Error Analysis for Image Inpainting

Image inpainting refers to restoring a damaged image with missing information. In recent years, there have been many developments on computational approaches to image inpainting problem [2, 4, 6, 9, 11–13, 27, 28]. While there are many effective algorithms available, there is still a lack of theoretical understanding on under what conditions these algorithms work well. In this paper, we take a step in this direction. We investigate an error bound for inpainting methods, by considering different image spaces such as smooth images, piecewise constant images and a particular kind of piecewise continuous images. Numerical results are presented to validate the theoretical error bounds. Tony F. Chan received the B.S. degree in engineering and the M.S. degree in aerospace engineering in 1973, from the California Institute of Technology, and the Ph.D. degree in computer science from Stanford University in 1978. He is Professor of Mathematics and currently also Dean of the division of Physical science at University of California, Los Angeles, where he has been a Professor since 1986. His research interests include mathematical and computational methods in image processing, multigrid, domain decomposition algorithms, iterative methods, Krylov subspace methods, and parallel algorithms. Sung Ha Kang received the Ph.D. degree in mathematics in 2002, from University of California, Los Angeles, and currently is Assistant Professor of Mathematics at University of Kentucky since 2002. Her research interests include mathematical and computational methods in image processing and computer vision.     

Total Variation Wavelet Inpainting

We consider the problem of filling in missing or damaged wavelet coefficients due to lossy image transmission or communication. The task is closely related to classical inpainting problems, but also remarkably differs in that the inpainting regions are in the wavelet domain. New challenges include that the resulting inpainting regions in the pixel domain are usually not geometrically well defined, as well as that degradation is often spatially inhomogeneous. We propose two related variational models to meet such challenges, which combine the total variation (TV) minimization technique with wavelet representations. The associated Euler-Lagrange equations lead to nonlinear partial differential equations (PDE’s) in the wavelet domain, and proper numerical algorithms and schemes are designed to handle their computation. The proposed models can have effective and automatic control over geometric features of the inpainted images including sharp edges, even in the presence of substantial loss of wavelet coefficients, including in the low frequencies. Existence and uniqueness of the optimal inpaintings are also carefully investigated. Research supported in part by grants ONR-N00014-03-1-0888, NSF DMS-9973341, DMS-0202565 and DMS-0410062, and NIH contract P 20 MH65166.     

Classification-based objective functions

Backpropagation, similar to most learning algorithms that can form complex decision surfaces, is prone to overfitting. This work presents classification-based objective functions, an approach to training artificial neural networks on classification problems. Classification-based learning attempts to guide the network directly to correct pattern classification rather than using common error minimization heuristics, such as sum-squared error (SSE) and cross-entropy (CE), that do not explicitly minimize classification error. CB1 is presented here as a novel objective function for learning classification problems. It seeks to directly minimize classification error by backpropagating error only on misclassified patterns from culprit output nodes. CB1 discourages weight saturation and overfitting and achieves higher accuracy on classification problems than optimizing SSE or CE. Experiments on a large OCR data set have shown CB1 to significantly increase generalization accuracy over SSE or CE optimization, from 97.86% and 98.10%, respectively, to 99.11%. Comparable results are achieved over several data sets from the UC Irvine Machine Learning Database Repository, with an average increase in accuracy from 90.7% and 91.3% using optimized SSE and CE networks, respectively, to 92.1% for CB1. Analysis indicates that CB1 performs a fundamentally different search of the feature space than optimizing SSE or CE and produces significantly different solutions. Editor: Risto Miikkulainen     

High-temperature quasi-continuous operation of optically pumpedCdZnTe/ZnTe multiple-quantum-well lasers at 620 nm

Low-lasing-threshold quasi-continuous optically pumped II-VI quantum-well lasers were operated well above room temperature at 620 nm. This result shows promise for high-repetition-rate, short-pulse generation by direct modulation of the injection current and also CW operation of future wide-gap II-VI diode lasers above room temperature     

Fluctuation statistics of millimeter-wave scattering fromdistributed targets

The applicability of the Rayleigh fading model for characterizing radar scattering from terrain is examined at 35 GHz for both backscattering and bistatic scattering. The model is found to be in excellent agreement with experimental observations for single-frequency observations of uniform targets such as asphalt and snow-covered ground. The use of frequency averaging to reduce signal fading variations was examined experimentally by sweeping the radar signal from 34-36 GHz in 401 steps. The results show that the formulation based on the Rayleigh model relating the reduction in signal fluctuation to the bandwidth used provides a reasonable estimate for the improvement provided by frequency averaging     

Quadrature Overlapped Raised-Cosine Modulation

A variation of an old but neglected pulse shaping technique, raised-cosine, is investigated. By overlapping raised-cosine pulses in each of two data streams and then by quadrature combining them, a simple QPSK/MSK type modulation results. This quadrature overlapped raised-cosine (QORC) modulation exhibits a hybrid structure of QPSK and MSK modulations. The power spectral density of QORC is shown to take on the form of the product of the power spectral densities of MSK and QPSK. The obvious consequences are that the power spectral density main lobe retains the width of the spectral density main lobe of QPSK, but the sidelobes drop off much faster(1/f^{6}). A simple QORC modulator can be implemented similar to an MSK modulator. Several correlation type receivers are investigated and their performances calculated. Computer simulation results are used to compare end-to-end system performance of QORC and staggered QORC (SQORC) with MSK, QPSK, and staggered QPSK (SQPSK) for both linear and nonlinear satellite channels. The performance of QORC and SQORC compares very favorably with QPSK, SQPSK, and MSK. QORC performs particularly well in the presence of a nonlinear channel. The effect of phase equalization of the channel filter was investigated with outstanding performance improvement. The simulation results show that sidelobe regeneration caused by the channel nonlinearity is much less for SQORC than it is for the other modulation formats considered.