[SciPy-Dev] Spline interpolation in ndimage

[Charles R Harris]

On Tue, Feb 27, 2018 at 4:59 AM, Jaime Fernández del Río <
jaime.frio at gmail.com> wrote:

> Hi all,
>
> We have been discussing in issue #8465
> <https://github.com/scipy/scipy/issues/8465> about the extension modes
> for the interpolation module of ndimage. As some other issues linked in
> that one show, this is an old problem, that has been recurringly discussed,
> and there mostly is agreement on what the correct behavior should be. But
> as all things ndimage, implementing any change is hard, in this case not
> only because the code is complicated, but also because of the mathematical
> subtleties of the interpolation method used.
>
> In trying to come up with a way forward for this I think I can handle the
> code complexity, but am having trouble being sure that I come up with a
> sound math approach. I think I have more or less figured it out, but I
> don't have a good academic background on signal processing, so was hoping
> that someone who does (I'm thinking of you, Chuck!) can read my longish
> description of things below and validate or correct it.
>
> My main source of knowledge has been:
>
> M. Unser, "Splines: a perfect fit for signal and image processing," in
> IEEE Signal Processing Magazine, vol. 16, no. 6, pp. 22-38, Nov 1999.
>
> Recommendations for bigger, better readings on the subject are also
> welcome!
>
> -----
>
> In what follows I'll assume the input image is 2-D and NxN for simplicity,
> but I think everything generalizes easily
>
> If I'm understanding it right, all of our interpolation functions use
> B-splines for interpolation. So instead of having NxN discrete values on a
> grid, we have NxN B-splines centered at the grid points, and we keep the
> NxN coefficients multiplying each spline to reproduce the original image at
> the grid points exactly. If the spline order is < 2, the spline
> coefficients are the same as the image values. But if order >= 2 (and we
> use 3 by default), these have to be calculated in a non-trivial fashion.
> This can be done efficiently by applying a separable filter, which is
> implemented in ndimage.spline_filter.
>
> Because B-splines have compact support, when using splines of order n we
> only need to consider the B-splines on an (n + 1)x(n + 1) grid neighborhood
> of the point being interpolated. This is more or less straightforward,
> until you move close to the edges and all of a sudden some of the points in
> your grid neighborhood fall outside the original image grid. We have our
> extend mode, which controls how this points outside should be mapped to
> points inside. But here is where things start getting tricky...
>
> When the spline coefficients are computed (i.e. when ndimage.spline_filter
> is called), assumptions have to be made about boundary conditions. But
> ndimage.spline_filter does not take a mode parameter to control this! So
> what I think ndimage does is compute the spline coefficients assuming
> "mirror symmetric" boundary conditions, i.e.:
>
> a b c d c b | a b c d | c b a b c d
>

For "a, b, c", I would write that as "..., a, b, c, b, a, b, c, ..."
extended to infinity in both directions because we are applying IIR
filters. Fortunately, the IIR filters fall off rapidly, so one need not
take too many extended points to start them up. Counter intuitively, I
think the interpolation "mode" only applies to the spline coefficients, not
to the original image, resulting in unexpected behavior at the edges.

If we want to have "mode" apply to the image rather than the filtered
results, we can do that, but the result of the filtering would best be a
named tuple containing the mode, the spline order, and,  in the case of a
constant extension, extra edge coefficients. The  coefficients for the
"wrap" and "constant" modes can then be obtained by appropriately extending
the data for starting up the filters. For the "wrap" and "reflect" modes,
the coefficents have the same symmetries, so that can be used to get the
needed coefficients outside the boundaries.

The spline_filter using IIR is just a clever way to solve A*coef = data.
For the cubic case and three data points, A depends on the mode as follows:

|4 2 0|
|1 4 1| x 1/6 (reflect about center of edge pixels -- we do this)
|0 2 4|

|5 1 0|
|1 4 1| x 1/6 (reflect about edge -- we don't do this)
|0 1 5|

|4 1 1|
|1 4 1| x 1/6 (wrap -- we don't do this)
|1 1 4|

Explicitly inverting such matrices also provides a good test to check that
things are done right if we go this way. As you have pointed out, the
proper coefficients will vary depending on the mode.

The spline coefficients have the same symmetry as the image pixels.
Unfortunately, `map_coordinates` reflects about the edges, which doesn't
match any of the coefficients that we compute.


> So if our interpolated point is within the image boundaries, but some of
> the grid points in its (n + 1)x(n + 1) neighborhood fall outside the
> boundary, the code uses a mirror symmetric extension mode to fill in those
> values. This approach has a funny smell to it, but even if it's formally
> incorrect I think it would only be marginally so, as the values are
> probably going to be very close to the correct ones.
>
> The problem comes when the point being interpolated is outside the
> boundaries of the image. We cannot use mirror-symmetric spline coefficients
> to extend if e.g. we have been asked to extend using wrap mode. So what
> ndimage does is first map the value outside the boundaries to a value
> within the boundaries, using the given extension mode, then interpolate it
> as before, using mirror-symmetric coefficients if needed because its (n +
> 1)x(n + 1) neighborhood extends outside. Again, this smells funny, but it
> is either correct or very close to correct.
>
> This is mostly all good and well, except for the "wrap" and "reflect"
> extension modes: in these cases the area within one pixel of the image
> boundaries is different from anything inside the image, so we cannot use
> that approach. So what ndimage does is basically make shit up and use
> something similar, but not quite right. "reflect" is mostly correct, except
> for within that pixel of the boundary, but "wrap" is a surprising and
> confusing mess.
>
> So how do we fix this? I see two ways forward:
>
>    1. What seems the more correct approach would be to compute the spline
>    coefficients taking into account the extension mode to be used, then use
>    the same extension mode to fill in the neighborhood values when
>    interpolating for a point outside the boundaries.
>    1. First question is whether this is doable? I need to work out the
>       math, but for "wrap" it looks like it should be, not 100% sure if also is
>       for "reflect".
>
>
Yes. The constant mode is actually the tricky one. I'd be tempted to make
it orthogonal to "reflect", "wrap", and "nearest", that way we don't need
to compute extra coefficients. Note that if the image isn't prefiltered, we
are running a smoothing filter over it.

>
>    1. Assuming it is it has the main advantage of doing things in a more
>       general and understandable way once you have enough background knowledge.
>       2. It does go a little bit against our API design: you can control
>       whether the input is spline-filtered automatically with a parameter, the
>       idea being that you may want to do the filtering yourself if you are going
>       to apply several different transformations to the same image. If the mode
>       of the filtering has to be synced with the mode of the transformation,
>       letting the user do it themselves is a recipe for disaster, because it's
>       going to lead to very hard to track bugs.
>       3. As elsewhere in ndimage, the current implementation does a lot
>       of caching, which works because it always interpolates for a point within
>       the image boundaries. If we started interpolating for points outside the
>       boundaries without first mapping to within there may be a performance hit
>       which has to be evaluated.
>    1. The other approach is,  for "wrap" and "reflect" modes, pad the
>    input image with an extra pixel in each direction, then compute our
>    current "mirror symmetric" spline coefficients, and leave things as they
>    are right now, aside from some changes to the mapping of values to take the
>    extra pixels into account.
>
>
Are "mirror" and "reflect" actually different? The function documentation
only mentions "reflect".


>
>    1. This looks like a nightmare of special cases everywhere and
>       potential off-by-one errors while putting it together, but it would just go
>       along with the ugliness of the existing code.
>       2. It's unclear what we would do if we are given an input with
>       prefilter=False, so this also breaks the current API, probably even more so.
>
> Any thoughts or recommendations are very welcome!
>

The docstrings need fixing, they are almost useless.

Chuck
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