Mathematical Models of Image Processing

The purpose of this thesis is to develop various advanced linear algebra techniques that apply to image processing. With the increasing use of computers and digital photography, being able to manipulate digital images efficiently and with greater freedom is extremely important. By applying the tools of linear algebra, we hope to improve the ability to process such images. We are especially interested in developing techniques that allow computers to manipulate images with the least amount of human guidance. In Chapter 2 and Chapter 3, we develop the basic definitions and linear algebra concepts that lay the foundation for later chapters. Then, in Chapter 4, we demonstrate techniques that allow a computer to rotate an image to the correct orientation automatically, and similarly, for the computer to correct a certain class of color distortion automatically. In both cases, we use certain properties of the eigenvalues and eigenvectors of covariance matrices. We then model color clashing and color variation in Chapter 5 using a powerful tool from linear algebra known as the Perron-Frobenius theorem. Finally, we explore ways to determine whether an image is a blur of another image using invariant functions. The inspiration behind these functions are recent applications of Lie Groups and Lie algebra to image processing