imresize - trying to understand the bicubic interpolation

Question

I\'m trying to understand the function:

function [weights, indices] = contributions(in_length, out_length, ...
                                            sc         


        

Answer 1

imresize accomplishes anti-aliasing when downsizing an image by simply broadening the cubic kernel, rather than a discrete pre-processing step.

For a kernel_width of 4 pixels (8 after re-scaled), where the contributions function utilizes 10 neighbors for each pixel, the kernel vs h (scaled kernel) look like (unnormalized, ignore x-axis):

This is easier than first performing a low-pass filter or Gaussian convolution in a separate pre-processing step.

The cubic kernel is defined at the bottom of imresize.m as:

function f = cubic(x)
% See Keys, "Cubic Convolution Interpolation for Digital Image
% Processing," IEEE Transactions on Acoustics, Speech, and Signal
% Processing, Vol. ASSP-29, No. 6, December 1981, p. 1155.

absx = abs(x);
absx2 = absx.^2;
absx3 = absx.^3;

f = (1.5*absx3 - 2.5*absx2 + 1) .* (absx <= 1) + ...
                (-0.5*absx3 + 2.5*absx2 - 4*absx + 2) .* ...
                ((1 < absx) & (absx <= 2));

PDF of the referenced paper.

The relevant part is equation (15):

This is a specific version of the general interpolation equations for a = -0.5 in the following equations:

a is usually set to -0.5, or -0.75. Note that a = -0.5 corresponds to the Cubic Hermite spline, which will be continuous and have a continuous first derivitive. OpenCV seems to use -0.75.

However, if you edit [OPENCV_SRC]\modules\imgproc\src\imgwarp.cpp and change the code :

static inline void interpolateCubic( float x, float* coeffs )
{
    const float A = -0.75f;
    ...

to:

static inline void interpolateCubic( float x, float* coeffs )
{
    const float A = -0.50f;
    ...

and rebuild OpenCV (tip: disable CUDA and the gpu module for short compile time), then you get the same results. See the matching output in my other answer to a related question by the OP.