Abstract-It is well known that spatial averaging can be realized (in space or frequency domain) using algorithms whose complexity does not scale with the size or shape of the filter. These fast algorithms are generally referred to as constant-time or (1) algorithms in the image-processing literature. Along with the spatial filter, the edge-preserving bilateral filter involves an additional range kernel. This is used to restrict the averaging to those neighborhood pixels whose intensity are similar or close to that of the pixel of interest. The range kernel operates by acting on the pixel intensities. This makes the averaging process nonlinear and computationally intensive, particularly when the spatial filter is large. In this paper, we show how the (1) averaging algorithms can be leveraged for realizing the bilateral filter in constant time, by using trigonometric range kernels. This is done by generalizing the idea presented by Porikli, i.e., using polynomial kernels. The class of trigonometric kernels turns out to be sufficiently rich, allowing for the approximation of the standard Gaussian bilateral filter. The attractive feature of our approach is that, for a fixed number of terms, the quality of approximation achieved using trigonometric kernels is much superior to that obtained by Porikli using polynomials.

A direct implementation of the bilateral filter [1] requires O(σ 2 s ) operations per pixel, where σ s is the (effective) width of the spatial kernel. A fast implementation of the bilateral filter was recently proposed in [19] that required O(1) operations per pixel with respect to σ s . This was done by using trigonometric functions for the range kernel of the bilateral filter, and by exploiting their so-called shiftability property. In particular, a fast implementation of the Gaussian bilateral filter was realized by approximating the Gaussian range kernel using raised cosines. Later, it was demonstrated in [24] that this idea could be extended to a larger class of filters, including the popular non-local means filter [2,3]. As already observed in [19], a flip side of this approach was that the run time depended on the width σ r of the range kernel. For an image with dynamic range [0, T ], the run time scaled as O(T 2 /σ 2 r ) with σ r . This made it difficult to implement narrow range kernels, particularly for images with large dynamic range. In this paper, we discuss this problem, and propose some simple steps to accelerate the implementation, in general, and for small σ r in particular. We provide some experimental results to demonstrate the acceleration that is achieved using these modifications.

Consider N points in R d and M local coordinate systems that are related through unknown rigid transforms. For each point we are given (possibly noisy) measurements of its local coordinates in some of the coordinate systems. Alternatively, for each coordinate system, we observe the coordinates of a subset of the points. The problem of estimating the global coordinates of the N points (up to a rigid transform) from such measurements comes up in distributed approaches to molecular conformation and sensor network localization, and also in computer vision and graphics.The least-squares formulation of this problem, though non-convex, has a well known closed-form solution when M = 2 (based on the singular value decomposition). However, no closed form solution is known for M ≥ 3.In this paper, we demonstrate how the least-squares formulation can be relaxed into a convex program, namely a semidefinite program (SDP). By setting up connections between the uniqueness of this SDP and results from rigidity theory, we prove conditions for exact and stable recovery for the SDP relaxation. In particular, we prove that the SDP relaxation can guarantee recovery under more adversarial conditions compared to earlier proposed spectral relaxations, and derive error bounds for the registration error incurred by the SDP relaxation.We also present results of numerical experiments on simulated data to confirm the theoretical findings. We empirically demonstrate that (a) unlike the spectral relaxation, the relaxation gap is mostly zero for the semidefinite program (i.e., we are able to solve the original non-convex least-squares problem) up to a certain noise threshold, and (b) the semidefinite program performs significantly better than spectral and manifold-optimization methods, particularly at large noise levels.

In this letter, we note that the denoising performance of Non-Local Means (NLM) can be improved at large noise levels by replacing the mean by the Euclidean median. We call this new denoising algorithm the Non-Local Euclidean Medians (NLEM). At the heart of NLEM is the observation that the median is more robust to outliers than the mean. In particular, we provide a simple geometric insight that explains why NLEM performs better than NLM in the vicinity of edges, particularly at large noise levels. NLEM can be efficiently implemented using iteratively reweighted least squares, and its computational complexity is comparable to that of NLM. We provide some preliminary results to study the proposed algorithm and to compare it with NLM.

Abstract-We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions-the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of , we then discuss a methodology for constructing 2-D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1-D counterpart, we relate the real and imaginary components of these complex wavelets using a multidimensional extension of the HT-the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient fast Fourier transform (FFT)-based filterbank algorithm for implementing the associated complex wavelet transform.