We propose a framework for the detection of junctions in images. Although the detection of edges and key points is a well examined and described area, the multiscale detection of junction centers, especially for odd orders, poses a challenge in pattern analysis. The goal of this paper is to build optimal junction detectors based on 2D steerable wavelets that are polar-separable in the Fourier domain. The approaches we develop are general and can be used for the detection of arbitrary symmetric and asymmetric junctions. The backbone of our construction is a multiscale pyramid with a radial wavelet function where the directional components are represented by circular harmonics and encoded in a shaping matrix. We are able to detect M -fold junctions in different scales and orientations. We provide experimental results on both simulated and real data to demonstrate the effectiveness of the algorithm.

In this paper, we propose a continuous-domain version of principal-component analysis, with the constraint that the underlying family of templates appears at arbitrary orientations. We show that the corresponding principal components are steerable. Our method can be used for designing steerable filters so that they best approximate a given collection of reference templates. We apply this framework to the detection and classification of micrometer-sized particles that are used in a microfluidic diagnostics system. This is done in two steps. First, we decompose the particles into a small number of templates and compute their steerable principal components. Then we use these principal components to automatically estimate the orientation and the class of each particle.

Kaiser-Bessel window functions are frequently used to discretize tomographic problems because they have two desirable properties: 1) their short support leads to a low computational cost and 2) their rotational symmetry makes their imaging transform independent of the direction. In this paper, we aim at optimizing the parameters of these basis functions. We present a formalism based on the theory of approximation and point out the importance of the partition-of-unity condition. While we prove that, for compact-support functions, this condition is incompatible with isotropy, we show that minimizing the deviation from the partition of unity condition is highly beneficial. The numerical results confirm that the proposed tuning of the Kaiser-Bessel window functions yields the best performance.

We present a multilevel extension of the popular "thresholded Landweber" algorithm for wavelet-regularized image restoration that yields an order of magnitude speed improvement over the standard fixed-scale implementation. The method is generic and targeted towards large-scale linear inverse problems, such as 3-D deconvolution microscopy. The algorithm is derived within the framework of bound optimization. The key idea is to successively update the coefficients in the various wavelet channels using fixed, subband-adapted iteration parameters (step sizes and threshold levels). The optimization problem is solved efficiently via a proper chaining of basic iteration modules. The higher level description of the algorithm is similar to that of a multigrid solver for PDEs, but there is one fundamental difference: the latter iterates though a sequence of multiresolution versions of the original problem, while, in our case, we cycle through the wavelet subspaces corresponding to the difference between successive approximations. This strategy is motivated by the special structure of the problem and the preconditioning properties of the wavelet representation. We establish that the solution of the restoration problem corresponds to a fixed point of our multilevel optimizer. We also provide experimental evidence that the improvement in convergence rate is essentially determined by the (unconstrained) linear part of the algorithm, irrespective of the type of wavelet. Finally, we illustrate the technique with some image deconvolution examples, including some real 3-D fluorescence microscopy data.

We propose a recursive data-driven risk-estimation method for non-linear iterative deconvolution. Our two main contributions are 1) a solution-domain risk-estimation approach that is applicable to non-linear restoration algorithms for ill- conditioned inverse problems; and 2) a risk estimate for a state-of-the-art iterative procedure, the thresholded Landweber iteration, which enforces a wavelet-domain sparsity constraint. Our method can be used to estimate the SNR improvement at every step of the algorithm; e.g., for stopping the iteration after the highest value is reached. It can also be applied to estimate the optimal threshold level for a given number of iterations.

We present a fast variational deconvolution algorithm that minimizes a quadratic data term subject to a regularization on the l(1)-norm of the wavelet coefficients of the solution. Previously available methods have essentially consisted in alternating between a Landweber iteration and a wavelet-domain soft-thresholding operation. While having the advantage of simplicity, they are known to converge slowly. By expressing the cost functional in a Shannon wavelet basis, we are able to decompose the problem into a series of subband-dependent minimizations. In particular, this allows for larger (subband-dependent) step sizes and threshold levels than the previous method. This improves the convergence properties of the algorithm significantly. We demonstrate a speed-up of one order of magnitude in practical situations. This makes wavelet-regularized deconvolution more widely accessible, even for applications with a strong limitation on computational complexity. We present promising results in 3-D deconvolution microscopy, where the size of typical data sets does not permit more than a few tens of iterations.

We present a generalization of the Daubechies wavelet family. The context is that of a non-stationary multiresolution analysis - i.e., a sequence of embedded approximation spaces generated by scaling functions that are not necessarily dilates of one another. The constraints that we impose on these scaling functions are: (1) orthogonality with respect to translation; (2) reproduction of a given set of exponential polynomials; and (3) minimal support. These design requirements lead to the construction of a general family of compactly-supported, orthonormal wavelet-like bases of L/sub 2/. If the exponential parameters are all zero, then one recovers Daubechies wavelets, which are orthogonal to the polynomials of degree (N-1) where N is the order (vanishing-moment property). A fast filterbank implementation of the generalized wavelet transform follows naturally; it is similar to Mallat's algorithm, except that the filters are now scale-dependent. The new transforms offer increased flexibility and are tunable to the spectral characteristics of a wide class of signals.