Entropy Encoding, Hilbert Space and Karhunen-Loeve Transforms

By introducing Hilbert space and operators, we show how probabilities, approximations and entropy encoding from signal and image processing allow precise formulas and quantitative estimates. Our main results yield orthogonal bases which optimize distinct measures of data encoding.Comment: 25 pages, 1 figur

Wavelet-based Multiresolution Local Tomography

We develop an algorithm to reconstruct the wavelet coefficients of an image from the Radon transform data. The proposed method uses the properties of wavelets to localize the Radon transform and can be used to reconstruct a local region of the cross section of a body, using almost completely local data which significantly reduces the amount of exposure and computations in X-ray tomography. This property which distinguishes our algorithm from the previous algorithms is based on the observation that for some wavelet basis with sufficiently many vanishing moments, the ramp-filtered version of the scaling function as well as the wavelet function has extremely rapid decay. We show that the variance of the elements of the null-space is negligible in the locally reconstructed image. Also we find an upper bound for the reconstruction error in terms of the amount of data used in the algorithm. To reconstruct a local region 20 pixels in radius in a 256 \Theta 256 image we require 12:5% of full e..

Wavelet-Based Multiresolution Local Tomography

We develop an algorithm to reconstruct the wavelet coefficients of an image from the Radon transform data. The proposed method uses the properties of wavelets to localize the Radon transform and can be used to reconstruct a local region of the cross section of a body, using almost completely local data which significantly reduces the amount of exposure and computations in X-ray tomography. This property which distinguishes our algorithm from the previous algorithms is based on the observation that for some wavelet basis with sufficiently many vanishing moments, the ramp-filtered version of the scaling function as well as the wavelet function has extremely rapid decay. We show that the variance of the elements of the null- space is negligible in the locally reconstructed image. Also we find an upper bound for the reconstruction error in terms of the amount of data used in the algorithm. To reconstruct a local region 20 pixels in radius in a 256 X 256 image we require 12.5% of full exposure data while the previous methods can reduce the amount of exposure only to 40% for the same case

Wavelet-based Multiresolution Local Tomography

We develop an algorithm to reconstruct the wavelet coefficients of an image from the Radon transform data. The proposed method uses the properties of wavelets to localize the Radon transform and can be used to reconstruct a local region of the cross section of a body, using almost completely local data which significantly reduces the amount of exposure and computations in X-ray tomography. The property which distinguishes our algorithm from the previous algorithms is based on the observation that for some wavelet bases with sufficiently many vanishing moments, the rampfiltered version of the scaling function as well as the wavelet function has extremely rapid decay. We show that the variance of the elements of the null-space is negligible in the locally reconstructed image. Also we find an upper bound for the reconstruction error in terms of the amount of data used in the algorithm. To reconstruct a local region 16 pixels in radius in a 256 \Theta 256 image we require 22% of full expos..