Approximation power of biorthogonal wavelet expansions

This paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions. The Strang-Fix conditions imply that the error for an orthogonal wavelet approximation at scale a=2^-i globally decays as a^N, where N is the order of the transform. This is why, for a given number of scales, higher order wavelet transforms usually result in better signal approximations. We prove that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone. We also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to zero. These results strengthen Sweldens earlier analysis and confirm that the approximation power of biorthogonal and (semi-)orthogonal wavelet expansions is essentially the same. Finally, we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines. We also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error     

/sub p/-multiresolution analysis: how to reduce ringing and sparsify the error

We propose to design the reduction operator of an image pyramid so as to minimize the approximation error in the /sub p/-sense (not restricted to the usual p=2), where p can take noninteger values. The underlying image model is specified using shift-invariant basis functions, such as B-splines. The solution is well-defined and determined by an iterative optimization algorithm based on digital filtering. Its convergence is accelerated by the use of first and second order derivatives. For p close to 1, we show that the ringing is reduced and that the histogram of the detail image is sparse as compared with the standard case, where p=2.     

Teaching image-processing programming in Java

Image processing (IP) can be taught very effectively by complementing the basic lectures with computer laboratories where the participants can actively manipulate and process images. This offering can be made even more attractive by allowing the students to develop their own IP code within a reasonable time frame. A designed system to be as "student friendly" as possible is presented. The software is built around ImageJ, a freely available, full-featured, and user-friendly program for image analysis. The students can walk away from the course with an IP system that is operational. Using the ImageAccess interface layer, they can easily program both ImageJ plug-ins and Internet applets. The system that we have described may also appeal to practitioners as it offers simple, full-proof way of developing professional level IP software.     

Monte-Carlo sure: a black-box optimization of regularization parameters for general denoising algorithms

We consider the problem of optimizing the parameters of a given denoising algorithm for restoration of a signal corrupted by white Gaussian noise. To achieve this, we propose to minimize Stein's unbiased risk estimate (SURE) which provides a means of assessing the true mean-squared error (MSE) purely from the measured data without need for any knowledge about the noise-free signal. Specifically, we present a novel Monte-Carlo technique which enables the user to calculate SURE for an arbitrary denoising algorithm characterized by some specific parameter setting. Our method is a black-box approach which solely uses the response of the denoising operator to additional input noise and does not ask for any information about its functional form. This, therefore, permits the use of SURE for optimization of a wide variety of denoising algorithms. We justify our claims by presenting experimental results for SURE-based optimization of a series of popular image-denoising algorithms such as total-variation denoising, wavelet soft-thresholding, and Wiener filtering/smoothing splines. In the process, we also compare the performance of these methods. We demonstrate numerically that SURE computed using the new approach accurately predicts the true MSE for all the considered algorithms. We also show that SURE uncovers the optimal values of the parameters in all cases.     

Discretization of the radon transform and of its inverse by spline convolutions

We present an explicit formula for B-spline convolution kernels; these are defined as the convolution of several B-splines of variable widths h(i) and degrees n(i). We apply our results to derive spline-convolution-based algorithms for two closely related problems: the computation of the Radon transform and of its inverse. First, we present an efficient discrete implementation of the Radon transform that is optimal in the least-squares sense. We then consider the reverse problem and introduce a new spline-convolution version of the filtered back-projection algorithm for tomographic reconstruction. In both cases, our explicit kernel formula allows for the use of high-degree splines; these offer better approximation performance than the conventional lower-degree formulations (e.g., piecewise constant or piecewise linear models). We present multiple experiments to validate our approach and to find the parameters that give the best tradeoff between image quality and computational complexity. In particular, we find that it can be computationally more efficient to increase the approximation degree than to increase the sampling rate.     

Four-dimensional wavelet compression of arbitrarily sized echocardiographic data

Wavelet-based methods have become most popular for the compression of two-dimensional medical images and sequences. The standard implementations consider data sizes that are powers of two. There is also a large body of literature treating issues such as the choice of the "optimal" wavelets and the performance comparison of competing algorithms. With the advent of telemedicine, there is a strong incentive to extend these techniques to higher dimensional data such as dynamic three-dimensional (3-D) echocardiography [four-dimensional (4-D) datasets]. One of the practical difficulties is that the size of this data is often not a multiple of a power of two, which can lead to increased computational complexity and impaired compression power. Our contribution in this paper is to present a genuine 4-D extension of the well-known zerotree algorithm for arbitrarily sized data. The key component of our method is a one-dimensional wavelet algorithm that can handle arbitrarily sized input signals. The method uses a pair of symmetric/antisymmetric wavelets (10/6) together with some appropriate midpoint symmetry boundary conditions that reduce border artifacts. The zerotree structure is also adapted so that it can accommodate noneven data splitting. We have applied our method to the compression of real 3-D dynamic sequences from clinical cardiac ultrasound examinations. Our new algorithm compares very favorably with other more ad hoc adaptations (image extension and tiling) of the standard powers-of-two methods, in terms of both compression performance and computational cost. It is vastly superior to slice-by-slice wavelet encoding. This was seen not only in numerical image quality parameters but also in expert ratings, where significant improvement using the new approach could be documented. Our validation experiments show that one can safely compress 4-D data sets at ratios of 128:1 without compromising the diagnostic value of the images. We also display some more extreme compression results at ratios of 2000:1 where some key diagnostically relevant key features are preserved.     

Feature extraction and decision procedure for automated inspection of textured materials

This paper proposes a general system approach applicable to the automatic inspection of textured material. First, the input image is preprocessed in order to be independent of non-uniformities. A tone-to-texture transform is then performed by mapping the original grey level picture on a multivariate local feature sequence, which turns out to be normally distributed. More specifically, features derived with the help of the Karhunen-Loève decomposition of a small neighbourhood of each pixel are used. A decision as to conformity with a reference texture is arrived at by thresholding the Mahalanobis distance for every realization of the feature vector. It is shown that this approach is optimum under the Gaussian assumption in the sense that it has a minimum acceptance region for a fixed probability of false rejection.     

Linear interpolation revitalized

We present a simple, original method to improve piecewise-linear interpolation with uniform knots: we shift the sampling knots by a fixed amount, while enforcing the interpolation property. We determine the theoretical optimal shift that maximizes the quality of our shifted linear interpolation. Surprisingly enough, this optimal value is nonzero and close to 1/5. We confirm our theoretical findings by performing several experiments: a cumulative rotation experiment and a zoom experiment. Both show a significant increase of the quality of the shifted method with respect to the standard one. We also observe that, in these results, we get a quality that is similar to that of the computationally more costly "high-quality" cubic convolution.     

Construction of biorthogonal wavelets starting from any twomultiresolutions

Starting from any two given multiresolution analyses of L₂ , {V_j¹}_j∈Z and {V_j²}_j∈Z, we construct biorthogonal wavelet bases that are associated with this chosen pair of multiresolutions. Thus, our construction method takes a point of view opposite to the one of Cohen-Daubechies-Feauveau (1992), which starts from a well-choosen pair of biorthogonal discrete filters. In our construction, the necessary and sufficient condition is the nonperpendicularity of the multiresolutions