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Geometric Transformation of Images using Spline-Interpolation

Module by: Sujesh Sreedharan

Summary: 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 φnxy φ x y n defined as

φnx-ky-l=βnx-kβny-l φ x k y l n β x k n β y l n (1)
Using these bases we can interpolate a discrete image fkl f k l and come up with a continuous function fxy f x y over ℝ2 2 .
fxy=∑k∑lcklβnx-kβny-l f x y k k l l c k l β x k n β y l n (2)
The coefficients ckl c k l are obtained by fitting the discrete image to the 2D spline basis.

Image Transformation

Let xy=Guv x y G u v be the transformation (rotation, scaling) of the domain, where uv u v describe the transformed plane and xy x y the original image plane. Then our problem is to find the samples of the transformed image gmn g m n from the samples of original image fkl f k l . Here is how we go about it:

  • Calculate ckl c k l from the pixel values fkl f k l
  • For each mn m n on transformed image find corresponding source location xy x y
  • Compute fxy f x y using the spline model, now gmn=fxy g m n f x y

Example 1

In Figure 1 we look at the rotation of an image by certain angle and compare the fits using φnmn φ m n n n=0 n 0 (nearest neighbour), n=1 n 1 (bilinear), n=3 n 3 (bicubic).

Figure 1: 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.
Figure 1 (thello.png)

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