Statistics of the contrast of coherent images

Contrast optimization, also known as image sharpening, is a method that can be used to estimate phase errors in coherent images. However, the contrast measure of a coherent image is a random variable because of the speckle present in coherent images. The variance of this measure puts a limit on the ability of contrast optimization to focus an image. We derive the probability distribution function of the most common contrast measure, the sum of the pixel intensities raised to a power. These statistics are then verified by a number of speckle simulations and compared with measured statistics from synthetic aperture sonar images. The developed statistics can be used as a tool to understand and improve the method of contrast optimization as well as assess its performance for a given imaging system. They can also be used to predict the effect of certain image processing operations on the contrast.