Affine transformations of images: a least squares formulation

We present a general framework for the design of discrete geometrical transformation operators, including rotations and scaling. The first step is to fit the discrete input image with a continuous model that provides an exact interpolation at the pixel locations. The corresponding image model is selected within a certain subspace V(φ)⊂L₂(R^P) that is generated from the integer translates of a generating function φ; particular examples of this construction include polynomial spline and bandlimited signal representations. Next, the geometrical transformation is applied to the fitted model, and the result is re-projected onto the representation space. This procedure yields a solution that is optimal in the least squares sense. We show that this method can be implemented exactly using a combination of digital filters and a re-sampling step that uses a modified sampling kernel. We then derive explicit implementation formulas for the piecewise constant and cubic spline image models. Finally, we consider image processing examples and show that the present method compares very favorably with a standard interpolation that uses the same model