We propose two new image compression-decompression methods that reproduce images with better visual fidelity, less blocking artifacts, and better PSNR, particularly in low bit rates, than those processed by the JPEG Baseline method at the same bit rates. The additional computational cost is small, i.e., linearly proportional to the number of pixels in an input image. The first method, the "full mode" polyharmonic local cosine transform (PHLCT), modifies the encoder and decoder parts of the JPEG Baseline method. The goal of the full mode PHLCT is to reduce the code size in the encoding part and reduce the blocking artifacts in the decoder part. The second one, the "partial mode" PHLCT (or PPHLCT for short), modifies only the decoder part, and consequently, accepts the JPEG files, yet decompresses them with higher quality with less blocking artifacts. The key idea behind these algorithms is a decomposition of each image block into a polyharmonic component and a residual. The polyharmonic component in this paper is an approximate solution to Poisson's equation with the Neumann boundary condition, which means that it is a smooth predictor of the original image block only using the image gradient information across the block boundary. Thus, the residual--obtained by removing the polyharmonic component from the original image block--has approximately zero gradient across the block boundary, which gives rise to the fast-decaying DCT coefficients, which, in turn, lead to more efficient compression-decompression algorithms for the same bit rates. We show that the polyharmonic component of each block can be estimated solely by the first column and row of the DCT coefficient matrix of that block and those of its adjacent blocks and can predict an original image data better than some of the other AC prediction methods previously proposed. Our numerical experiments objectively and subjectively demonstrate the superiority of PHLCT over the JPEG Baseline method and the improvement of the JPEG-compressed images when decompressed by PPHLCT.

We describe a new technique for detecting and characterizing ellipsoidal shapes automatically from any type of image. This technique is a single pass algorithm which can extract any group of ellipse parameters or characteristics which can be computed from those parameters without having to detect all five parameters for each ellipsoidal shape. Moreover, the method can explicitly incorporate any a priori knowledge the user may have concerning ellipse parameters. The method is based on techniques from projective geometry and on the Hough transform. This technique can significantly reduce interpretation and computation time by automatically extracting only those features or geometric parameters of interest from images and making exact use of a priori information.

We describe extensions to the "best-basis" method which select orthonormal bases suitable for signal classification and regression problems from a large collection of orthonormal bases. For classification problems, we select the basis which maximizes relative entropy of time-frequency energy distributions among classes. For regression problems, we select the basis which tries to minimize the regression error. Once these bases are selected, a small number of most significant coordinates are fed into a traditional classifier or regression method such as linear discriminant analysis (LDA) or classification and regression tree (CART). The performance of these statistical methods is enhanced since the proposed methods reduce the dimensionality of the problems without losing important information for the problem at hand. The basis functions which are well-localized in the time-frequency plane are used as feature extractors. We also compare their performance with the traditional methods using a synthetic example.

A shift-invariant multiresolution representation of signals or images using dilations and translations of the autocorrelation functions of compactly supported wavelets is proposed. Although this set of functions does not form an orthonormal basis, a number of properties of the autocorrelation functions of the compactly supported wavelets make them useful for signal and image analysis. Unlike wavelet-based orthonormal representations, the representation has symmetric analyzing functions, shift invariance, natural and simple iterative interpolation schemes, and a simple algorithm for finding the locations of the multiscale edges as zero crossings. A noniterative method is developed for reconstructing signals from their zero crossings (and slopes at these zero crossings) in the representation. This method reduces the problem to that of solving a system of linear equations.>

Recovering a broadband object from noisy bandlimited observations is an ill-posed problem. If the object has discontinuities, the solution by the conventional Tikhonov regularization method loses them, although the solution can be stably obtained. It is shown that a structural constraint such as the number of spikes/discontinuities in the solution plays an important role in recovering the missing frequency information in the presence of noise. This nonlinear constrained minimization problem has been solved by data adaptive regularization. Applications of this method to synthetic data have shown its superior resolving power and stability compared to the conventional Tikhonov regularization method.>

The E-filter is a homomorphic filter proposed by D.J.H. Moore and D.J. Parker (1973) which differentiates the small and large amplitude components of a signal. A description is given of a generalization of the E-filter to include a control parameter. Selective tuning of this parameter causes the behavior of the E-filter to range from linear to highly nonlinear. It is found that the E-filter can be simply implemented and has several properties which make it attractive for use in edge detection problems. The superiority of the E-filter to sale-space filtering for edge detection in one-dimensional systems is shown.>