Why is the output of corr2 of two matrixes not same dimension as the input matrix?

Hi, I am currently implementing an algorithm of segmentation based on flow orientation, and one of the equations in the paper uses autocorrelation and crosscorrelation of image gradient.

The code runs fine, but when I try to reshape the final values into a matrix with same dimension as the input image, I receive error. The following is my code:

close all
x = imread(imfile);
x = im2double(x);
%%Gradient - orientation estimate
[Gx,Gy] = imgradientxy(x);
gxx = xcorr2(Gx(:,:));
gyy = xcorr2(Gy(:,:));
gxy = xcorr2(Gx(:,:), Gy(:,:));
angle = 0.5.*atan((2.*gxy)./(gxx - gyy));    % Equation 4
temp = (angle< 0);
angle(temp) = angle(temp) + 2.*pi;     % angle within 0 to 2pi
angle = angle / pi *180;

Equation 4:

I spent some time looking at the length of each entry, and it turned out that all the values after processing by corr2 are double of the initial size; i.e.

input_image = 450 x 369

gx (after imgradientxy) = 450 x 450

gxx = 899 x 750

angle = 899 x 737

My questions:

1) How to make the matrix after autocorrelation & cross correlation same as the initial image dimension?

2) Why is there a difference in the matrix dimension after processing by imgradientxy ?

Any help would be greatly appreciated, thanks!

Answer 1

[Rahul Goel]

Hi Alexander, I tried your code with a few images which ship with MATLAB, for example “pout.tif” image file. But I could not reproduce this issue. However, I observed the following:

1. Output of imgradientxy should not have different dimensions from that of input matrix.

As the documentation states,

[Gx,Gy] = imgradientxy(I)

returns the directional gradients, Gx and Gy, the same size as the input image I.

For image “pout.tif”,

close all
x = imread('pout.tif');
x = im2double(x);
[Gx,Gy] = imgradientxy(x);

>> whos x
  Name        Size              Bytes  Class     Attributes

    x         291x240            558720  double              

>> whos Gx
  Name        Size              Bytes  Class     Attributes

    Gx        291x240            558720  double

2. Output of xcorr2(A) has different dimensions from that of the input matrix. If A is a matrix of size P X Q, "xcorr2(A)" will produce a matrix of size (P+P-1) X (Q+Q-1).

>> gxx = xcorr2(Gx(:,:));
>> gyy = xcorr2(Gy(:,:));
>> gxy = xcorr2(Gx(:,:), Gy(:,:));
>> whos gxx
  Name        Size               Bytes  Class     Attributes

    gxx       581x479            2226392  double              

>> whos gxy
  Name        Size               Bytes  Class     Attributes

    gxy       581x479            2226392  double              

>> whos gyy
  Name        Size               Bytes  Class     Attributes

    gyy       581x479            2226392  double

So, the change in dimensions after “xcorr2” is an expected behavior. You mentioned that you wanted the dimensions of output of “xcorr2” to be similar as that of the input matrix. However, as we have seen, it is not possible with “xcorr2” function. It would be more insightful if you could state the motivation behind that requirement. The user community will be able to assist you better with that information.

[Alexander Hum]

Thanks for your reply! The explanation is really helpful (since I don't have too much knowledge regarding autocorrelation).

I am trying to feed in the orientation values to trace boundary by using an average of flow direction, essentially the algorithm is here: http://home.cc.umanitoba.ca/~thomasg/MP/Research/DIP/samiee.htm

I tried downsampling, but it doesn't give me the same dimension either (with x-axis still significantly longer).

Currently I skipped the equation, and replaced by the Gdir values obtained by imgredient; but it would be great if I get some pointers (or can this equation by replaced by Gdir of imgredient?)

Thanks for your help!