isotropic polyharmonic B-spline wavelet converges to a combination of four Gabor atoms, which are well separated in the frequency domain. We also show that these wavelets are nearly isotropic and that they behave as an iterated Laplacian operator at low frequencies. We describe an efficient fast Fourier transform-based implementation of the discrete wavelet transform based on polyharmonic B-splines.

Use of multiresolution wavelet feature pyramids for automatic registration of multisensor imagery

Abstract: The problem of image registration, or the alignment of two or more images representing the same scene or object, has to be addressed in various disciplines that employ digital imaging. In the area of remote sensing, just like in medical imaging or computer vision, it is necessary to design robust, fast, and widely applicable algorithms that would allow automatic registration of images generated by various imaging platforms at the same or different times and that would provide subpixel accuracy. One of the main issues that needs to be addressed when developing a registration algorithm is what type of information should be extracted from the images being registered, to be used in the search for the geometric transformation that best aligns them. The main objective of this paper is to evaluate several wavelet pyramids that may be used both for invariant feature extraction and for representing images at multiple spatial resolutions to accelerate registration. We find that the bandpass wavelets obtained from the steerable pyramid due to Simoncelli performs best in terms of accuracy and consistency, while the low-pass wavelets obtained from the same pyramid give the best results in terms of the radius of convergence. Based on these findings, we propose a modification of a gradient-based registration algorithm that has recently been developed for medical data. We test the modified algorithm on several sets of real and synthetic satellite imagery.

Feature-based wavelet shrinkage algorithm for image denoising

Abstract: A selective wavelet shrinkage algorithm for digital image denoising is presented. The performance of this method is an improvement upon other methods proposed in the literature and is algorithmically simple for large computational savings. The improved performance and computational speed of the proposed wavelet shrinkage algorithm is presented and experimentally compared with established methods. The denoising method incorporated in the proposed algorithm involves a two-threshold validation process for real-time selection of wavelet coefficients. The two-threshold criteria selects wavelet coefficients based on their absolute value, spatial regularity, and regularity across multiresolution scales. The proposed algorithm takes image features into consideration in the selection process. Statistically, most images have regular features resulting in connected subband coefficients. Therefore, the resulting subbands of wavelet transformed images in large part do not contain isolated coefficients. In the proposed algorithm, coefficients are selected due to their magnitude, and only a subset of those selected coefficients which exhibit a spatially regular behavior remain for image reconstruction. Therefore, two thresholds are used in the coefficient selection process. The first threshold is used to distinguish coefficients of large magnitude and the second is used to distinguish coefficients of spatial regularity. The performance of the proposed wavelet denoising technique is an improvement upon several other established wavelet denoising techniques, as well as being computationally efficient to facilitate real-time image-processing applications.

A multiresolution method of moments for triangular meshes

Abstract: This paper presents the construction, use, and properties of a multiresolution (wavelet) basis for the method of moments (MoM) analysis of metal antennas, scatterers, and microwave circuits discretized by triangular meshes. Several application examples show fast convergence of iterative solvers and accurate solutions with highly sparse MoM matrices. The proposed basis is organized in hierarchical levels, and keeps the different scales of the problem directly into the basis functions representation; the current is divided into a solenoidal and a quasi-irrotational part, which allows mapping these two vector parts onto fully scalar quantities, where the wavelets are defined. As a byproduct, this paper also presents a way to construct hierarchical sets of Rao-Wilton-Glisson (RWG) functions on a family of meshes obtained by subsequent refinement, i.e., with the RWG of coarser meshes expressed as linear combinations of those of finer meshes.

The wavelet-NARMAX representation: a hybrid model structure combining polynomial models with multiresolution wavelet decompositions

Abstract: A new hybrid model structure combing polynomial models with multiresolution wavelet decompositions is introduced for nonlinear system identification. Polynomial models play an important role in approximation theory and have been extensively used in linear and non-linear system identification. Wavelet decompositions, in which the basis functions have the property of localization in both time and frequency, outperform many other approximation schemes and offer a flexible solution for approximating arbitrary functions. Although wavelet representations can approximate even severe nonlinearities in a given signal very well, the advantage of these representations can be lost when wavelets are used to capture linear or low-order nonlinear behaviour in a signal. In order to sufficiently utilize the global property of polynomials and the local property of wavelet representations simultaneously, in this study polynomial models and wavelet decompositions are combined together in a parallel structure to represent nonlinear input-output systems. As a special form of the NARMAX model, this hybrid model structure will be referred to as the WAvelet-NARMAX model, or simply WANARMAX. Generally, such a WANARMAX representation for an input-output system might involve a large number of basis functions and therefore a great number of model terms. Experience reveals that only a small number of these model terms are significant to the system output. A new fast orthogonal least-squares algorithm, called the matching pursuit orthogonal least squares (MPOLS) algorithm, is also introduced in this study to determine which terms should be included in the final model.

A spatially adaptive nonparametric regression image deblurring

Abstract: We propose a novel nonparametric regression method for deblurring noisy images. The method is based on the local polynomial approximation (LPA) of the image and the paradigm of intersecting confidence intervals (ICI) that is applied to define the adaptive varying scales (window sizes) of the LPA estimators. The LPA-ICI algorithm is nonlinear and spatially adaptive with respect to smoothness and irregularities of the image corrupted by additive noise. Multiresolution wavelet algorithms produce estimates which are combined from different scale projections. In contrast to them, the proposed ICI algorithm gives a varying scale adaptive estimate defining a single best scale for each pixel. In the new algorithm, the actual filtering is performed in signal domain while frequency domain Fourier transform operations are applied only for calculation of convolutions. The regularized inverse and Wiener inverse filters serve as deblurring operators used jointly with the LPA-design directional kernel filters. Experiments demonstrate the state-of-art performance of the new estimators which visually and quantitatively outperform some of the best existing methods.

Multiresolution direction filterbanks: theory, design, and applications

Abstract: This paper presents new developments of directional filterbanks (DFBs) The motivation for the paper is the existence of multiresolution and multidirection orthogonal transform for two-dimensional (2-D) discrete signals Based on the frequency supports of the ideal transform, a new uniformly, maximally decimated DFB with six highpass directional subbands and two lowpass subands is introduced The uniform DFB (uDFB) can be implemented by a binary tree structure of two-channel filterbanks The filterbank employed in the tree is shown to be alias-free decimation and permissible The uDFB is then extended to a nonuniform case (nuDFB), which is still maximally decimated, by combining the two lowpass subbands The nuDFB yields nonuniform frequency division, which composes of one lowpass filter with a decimation factor of one fourth and six highpass directional filters with a decimation factor of one eighth The new DFBs offer alternative image decompositions, which overcome the limited directional selectivity of the separable wavelets and the limited multiresolution of the conventional DFB The lowpass subband of the nuDFB can be used to obtain a multiresolution representation by simply reiterating the same nuDFB decomposition On the other hand, the directional subbands can also be further refined by simply applying a two-channel conventional DFB at each highpass component A simple design method yielding near orthogonal uniform and nonuniform multidimensional filterbanks is presented Finally, the performances of the newly proposed nuDFB are compared with other conventional transforms in nonlinear approximation, image denoising, and texture classification to demonstrate its potential

Orthogonal wavelets with compact support on locally compact Abelian groups

Abstract: We extend and improve the results of W. Lang (1998) on the wavelet analysis on the Cantor dyadic group . Our construction is realized on a locally compact Abelian group which is defined for an integer and coincides with when . For any integers we determine a function in which 1) is the sum of a lacunary series by generalized Walsh functions, 2) has orthonormal integer shifts in , 3) satisfies the scaling equation with numerical coefficients, 4) has compact support whose Haar measure is proportional to , 5) generates a multiresolution analysis in . Orthogonal wavelets with compact supports on are defined by such functions . The family of these functions is in many respects analogous to the well-known family of Daubechies' scaling functions. We give a method for estimating the moduli of continuity of the functions , which leads to sharp estimates for small and . We also show that the notion of adapted multiresolution analysis recently suggested by Sendov is applicable in this situation.

Variational image reconstruction from arbitrarily spaced samples: a fast multiresolution spline solution

Abstract: We propose a novel method for image reconstruction from nonuniform samples with no constraints on their locations. We adopt a variational approach where the reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms: 1) the sum of squared errors at the specified points and 2) a quadratic functional that penalizes the lack of smoothness. We search for a solution that is a uniform spline and show how it can be determined by solving a large, sparse system of linear equations. We interpret the solution of our approach as an approximation of the analytical solution that involves radial basis functions and demonstrate the computational advantages of our approach. Using the two-scale relation for B-splines, we derive an algebraic relation that links together the linear systems of equations specifying reconstructions at different levels of resolution. We use this relation to develop a fast multigrid algorithm. We demonstrate the effectiveness of our approach on some image reconstruction examples.

Motion-Free Super-Resolution

Abstract: 1. Image Super-Resolution 2. Review of Current Status 3. Use of Defocus Cue Introduction Defocus Phenomenon Low-resolution image formation Modeling of Image and Scene Depth MAP Estimation Demonstration of Results Discussions 4. Photometry Based Method Introduction Photometry and Scene Structure Generalized Interpolation Inconsistency handling Super-resolution Estimation Experimental Demonstration Discussions 5. Blind Restoration of Photometric Observartions Introduction Blind Restoration Observation Model Joint Restoration and Structure Recovery Demonstration Discussions 6. Wavelet Based Method Introduction Single-Frame Super-Resolution Wavelets and Multiresolution Analysis Learning of Wavelet Coefficients Super-resolution Restoration Experimental Performance Discussions 7. Use of Zoom Cue Introduction Problem Definition Zoom Estimation Super-resolving a Scene with a Known Model Learning of Model Parameters Experimental Demonstration Discussions 8. Looking Ahead

Wavelet-based enhancement of lung and bowel sounds using fractal dimension thresholding-part I: methodology

Abstract: An efficient method for the enhancement of lung sounds (LS) and bowel sounds (BS), based on wavelet transform (WT), and fractal dimension (FD) analysis is presented in this paper. The proposed method combines multiresolution analysis with FD-based thresholding to compose a WT-FD filter, for enhanced separation of explosive LS (ELS) and BS (EBS) from the background noise. In particular, the WT-FD filter incorporates the WT-based multiresolution decomposition to initially decompose the recorded bioacoustic signal into approximation and detail space in the WT domain. Next, the FD of the derived WT coefficients is estimated within a sliding window and used to infer where the thresholding of the WT coefficients has to happen. This is achieved through a self-adjusted procedure that iteratively "peels" the estimated FD signal and isolates its peaks produced by the WT coefficients corresponding to ELS or EBS. In this way, two new signals are constructed containing the useful and the undesired WT coefficients, respectively. By applying WT-based multiresolution reconstruction to these two signals, a first version of the desired signal and the background noise is provided, accordingly. This procedure is repeated until a stopping criterion is met, finally resulting in efficient separation of the ELS or EBS from the background noise. The proposed WT-FD filter introduces an alternative way to the enhancement of bioacoustic signals, applicable to any separation problem involving nonstationary transient signals mixed with uncorrelated stationary background noise. The results from the application of the WT-FD filter to real bioacoustic data are presented and discussed in an accompanying paper.

A multiresolution manifold distance for invariant image similarity

Abstract: Accounting for spatial image transformations is a requirement for multimedia problems such as video classification and retrieval, face/object recognition or the creation of image mosaics from video sequences. We analyze a transformation invariant metric recently proposed in the machine learning literature to measure the distance between image manifolds - the tangent distance (TD) - and show that it is closely related to alignment techniques from the motion analysis literature. Exposing these relationships results in benefits for the two domains. On one hand, it allows leveraging on the knowledge acquired in the alignment literature to build better classifiers. On the other, it provides a new interpretation of alignment techniques as one component of a decomposition that has interesting properties for the classification of video. In particular, we embed the TD into a multiresolution framework that makes it significantly less prone to local minima. The new metric - multiresolution tangent distance (MRTD) - can be easily combined with robust estimation procedures, and exhibits significantly higher invariance to image transformations than the TD and the Euclidean distance (ED). For classification, this translates into significant improvements in face recognition accuracy. For video characterization, it leads to a decomposition of image dissimilarity into "differences due to camera motion" plus "differences due to scene activity" that is useful for classification. Experimental results on a movie database indicate that the distance could be used as a basis for the extraction of semantic primitives such as action and romance.