Interpolating/crossfading a stack of matrices
 

Hi,

I want to interpolate (with quadratic splines) a stack of 2D-arrays/matrices y1, y2, y3, ... in a third dimension (which I call x) e.g. for crossfading images. I already have a working code which unfortunately still contains two explicit loops over the rows and colums of the matrices. Inside these loops I simply use 'interp1d' from scipy suitable for 1D-interpolations. Is anybody here aware of a better, more efficient solution of my problem? Maybe somewhere out there a compiled routine for my problem already exists in apython library... :-)

My code:

-----============================================-----
from scipy.interpolate import interp1d
from numpy import array, empty_like, dstack

x = [0.0, 0.25, 0.5, 0.75, 1.0]

y1 = array([[1, 10, 100, 1000], [1, 10, 100, 1000]], float)
y2 = array([[2, 20, 200, 2000], [2, 20, 200, 2000]], float)
y3 = array([[3, 30, 300, 3000], [4, 40, 400, 4000]], float)
y4 = array([[4, 40, 400, 4000], [8, 80, 800, 8000]], float)
y5 = array([[5, 50, 500, 5000], [16, 160, 1600, 16000]], float)

y = dstack((y1, y2, y3, y4, y5))

y_interpol = empty_like(y[:, :, 0])
i_range, j_range = y.shape[:2]

for i in xrange(i_range):
for j in xrange(j_range):
# interpolated value for x = 0.2
y_interpol[i,j] = interp1d(x, y[i, j,:], kind='quadratic')(0.2)

print y_interpol
-----============================================-----

Cheers, Raphael

Re: Interpolating/crossfading a stack of matrices
 

On 9 January 2013 16:02, <raphael@mameghani.de> wrote:
> Hi,
>
> I want to interpolate (with quadratic splines) a stack of 2D-arrays/matrices y1, y2, y3, ... in a third dimension (which I call x) e.g. for crossfading images. I already have a working code which unfortunately still contains two explicit loops over the rows and colums of the matrices. Inside theseloops I simply use 'interp1d' from scipy suitable for 1D-interpolations. Is anybody here aware of a better, more efficient solution of my problem? Maybe somewhere out there a compiled routine for my problem already exists ina python library... :-)

It's possible. I wouldn't be surprised if there wasn't any existing
code ready for you to use.

>
> My code:
>
> -----============================================-----
> from scipy.interpolate import interp1d
> from numpy import array, empty_like, dstack
>
> x = [0.0, 0.25, 0.5, 0.75, 1.0]
>
> y1 = array([[1, 10, 100, 1000], [1, 10, 100, 1000]], float)
> y2 = array([[2, 20, 200, 2000], [2, 20, 200, 2000]], float)
> y3 = array([[3, 30, 300, 3000], [4, 40, 400, 4000]], float)
> y4 = array([[4, 40, 400, 4000], [8, 80, 800, 8000]], float)
> y5 = array([[5, 50, 500, 5000], [16, 160, 1600, 16000]], float)
>
> y = dstack((y1, y2, y3, y4, y5))
>
> y_interpol = empty_like(y[:, :, 0])
> i_range, j_range = y.shape[:2]
>
> for i in xrange(i_range):
> for j in xrange(j_range):
> # interpolated value for x = 0.2
> y_interpol[i,j] = interp1d(x, y[i, j,:], kind='quadratic')(0.2)
>
> print y_interpol
> -----============================================-----

Since numpy arrays make it so easy to form linear combinations of
arrays without loops I would probably eliminate the loops and just
form the appropriate combinations of the image arrays. For example, to
use linear interpolation you could do:

def interp_frames_linear(times, frames, t):
'''times is a vector of floats
frames is a 3D array whose nth page is the image for time t[n]
t is the time to interpolate for
'''
# Find the two frames to interpolate between
# Probably a better way of doing this
for n in range(len(t)-1):
if times[n] <= t < times[n+1]:
break
else:
raise OutOfBoundsError

# Interpolate between the two images
alpha = (t - times[n]) / (times[n+1] - times[n])
return (1 - alpha) * frames[:, :, n] + alpha * frames[:, :, n+1]

I'm not really sure how quadratic interpolation is supposed to work
(I've only ever used linear and cubic) but you should be able to do
the same sort of thing.


Oscar

Re: Interpolating/crossfading a stack of matrices
 

>> Hi,
>>
>> I want to interpolate (with quadratic splines) a stack of 2D-arrays/matrices
>> y1, y2, y3, ... in a third dimension (which I call x) e.g. for crossfading
>> images. I already have a working code which unfortunately still contains two
>> explicit loops over the rows and colums of the matrices. Inside these loops I
>> simply use 'interp1d' from scipy suitable for 1D-interpolations. Is anybody
>> here aware of a better, more efficient solution of my problem? Maybe
>> somewhere out there a compiled routine for my problem already exists in a
>> python library... :-)

> Since numpy arrays make it so easy to form linear combinations of
> arrays without loops I would probably eliminate the loops and just
> form the appropriate combinations of the image arrays. For example, to
> use linear interpolation you could do:
>
>
>
> def interp_frames_linear(times, frames, t):
>
> '''times is a vector of floats
>
> frames is a 3D array whose nth page is the image for time t[n]
>
> t is the time to interpolate for
>
> '''
>
> # Find the two frames to interpolate between
>
> # Probably a better way of doing this
>
> for n in range(len(t)-1):
>
> if times[n] <= t < times[n+1]:
>
> break
>
> else:
>
> raise OutOfBoundsError
>
>
>
> # Interpolate between the two images
>
> alpha = (t - times[n]) / (times[n+1] - times[n])
>
> return (1 - alpha) * frames[:, :, n] + alpha * frames[:, :, n+1]
>
>
>
> I'm not really sure how quadratic interpolation is supposed to work
> (I've only ever used linear and cubic) but you should be able to do
> the same sort of thing.
>
> Oscar

Indeed, the 'manual' reimplementation of the interpolation formula using numpy arrays significantly sped up the code. The numexpr package made it even faster. Thanks a lot for your advice!

Raphael

Re: Interpolating/crossfading a stack of matrices
 

>> Hi,
>>
>> I want to interpolate (with quadratic splines) a stack of 2D-arrays/matrices
>> y1, y2, y3, ... in a third dimension (which I call x) e.g. for crossfading
>> images. I already have a working code which unfortunately still contains two
>> explicit loops over the rows and colums of the matrices. Inside these loops I
>> simply use 'interp1d' from scipy suitable for 1D-interpolations. Is anybody
>> here aware of a better, more efficient solution of my problem? Maybe
>> somewhere out there a compiled routine for my problem already exists in a
>> python library... :-)

> Since numpy arrays make it so easy to form linear combinations of
> arrays without loops I would probably eliminate the loops and just
> form the appropriate combinations of the image arrays. For example, to
> use linear interpolation you could do:
>
>
>
> def interp_frames_linear(times, frames, t):
>
> '''times is a vector of floats
>
> frames is a 3D array whose nth page is the image for time t[n]
>
> t is the time to interpolate for
>
> '''
>
> # Find the two frames to interpolate between
>
> # Probably a better way of doing this
>
> for n in range(len(t)-1):
>
> if times[n] <= t < times[n+1]:
>
> break
>
> else:
>
> raise OutOfBoundsError
>
>
>
> # Interpolate between the two images
>
> alpha = (t - times[n]) / (times[n+1] - times[n])
>
> return (1 - alpha) * frames[:, :, n] + alpha * frames[:, :, n+1]
>
>
>
> I'm not really sure how quadratic interpolation is supposed to work
> (I've only ever used linear and cubic) but you should be able to do
> the same sort of thing.
>
> Oscar

Indeed, the 'manual' reimplementation of the interpolation formula using numpy arrays significantly sped up the code. The numexpr package made it even faster. Thanks a lot for your advice!

Raphael