[Numpy-discussion] nd_image.affine_transform edge effects

[Stefan van der Walt]

On Sat, Mar 24, 2007 at 01:41:21AM -0400, James Turner wrote:
> That's hard to say. Just because it's mainly a continuous-tone image
> doesn't necessarily mean it is well sampled everywhere. This depends
> both on the subject and the camera optics. Unlike the data I usually
> work with, I think everyday digital photographs (probably a photo
> scan in the case of Lena) do not generally have the detector sampling
> frequency matched to the optical resolution of the image. If that's
> true, the presence of aliasing in interpolated images depends on the
> structure of the subject and whether the scene has edges or high-
> frequency patterns in it.

Agreed, but the aliasing effects isn't not the problem here, as it
should be visible in the input image as well.  I'd expect a
third-order spline interpolation to be more smooth than a first-order
interpolant, but in the resulting images this isn't the case.

See

http://mentat.za.net/results/lena_small.png
http://mentat.za.net/results/img_rot_30_1.png (1st order spline)
http://mentat.za.net/results/img_rot_30_3.png (3rd order spline)

> Lena has been decimated (reduced in size) prior to the rotation. That
> is definitely a good way to get artefacts, unless an anti-aliasing
> filter is applied before shrinking the image. My impression is that
> this image is probably somewhat undersampled (to understand exactly
> what that means, read up on the Sampling Theorem).

The artefacts arn't visible in the source image (url above).  The
image definately is a scaled down version of the original Lena -- very
interesting, btw, see

http://www.cs.cmu.edu/~chuck/lennapg/lenna.shtml

> investigating it further. One experiment might be to blur the original
> Lena with a Gaussian whose sigma is 1 pixel of the shrunken image
> before actually shrinking her, then do the rotation.

A rotation should take place without significant shifts in colour.
This almost looks like a value overflow problem.

> So I do wonder if the algorithm in nd_image is making this worse
> than it needs to be.

That is my suspicion, too.

> compare the results? I just tried doing a similar rotation in PyRAF on
> a monochrome image with a bicubic spline, and see considerably smaller
> artefacts (just a compact overshoot of probably a few % at the
> edge).

Could you apply the PyRAF rotation on the Lena given above and post
the result?

I always thought we could simply revert to using bilinear and bicubic
polygon interpolation (instead of spline interpolation), but now I
read on wikipedia:

"""
In the mathematical subfield of numerical analysis, spline
interpolation is a form of interpolation where the interpolant is a
special type of piecewise polynomial called a spline. Spline
interpolation is preferred over polynomial interpolation because the
interpolation error can be made small even when using low degree
polynomials for the spline. Thus, spline interpolation avoids the
problem of Runge's phenomenon which occurs when using high degree
polynomials.
"""

http://en.wikipedia.org/wiki/Spline_interpolation

also take a look at

http://en.wikipedia.org/wiki/Runge%27s_phenomenon

So much for side-stepping Runge's phenomenon :)

Cheers
Stéfan