Geometric Transformation of Images using Spline-Interpolation 2.1 2003/05/05 2003/05/07 Sujesh Sreedharan sujesh@rice.edu Alena Ixmocane Scott oetting@stat.rice.edu Sujesh Sreedharan sujesh@rice.edu Rebecca Willett willett@rice.edu Adan Galvan jago@rice.edu Charlet Reedstrom charlet@rice.edu B-splines Image transforms interpolation splines To perform geometric transforms on discrete images such as a rotation or zooming we need to first fit the discrete data to a continuous function. This can be done using splines. B-splines can be used to interpolate and form the continuos image from the discrete samples. Here we illustrate the use of splines in two-dimensions for transforming discrete images. When we apply a transformation (such as rotation or zooming) it becomes necessary to know the image intensity at a location in between the sample points. This is an interpolation problem and splines come in handy.

Spline Tensor Product Basis We extend the 1D B-spline basis to 2D using tensor products φ x y n defined as φ x k y l n β x k n β y l n Using these bases we can interpolate a discrete image f k l and come up with a continuous function f x y over 2 . f x y k k l l c k l β x k n β y l n The coefficients c k l are obtained by fitting the discrete image to the 2D spline basis.

Image Transformation Let x y G u v be the transformation (rotation, scaling) of the domain, where u v describe the transformed plane and x y the original image plane. Then our problem is to find the samples of the transformed image g m n from the samples of original image f k l . Here is how we go about it: Calculate c k l from the pixel values f k l For each m n on transformed image find corresponding source location x y Compute f x y using the spline model, now g m n f x y In we look at the rotation of an image by certain angle and compare the fits using φ m n n n 0 (nearest neighbour), n 1 (bilinear), n 3 (bicubic).

Illustration of how the jagged edges appear in nearest neighbour interpolation and they progressively smooth out in higher order spline-interpolation. Bicubic spline interpolation is highly popular as cubic splines appear smooth to the human eye.