I will be working on these subjects for a term to get more idea on image processing. You should know some numerical analyses then dig into Finite Elements and by playing with B-Splines, you will get good idea on how image processing actually works. Reducing degree of complex differential equations with B-Splines can be done easily. Following is a part of an article I found good for starting point.
Consequently, reconstruction methods better than linear interpolation became an attractive filed of research. Hou and Andrews  investigated, at the end of the seventies, the applicability of B-splines for image processing tasks like interpolation, smoothing, filtering, enlargement, and reduction. They concentrate on how to choose an optimal and yet easy to implement basis function, the so-called B-spline, and use it for interpolation and data smoothing. Only a visual comparison of B-spline interpolation to other interpolation methods is given by the authors, whereby they compare it only to nearest neighbor and linear interpolation. For testing purposes they performed magnification and minification on digital pictures and come to the result that B-spline interpolation is superior. Read More . . .