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Abstract-We present a couple of new algorithmic procedures for the detection of ridges in the modulus of the (continuous) wavelet transform of one-dimensional (1-D) signals. These detection procedures are shown to be robust to additive white noise. We also derive and test a new reconstruction procedure. The latter uses only information from the restriction of the wavelet transform to a sample of points from the ridge. This provides a very efficient way to code the information contained in the signal. I. INTRODUCTIONThe characterization and the separation of amplitude and frequency modulated signals is a classical problem of signal analysis and signal processing. Applications can be found in many situations, such as, for instance, radar/sonar detection and speech processing [10]. In 1990, the Marseille group proposed a new algorithm (see [6] for a survey) based on the study of the phase of the wavelet (or Gabor) transform. The present work is an attempt to extend the latter to noisy situations. The main thrust of this correspondence is to use the localization properties of the modulus of the transform (which is generally more robust than the phase, even though the latter provides more precise estimates [6]). In the case of frequencymodulated signals, the wavelet transform is "concentrated" in the neighborhood of curves (the ridges of the transform). We develop a scheme in which these curves are searched as such, in a (high dimensional) space of ridges, via a stochastic relaxation procedure. This alternate characterization of the ridges is better suited to the needs of noisy signal analyzes. We also propose a stable method for signal reconstruction from the numerically computed ridges. This method is also based on an L 2 -minimization procedure.For the sake of simplicity, our discussion is restricted to the case of the wavelet transform, but since our algorithms deal only with postprocessing of time-frequency transforms, they can be extended to any time-frequency energetic representations. The case of the Gabor transform will be considered in the companion paper [4], where still another stochastic search algorithm, adapted to different situations, will be introduced.We close this introduction with a short summary of the contents of the paper. Section II is devoted to the statement of the problem and the definition of the ridges. Section III presents the main features of the variational problems we propose and solve to estimate the ridges. We also give a Bayesian interpretation of this approach, and Manuscript received June 13, 1996; revised October 14, 1996. This work was supported in part by ONR under Grant N00014-91-1010 and by NSF under Grant IBN 9405146. The associate editor coordinating the review of this paper and approving it for publication was Dr. Jelena Kovacević. R. A. Carmona is with the Statistics and Operations Research Program,

The ridges of the wavelet transform, the Gabor transform or any time-frequency representation of a signal contain crucial information on the characteristics of the signal. Indeed they mark the regions of the time-frequency plane where the signal concentrates most of its energy. We introduce a new algorithm to detect and identify these ridges. The procedure is based on an original form of Markov Chain Monte Carlo algorithm specially adapted to the present situation. We show that this detection algorithm is especially useful for noisy signals with multi-ridge transforms. It is a common practice among practitioners to reconstruct a signal from the skeleton of a transform of the signal (i.e. the restriction of the transform to the ridges). After reviewing several known procedures we introduce a new reconstruction algorithm and we illustrate its efficiency on speech signals.

Abstract-Complex-valued wavelets are normally used to measure instantaneous frequencies, while real wavelets are normally used to detect singularities. We prove that the wavelet modulus maxima with a complex-valued wavelet can detect and characterize singularities. This is an extension of the previous wavelet work of Mallat and Hwang on modulus maxima using a real wavelet. With this extension, we can simultaneously detect instantaneous frequencies and singularities from the wavelet modulus maxima of a complex-valued wavelet. Some results of singularity detection with the modulus maxima from a real wavelet and an analytic complex-valued wavelet are compared. We also demonstrate that singularity detection methods can be employed to detect the corners of a planar object.

In this correspondence, a method is proposed for estimating the surface orientation of a planar texture under perspective projection based on the ridge of a two-dimensional (2-D) continuous wavelet transform (CWT). We show that an analytical solution of the surface orientation can be derived from the scales of the ridge surface. A comparative study with an existing method is given.

In this paper, we propose a spatially-varying deblurring method to remove the out-of-focus blur. Our proposed method mainly contains three parts: blur map generation, image deblurring, and scale selection. First, we derive a blur map using local contrast prior and the guided filter. Second, we propose our image deblurring method with L1-2 optimization to obtain a better image quality. Finally, we adopt the scale selection to ensure our output free from ringing artifacts. The experimental results demonstrate our proposed method is promising.Index Terms-out-of-focus blur, spatially-varying deblurring, L1-2 optimization, guided blur map.

The operator-based signal separation approach uses an adaptive operator to separate a signal into additive subcomponents. The approach can be formulated as an optimization problem whose optimal solution can be derived analytically. However, the following issues must still be resolved: estimating the robustness of the operator's parameters and the Lagrangian multipliers, and determining how much of the information in the null space of the operator should be retained in the residual signal. To address these problems, we propose a novel optimization formula for operator-based signal separation and show that the parameters of the problem can be estimated adaptively.We demonstrate the efficacy of the proposed method by processing several signals, including real-life signals.

Abstract-Image mosaicing is the act of combining two or more images and is used in many applications in computer vision, image processing, and computer graphics. It aims to combine images such that no obstructive boundaries exist around overlapped regions and to create a mosaic image that exhibits as little distortion as possible from the original images. In the proposed technique, the to-be-combined images are first projected into wavelet subspaces. The images projected into the same wavelet space are then blended. Our blending function is derived from an energy minimization model which balances the smoothness around the overlapped region and the fidelity of the blended image to the original images. Experiment results and subjective comparison with other methods are given.