Spatial transformations: Inverse mapping

Posted by Steve Eddins, May 5, 2006

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I wrote last week about the forward mapping method of spatially transforming images. Because of the disadvantages of the forward mapping method, most of the practical implementations use a different technique, called inverse mapping.

Here's how it works:

Locate your output image pixel grid somewhere in output space. Then for each output pixel on the grid:

Apply the inverse spatial transformation to determine the corresponding location in input space: (u_k,v_k) = T^-1{(x_k,y_k)}.
Using the input image pixels nearest to (u_k,v_k), interpolate to get an approximate value for the input image at (u_k,v_k).
Use that value for the k-th output pixel.

This diagram illustrates the procedure:

This method completely avoids problems with gaps and overlaps. You just have to decide which interpolation method to you, and you also have to establish a rule for what value to use when (u_k,v_k) isn't within the input image pixel grid.

The inverse mapping does have its own disadvantages. I'll talk about those later, when I discuss more of the design details of specific Image Processing Toolbox functions.