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While studying the computable real numbers as a professional mathematician, I came to see the computable reals, and not the real numbers as usually presented in undergraduate real analysis classes, as the natural culmination of my evolving understanding of numbers as a schoolchild. This paper attempts to trace and explain that evolution. The first part recounts the nature of numbers as they were presented to us grade-school children. In particular, the introduction of square roots induced a step change in my understanding of numbers. Another incident gave me insight into the brilliance of Alan Turing in his paper introducing both the computable real numbers and his famous 'Turing machine". The final part of this paper describes the computable real numbers in enough detail to supplement the usual undergraduate real analysis class. An appendix presents programs that implement the examples in the text. [ABSTRACT FROM AUTHOR]

A mathematical structure for iterated translation-invariant wavelet shrinkage is presented. It is demonstrated that with orthogonal wavelets it is equal to gradient descent in L2(I) along the semi-norm for the Besov space B1/1(L1(I)).

The relationship between variational problems and wavelet-based image processing algorithms is analyzed. Wavelets are then used to solve various groups of variational problems. A deterministic smoothness model for images is also developed for use in image compression. Use of the model reaps good results in analyzing noise removal and also leads to the development of the signal-to-noise ration that is useful in estimating the size of the shrinkage parameter.

An attempt to motivate a class of engineering students leads to insights both personal and pedagogical. [ABSTRACT FROM AUTHOR]

Several methods have been proposed to reduce boundary artifacts in image deblurring. Some of those methods impose certain assumptions on image pixels outside the field-of-view; the most important of these assume reective or anti-reective boundary conditions. Boundary condition meth- ods, including reective and anti-reective ones, however, often fail to reduce boundary artifacts, and, in some cases, generate their own artifacts, especially when the image to be deblurred does not accurately satisfy the imposed condi- tion. To overcome these difficulties, we suggest using free boundary conditions, which do not impose any restrictions on image pixels outside the field-of-view, and preconditioned conjugate gradient methods, where preconditioners are de- signed to compensate for the non-uniformity in contributions from image pixels to the observation. Our simulation studies show that the proposed method out- performs reective and anti-reective boundary condition methods in removing boundary artifacts. The simulation studies also show that the proposed method can be applicable to arbitrarily shaped images and has the benefit of recovering damaged parts in blurred images. [ABSTRACT FROM AUTHOR]

Wavelet methods to invert radon transform with noisy data are examined.

We bound the difference between the solution to the continuous Rudin-Osher Fatemi (ROF) image smoothing model and the solutions to various finite-difference approximations to this model. These bounds apply to "typical" images, i.e., images with edges or with fractal structure. These are the first bounds on the error in numerical methods for ROF smoothing. [ABSTRACT FROM AUTHOR]

Deals with a study which discussed the use of wavelet methods for inverting the radon transform with noisy data in image processing. Review of notations and definitions in the study; Information on wavelet theory; Details on linear homogenous equations used in the study; Process of wavelet-vaguelette decompositions.

The previously described optimized binary compressive detection (OB-CD) strategy enables fast hyperspectral Raman (and fluorescence) spectroscopic analysis of systems containing two or more chemical components. However, each OB-CD filter collects only a fraction of the scattered photons and the remainder of the photons are lost. Here, we present a refinement of OB-CD, the OB-CD2 strategy, in which all of the collected Raman photons are detected using a pair of complementary binary optical filters that direct photons of different colors to two photon counting detectors. The OB-CD2 filters are generated using a new optimization algorithm described in this work and implemented using a holographic volume diffraction grating and a digital micromirror device (DMD) whose mirrors are programed to selectively direct photons of different colors either to one or the other photon-counting detector. When applied to pairs of pure liquids or two-component solid powder mixtures, the resulting OB-CD2 strategy is shown to more accurately estimate Raman scattering rates of each chemical component, when compared to the original OB-CD, thus facilitating chemical classification at speeds as fast as 3 μs per measurement and the collection of Raman images in under a second. [ABSTRACT FROM AUTHOR]