Advanced deconvolution techniques and medical radiography Adrian Leslie

Advanced deconvolution techniques and medical radiography Adrian Leslie Jannetta Ph. D. Dissertation in 2005

Measure of Image Quality • MTF(Modulation Transfer Function) and Spatial Resolution – Point Spread Function

• Signal to Noise Ratio – For a pair of image – For a single image

• Contrast resolution and windowing – Contrast resolution: ability of an image to discern the contrast difference • ex. 8 bits fir 256 levels of contrast difference – Windowing techniques for data with 1024 levels to display 256 levels 1 window 0

• Contrast, signal and noise – Contrast to noise ratio

Image Restoration • Difference between image enhancement and image restoration – Enhancement: • does not retrieve any new information from an image but makes detail already in an image easier to see. – Restoration • Undo the degradation process, which involves the form of missing data, blurring, noise or combinations ofthose effects.

Types of deconvolution methods • Image formation model • Types of deconvolution methods – – – Algebraic deconvolution Inverse or pseudo-inverse filtering Wiener filtering Regularized deconvolution Other method

Algebraic deconvolution

• Matrix representation of PSF – Sparse Block-Circulant matrix A

• Singular Value Decomposition

Generalized inverse for huge matrix is necessary even though there are several fast algorithms.

Inverse and pseudoinverse filtering

Wiener Filtering Adjustable parameter is chosen by the user to balance sharpness against noise in the restored image.

Regularized Deconvolution • From the original to degraded image is many to one mapping so that estimation problem from observed image is ill-posed. – Need constraint to choose an estimate from many alternatives – Can be formulated by Bayesian estimation of latent true image from observed image with prior information – Minimize the objective function with Lagrange multiplier

• Constrained Least Square – Tendency to fit the data too strongly in least mean squared error – Noisy observed data is appeared in the estimate – Apply the smoothness constraint • Minimize the Laplacian values of image

Iterative Deconvolution • Basic Algorithm – Assume that PSF of degradation is known – Iterate – Problem • Not robust when applied noisy image • Hard to control the algorithm to converge

• RL(Richardson-Lucy) deconvolution – Although RL decon-volution converges to the mathematically correct solution, a better quality restoration is usually arrived at before the method converges. – RL deconvolution has a tendency to overfit the restored image to the data (and therefore, the noise).

Bayesian Approach for Image Restoration

MAP is 1/20, but the probability [0, 1/10] is 0. 1. MAP is Ok but rarely happens. The expected value is ½.

Restoration is an ill-posed Problem • Ill posed problem if, under appropriate conditions, the solution fails to satisfy one or more of the following statements: – The solution exists – The solution is unique – The solution is stable • If the noise is involved, restoration problem is many to one mapping even though the degradation process is known. • It can be modelled by Bayesian network.

Maximum Entropy Deconvolution • An image solution is sought which has the least uncertain structure of all the solutions that are consistent with the measured data, equivalently to choosing the image having maximum entropy.

• • Images restored by Historic MEM tend to be grainy in appearance at the very finest scales: there can often be very little correlation between neighboring pixels within the image, because pixel correlations are not be introduced into the restored image. Allow pixel correlations in the restored image. • Hybrid Maximum Entropy Method with Intrinsic Correlation Function

TOTAL VARIATION-BASED IMAGE DECONVOLUTION: A MAJORIZATION-MINIMIZATION APPROACH Jos´e M. Bioucas-Dias, M´ario A. T. Figueiredo, and Jo˜ao P. Oliveira