[SciPy-Dev] Spline interpolation in ndimage

[Charles R Harris]

On Thu, Mar 1, 2018 at 9:15 AM, Charles R Harris <charlesr.harris at gmail.com>
wrote:

>
>
> 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
>>
>> 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.
>>
>
> The problem with the factorization is that it assumes infinite data points
> in all dimensions, i.e, no explicit boundary conditions. With finite data
> there are edge effects when the data is deconvolved to get the spline
> coefficients.  The way that ndimage deals with that is to extend the data
> using reflection and start far enough away that the edge effects have died
> away by the time the "real" data has been reached. How far away that is, is
> heuristic. I think it should not be too difficult to extend the data in the
> other ways, but note that since uniform splines are being used, the
> relevant coefficients lie outside the boundary and need to be picked up
> from the correct spots in the interior using the symmetries.
>
> IIRC, the b-splines are always centered at zero so that they are
> symmetrical. For odd order splines that will be at the center of a pixel
> (pixel points), for even order splines at an edge, pixel centers at half
> integer points. The data can always be considered to be at pixel centers,
> but the splines are displaced. I don't remember if ndimage treats that
> correctly.
>
>
>
>> 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".
>>       2. Assuming it is it has the main advantage of doing things in a
>>       more general and understandable way once you have enough background
>>       knowledge.
>>       3. 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.
>>       4. 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.
>>    2. 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.
>>       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!
>>
>
>
To explicate a bit more as to why the b-splines are centered, consider
quadratic and cubic splines when the data points are considered to occur at
the pixel centers.

*Quadratic (even order)*


1. If the data points are taken to be halfway between  knot points, we need
to deconvolve the sequence `array([1, 6, 1])/8`. The Fourier transform of
that has no zeros, indeed, is rather smooth, and two extra coefficients,
one at each end, are required to interpolate out to the pixel edges. It is
all nicely symmetrical.

2. If the data points are taken to correspond to the knot points, we need
to deconvolve the sequence `array([1, 1])/2`. The Fourier transform of that
sequence has a zero at the Nyquist, not good, and one extra coefficient is
needed at some arbitrary end in order to get interpolation out to the pixel
edges.

*Cubic (odd order)*

1. Taking the data points to correspond to the knot points, we need to
deconvolve the sequence `array([1, 4, 1])/6`, whose Fourier transform is
nicely behaved. However *four* extra extra coefficients, two at each end,
are required to interpolate out to the pixel edges. The "extra"
coefficients are needed in order to cover the outer half of the edge
pixels, otherwise we are extrapolating. I haven't looked, but I wonder if
ndimage gets that right?


In both cases, the corner pixels may be a bit of a problem. I haven't
thought through that bit.

Chuck
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