An Inner-Product Calculus for Periodic Functions and Curves

Our motivation is the design of efficient algorithms to process closed curves represented by basis functions or wavelets. To that end, we introduce an inner-product calculus to evaluate correlations and $L _{ 2 }$ distances between such curves. In particular, we present formulas for the direct and exact evaluation of correlation matrices in the case of closed (i.e., periodic) parametric curves and periodic signals. We give simplifications for practical cases that involve B-splines. To illustrate this approach, we also propose a least-squares approximation scheme that is able to resample curves while minimizing aliasing artifacts. Another application is the exact calculation of the enclosed area

Operator-Like Wavelet Bases of L2 L _{ 2 } (Rd (\mathbb{R} ^{ d } )

The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator

A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N = 6, ..., 20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases

Polyharmonic Smoothing Splines and the Multidimensional Wiener Filtering of Fractal-Like Signals

Motivated by the fractal-like behavior of natural images, we develop a smoothing technique that uses a regularization functional which is a fractional iterate of the Laplacian. This type of functional was initially introduced by Duchon for the approximation of nonuniformily sampled, multidimensional data. He proved that the general solution is a smoothing spline that is represented by a linear combination of radial basis functions (RBFs). Unfortunately, this is tedious to implement for images because of the poor conditioning of RBFs and their lack of decay. Here, we present a much more efficient method for the special case of a uniform grid. The key idea is to express Duchon's solution in a fractional polyharmonic B-spline basis that spans the same space as the RBFs. This allows us to derive an algorithm where the smoothing is performed by filtering in the Fourier domain. Next we prove that the above smoothing spline can be optimally tuned to provide the MMSE estimation of a fractional Brownian field corrupted by white noise. This is a strong result that not only yields the best linear filter (Wiener solution), but also the optimal interpolation space, which is not bandlimited. It also suggests a way of using the noisy data to identify the optimal parameters (order of the spline and smoothing strength), which yields a fully automatic smoothing procedure. We evaluate the performance of our algorithm by comparing it against an oracle Wiener filter, which requires the knowledge of the true noiseless power spectrum of the signal. We find that our approach performs almost as well as the oracle solution over a wide range of conditions

Parametric shape processing in biomedical imaging

In this thesis, we present a coherent and consistent approach for the estimation of shape and shape attributes from noisy images. As compared to the traditional sequential approach, our scheme is centered on a shape model which drives the feature extraction, shape optimization, and the attribute evaluation modules. In the first section, we deal with the detection of image features that guide the shape-extraction process. We propose a general approach for the design of 2-D feature detectors from a class of steerable functions, based on the optimization of a Canny-like criterion. As compared to previous computational designs, our approach is truly 2-D and yields more orientation selective detectors. We then address the estimation of the global shape from an image. Specifically, we propose to use cubic-spline-based parametric active contour models to solve two shape-extraction problems: (i) the segmentation of closed objects and (ii) the 3-D reconstruction of DNA filaments from their stereo cryo-electron micrographs. We present several enhancements of existing snake algorithms for segmentation. For the detection of 3-D DNA filaments from their orthogonal projections, we introduce the concept of projection-steerable matched filtering. We then use a 3-D snake algorithm to reconstruct the shape. Next, we analyze the efficiency of curve representations using refinable basis functions for the description of shape boundaries. We derive an exact expression for the error when we approximate a periodic signal in a scaling-function basis. Finally, we present a method for the exact computation of the area moments of such shapes