One of the main techniques used to de-noise and deblur signals and images is regularization, which is based on the fact that signals and images are usually smoother than noise. Traditional Tikhonov regularization assumes that signals and images are differentiable, but, as Mandelbrot has shown in his fractal theory, many signals and images are not differentiable. To de-noise and de-blur such images, researchers have designed a heuristic method of ℓp-regularization. ℓp-regularization leads to good results, but it is not used as widely as should be, because it lacks a convincing theoretical explanation - and thus, practitioners are often reluctant to use it, especially in critical situations. In this paper, we show that fuzzy techniques provide a theoretical explanation for the ℓp-regularization. Fuzzy techniques also enables us to come up with natural next approximations to be used when the accuracy of the ℓp-based denoising and de-blurring is not sufficient.

In many real-life situations, we need to reconstruct a blurred image in situations when no information about the blurring is available. This problem is known as the problem of blind deconvolution. There exist techniques for solving this problem, but these techniques are not rotation-invariant. Thus, the result of using this technique may change with rotation. So, if we rotate the image a little bit, the method, in general, leads to a different deconvolution result. Therefore, even when the original reconstruction is optimal, the reconstruction of a rotated image will be different and, thus, not optimal. To improve the quality of image decomposition, it is desirable to modify the current state-of-the art techniques by making them rotation-invariant. In this paper, we show how this can be done, and we show that this indeed improves the quality of blind deconvolution.

This paper demonstrates the design of separable complex orthogonal perfect reconstruction filters (OPRFs), which are defined to be complex OPRPs in which the real or imaginary part is also a valid OPRF. Separable filters greatly simplify the task of designing complex OPRFs by reducing it to a two step process: design a real OPRF (using now well known methods), then finding a "complex complementary" filter that results in an overall complex OPRF This paper parameterizes, given any real OPRF, filters complex complementary to it. The parameterization is based on simple angle relationships in the lattice representation of each filter. Also shown is a simple method for designing complex wavelets from separable complex OPRFs.

Computer modeling programs that generate three-dimensional (3-D) data on fine grids are capable of generating very large amounts of information. These data sets, as well as 3-D sensor/measured data sets, are prime candidates for the application of data compression algorithms. A very flexible and powerful compression algorithm for imagery data is the newly released JPEG 2000 standard. JPEG 2000 also has the capability to compress volumetric data, as described in Part 2 of the standard, by treating the 3-D data as separate slices. As a decoder standard, JPEG 2000 does not describe any specific method to allocate bits among the separate slices. This paper proposes two new bit allocation algorithms for accomplishing this task. The first procedure is rate distortion optimal (for mean squared error), and is conceptually similar to postcompression rate distortion optimization used for coding codeblocks within JPEG 2000. The disadvantage of this approach is its high computational complexity. The second bit allocation algorithm, here called the mixed model (MM) approach, mathematically models each slice's rate distortion curve using two distinct regions to get more accurate modeling at low bit rates. These two bit allocation algorithms are applied to a 3-D Meteorological data set. Test results show that the MM approach gives distortion results that are nearly identical to the optimal approach, while significantly reducing computational complexity.

Summary form only given. Many scientific applications require that image data be stored in floating point format due to the large dynamic range of the data. These applications pose a problem if the data needs to be compressed since modern image compression standards, such as JPEG2000, are only defined to operate on fixed point or integer data. This paper proposes straightforward extensions to the JPEG2000 image compression standard which allow for the efficient coding of floating point data. The extensions are based on the idea of representing floating point values as "extended integers", and these extensions maintain desirable properties of JPEG2000, such as scalable embedded bit streams and rate distortion optimality. Like JPEG2000, the proposed methods can be applied to both lossy and lossless compression. However, the discussion in this paper focuses on, and the test results are limited to, the lossless case. Test results show that one of the proposed lossless methods improve upon the compression ratio of standard methods such as gzip by an average of 16%.

The problem of optimal bit rate allocation for 3-D JPEG2000 compression is dicussed. In this paper we consider data where the 2-D slices are compressed using JPEG2000, and the third dimension is decorrelated using the Karhunen-Loeve Transform. Here two new methods are proposed. The first approach is called the Rate Distortion Optimal (RDO) method and is based on Post-Compression Rate-Distortion (PCRD) optimization concept. The second approach is here called the Mixed Model (MM) approach and consists of extending the traditional high-resolution model with a region that is accurate for low bit rates. The proposed bit allocation methods are tested by applying them to Meteorological (Met) data. The specific data set used was generated by the Battlescale Forecast Model (BFM). The test results shows that these approach significantly reduces the computational complexity.

Summary form only given. Modern image compression standards such as JPEG 2000 are only defined to operate on integer data and, thus, they pose a problem if data need compressing. The typical way to handle floating point data is to do an initial quantization. This approach allows standard techniques to be used on the quantized data and may be acceptable to researchers due to the irreversible data loss. Another compression approach is through the use of strictly lossless methods such as LZW based techniques. The disadvantages of this approach include low compression ratios, no capability for lossy compression, and low flexibility of the compressed data stream. Extensions to the JPEG2000 standard were proposed that allow efficient floating point data coding. These extensions have the advantage of maintaining desirable properties of JPEG2000, including flexible embedded bit streams, lossy and lossless compression image scalability. Bit plane coding is needed for an overall floating point algorithm and can be applied when acceptable wavelet and color transforms are derived. The derivation of the floating point bit plane coding algorithms considers only the narrow problem of JPEG2000. The proposed approach for bit plane coding represents floating point values as "big integers" and codes these integers directly as in JPEG2000. A separate eligibility pass every several bit planes was introduced to lower the number of coefficients coded in each coding pass resulting in improved efficiency.

Shannon (1948) showed that the problems of source and channel coding could be considered separately, without loss of performance, if complexity is not an issue. Previous investigations have shown a performance gain for more practical problems where channel and source coding have been jointly designed. This paper investigates the joint source/channel coding problem when using trellis coded quantization (TCQ). TCQ has shown to offer improved source coding performance (when compared to Lloyd-Max quantization) while additionally offering some error immunity. When using TCQ in practical transmission systems, the question arises as to the number of bits that should be allocated to source and channel coding for overall best performance. Results here show that for binary symmetric channels with medium SNRs, better overall mean square error (MSE) performance results from taking bits from the source coder and allocating them to the channel coder. However, for high and low SNRs better MSE results from allocating all bits to source coder.

This paper shows how to derive bit allocations for biorthogonal wavelet coders which result in minimum reconstruction error. It is shown that the effect of using biorthogonal filters is to weight the amount of quantization error which appears in the reconstructed output. The amount of weighting can be determined from the biorthogonal filters used and is a measure of the closeness of the biorthogonal filters to the class of orthogonal filters. Once the weights are computed, they can be used to derive bit allocations which minimize the output mean square error (MSE). These results extend previous results since no bit allocation procedures have been proposed for biorthogonal wavelet coders which minimize the MSE.