#### Technical Report USC-IPI-730

“The Degrees of Freedom of Sampled Images”

by Dennis Grant McCaughey

June 1977

This dissertation presents a degree of freedom or information content analysis of images and imaging systems in the context of digital image processing. As such it represents an attempt to quantify the number of truly independent samples one gathers with imaging devices.

In quantifying the degrees of freedom of an imaging system it is necessary to develop an appropriate model. In this work the imaging system is modeled as a linear system through the continuous-discrete imaging equation. The associated gram matrix is then employed as an aid in defining the system degrees of freedom. The gram matrix eigenvalues are shown to be related to those of the associated continuous-continuous model and can be used to predict the discretized system performance. These ideas are then applied to the tomographic or projection imaging system, and result in the ability to predict the performance of this system by indicating where redundant data is achieved, and the best ways of increasing the degrees of freedom with a minimum sample increase.

The degrees of freedom of a sampled image itself are developed as an approximation problem. Here bicubic splines with variable knots are employed in an attempt to answer the question as to what extent images are finitely representable in the context of digital computer.

Relatively simple algorithms for good knot placement are given, and result in spline approximations that achieve significant parameter reductions at acceptable error levels. The knots themselves are shown to be useful as an indicator of image activity, and have potential as an image segmentation device.

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