the next code gets an image of a grape that I photograph (is called: 'full_img') and calculate the area of the grape:
RGB = imread(full_img);
GRAY = rgb2gray(RGB);
threshold = graythresh(GRAY);
originalImage = im2bw(GRAY, threshold);
originalImage = bwareaopen(originalImage,250);
SE = strel('disk',10);
IM2 = imclose(originalImage,SE);
originalImage = IM2;
labeledImage = bwlabel(originalImage, 8); % Label each blob so we can make measurements of it
blobMeasurements = regionprops(labeledImage, originalImage, 'all');
numberOfBlobs = length(blobMeasurements);
pixperinch=get(0,'ScreenPixelsPerInch'); %# find resolution of your display
dpix=blobMeasurements(numberOfBlobs).Area; %# calculate distance in pixels
dinch=dpix/pixperinch; %# convert to inches from pixels
dcm=dinch*2.54; %# convert to cm from inches
blobArea = dcm; % Get area.
If I photograph the same grape with the same conditions by different cameras (photographed it from the same distance and the same lightning), will I get the same results? (what if I have a camera of 5 Mega Pixel and 12 Mega Pixel?).
No, it won't. You go from image coordinates to world coordinates using dpix/pixperinch. In general this is wrong. It will only work for a specific image (and that alone), if you know the pixperinch. In order to get the geometric characteristics of an object in an image (eg length, area etc), you must back-project the image pixels in the Cartesian space using the Camera matrix and the inverse projective transformation, in order to get Cartesian coordinates (let along calibrating the camera for lens distortion, which is a nonlinear problem). Then, you can perform the calculations. You code won't work even for the same camera.
See this for more.
Related
I'm trying to resize and reposition a ROI (region of interest) correctly from a low resolution image (256x256) to a higher resolution image (512x512). It should also be mentioned that the two images cover different field of view - the low and high resolution image have 330mm x 330mm and 180mm x 180mm FoV, respectively.
What I've got at my disposal are:
Physical reference point (in mm) in the 256x256 and 512x512 image, which are refpoint_lowres=(-164.424,-194.462) and refpoint_highres=(-94.3052,-110.923). The reference points are located in the top left pixel (1,1) in their respective images.
Pixel coordinates of the ROI in the 256x256 image (named pxX and pxY). These coordinates are positioned relative to the reference point of the lower resolution image, refpoint_lowres=(-164.424,-194.462).
Pixel spacing for the 256x256 and 512x512 image, which are 0.7757 pixel/mm and 2.8444 pixel/mm respectively.
How can I rescale and reposition the ROI (the binary mask) to correct pixel location in the 512x512 image? Many thanks in advance!!
Attempt
% This gives correctly placed and scaled binary array in the 256x256 image
mask_lowres = double(poly2mask(pxX, pxY, 256., 256.));
% Compute translational shift in pixel
mmShift = refpoint_lowres - refpoint_highres;
pxShift = abs(mmShift./pixspacing_highres)
% This produces a binary array that is only positioned correctly in the
% 512x512 image, but it is not upscaled correctly...(?)
mask_highres = double(poly2mask(pxX + pxShift(1), pxY + pxShift(2), 512.,
512.));
So you have coordinates pxX, and pxY in pixels with respect to the low-resolution image. You can transform these coordinates to real-world coordinates:
pxX_rw = pxX / 0.7757 - 164.424;
pxY_rw = pxY / 0.7757 - 194.462;
Next you can transform these coordinates to high-res coordinates:
pxX_hr = (pxX_rw - 94.3052) * 2.8444;
pxY_hr = (pxY_rw - 110.923) * 2.8444;
Since the original coordinates fit in the low-res image, but the high-res image is smaller (in physical coordinates) than the low-res one, it is possible that these new coordinates do not fit in the high-res image. If this is the case, cropping the polygon is a non-trivial exercise, it cannot be done by simply moving the vertices to be inside the field of view. MATLAB R2017b introduces the polyshape object type, which you can intersect:
bbox = polyshape([0 0 180 180] - 94.3052, [180 0 0 180] - 110.923);
poly = polyshape(pxX_rw, pxY_rw);
poly = intersect([poly bbox]);
pxX_rw = poly.Vertices(:,1);
pxY_rw = poly.Vertices(:,2);
If you have an earlier version of MATLAB, maybe the easiest solution is to make the field of view larger to draw the polygon, then crop the resulting image to the right size. But this does require some proper calculation to get it right.
I am trying to allign two images - one rgb and another depth using MATLAB. Please note that I have checked several places for this - like here , here which requires a kinect device, and here here which says that camera parameters are required for calibration. I was also suggested to use EPIPOLAR GEOMETRY to match the two images though I do not know how. The dataset I am referring to is given in rgb-d-t face dataset. One such example is illustrated below :
The ground truth which basically means the bounding boxes which specify the face region of interest are already provided and I use them to crop the face regions only. The matlab code is illustrated below :
I = imread('1.jpg');
I1 = imcrop(I,[218,198,158,122]);
I2 = imcrop(I,[243,209,140,108]);
figure, subplot(1,2,1),imshow(I1);
subplot(1,2,2),imshow(I2);
The two cropped images rgb and depth are shown below :
Is there any way by which we can register/allign the images. I took the hint from
here where basic sobel operator has been used on both the rgb and depth images to generate an edge map and then keypoints will need to be generated for matching purposes. The edge maps for both the images are generated here.
.
However they are so noisy that I do not think we will be able to do keypoint matching for this images.
Can anybody suggest some algorithms in matlab to do the same ?
prologue
This answer is based on mine previous answer:
Does Kinect Infrared View Have an offset with the Kinect Depth View
I manually crop your input image so I separate colors and depth images (as my program need them separated. This could cause minor offset change by few pixels. Also as I do not have the depths (depth image is 8bit only due to grayscale RGB) then the depth accuracy I work with is very poor see:
So my results are affected by all this negatively. Anyway here is what you need to do:
determine FOV for both images
So find some measurable feature visible on both images. The bigger in size the more accurate the result. For example I choose these:
form a point cloud or mesh
I use depth image as reference so my point cloud is in its FOV. As I do not have the distances but 8bit values instead I converted that to some distance by multiplying by constant. So I scan whole depth image and for every pixel I create point in my point cloud array. Then convert the dept pixel coordinate to color image FOV and copy its color too. something like this (in C++):
picture rgb,zed; // your input images
struct pnt3d { float pos[3]; DWORD rgb; pnt3d(){}; pnt3d(pnt3d& a){ *this=a; }; ~pnt3d(){}; pnt3d* operator = (const pnt3d *a) { *this=*a; return this; }; /*pnt3d* operator = (const pnt3d &a) { ...copy... return this; };*/ };
pnt3d **xyz=NULL; int xs,ys,ofsx=0,ofsy=0;
void copy_images()
{
int x,y,x0,y0;
float xx,yy;
pnt3d *p;
for (y=0;y<ys;y++)
for (x=0;x<xs;x++)
{
p=&xyz[y][x];
// copy point from depth image
p->pos[0]=2.000*((float(x)/float(xs))-0.5);
p->pos[1]=2.000*((float(y)/float(ys))-0.5)*(float(ys)/float(xs));
p->pos[2]=10.0*float(DWORD(zed.p[y][x].db[0]))/255.0;
// convert dept image x,y to color image space (FOV correction)
xx=float(x)-(0.5*float(xs));
yy=float(y)-(0.5*float(ys));
xx*=98.0/108.0;
yy*=106.0/119.0;
xx+=0.5*float(rgb.xs);
yy+=0.5*float(rgb.ys);
x0=xx; x0+=ofsx;
y0=yy; y0+=ofsy;
// copy color from rgb image if in range
p->rgb=0x00000000; // black
if ((x0>=0)&&(x0<rgb.xs))
if ((y0>=0)&&(y0<rgb.ys))
p->rgb=rgb2bgr(rgb.p[y0][x0].dd); // OpenGL has reverse RGBorder then my image
}
}
where **xyz is my point cloud 2D array allocated t depth image resolution. The picture is my image class for DIP so here some relevant members:
xs,ys is the image resolution in pixels
p[ys][xs] is the image direct pixel access as union of DWORD dd; BYTE db[4]; so I can access color as single 32 bit variable or each color channel separately.
rgb2bgr(DWORD col) just reorder color channels from RGB to BGR.
render it
I use OpenGL for this so here the code:
glBegin(GL_QUADS);
for (int y0=0,y1=1;y1<ys;y0++,y1++)
for (int x0=0,x1=1;x1<xs;x0++,x1++)
{
float z,z0,z1;
z=xyz[y0][x0].pos[2]; z0=z; z1=z0;
z=xyz[y0][x1].pos[2]; if (z0>z) z0=z; if (z1<z) z1=z;
z=xyz[y1][x0].pos[2]; if (z0>z) z0=z; if (z1<z) z1=z;
z=xyz[y1][x1].pos[2]; if (z0>z) z0=z; if (z1<z) z1=z;
if (z0 <=0.01) continue;
if (z1 >=3.90) continue; // 3.972 pre vsetko nad .=3.95m a 4.000 ak nechyti vobec nic
if (z1-z0>=0.10) continue;
glColor4ubv((BYTE* )&xyz[y0][x0].rgb);
glVertex3fv((float*)&xyz[y0][x0].pos);
glColor4ubv((BYTE* )&xyz[y0][x1].rgb);
glVertex3fv((float*)&xyz[y0][x1].pos);
glColor4ubv((BYTE* )&xyz[y1][x1].rgb);
glVertex3fv((float*)&xyz[y1][x1].pos);
glColor4ubv((BYTE* )&xyz[y1][x0].rgb);
glVertex3fv((float*)&xyz[y1][x0].pos);
}
glEnd();
You need to add the OpenGL initialization and camera settings etc of coarse. Here the unaligned result:
align it
If you notice I added ofsx,ofsy variables to copy_images(). This is the offset between cameras. I change them on arrows keystrokes by 1 pixel and then call copy_images and render the result. This way I manually found the offset very quickly:
As you can see the offset is +17 pixels in x axis and +4 pixels in y axis. Here side view to better see the depths:
Hope It helps a bit
Well I have tried doing it after reading lots of blogs and all. I am still not sure whether I am doing it correct or not. Please feel free to give comments if something is found amiss. For this I used a mathworks fex submission that can be found here : ginputc function.
The matlab code is as follows :
clc; clear all; close all;
% no of keypoint
N = 7;
I = imread('2.jpg');
I = rgb2gray(I);
[Gx, Gy] = imgradientxy(I, 'Sobel');
[Gmag, ~] = imgradient(Gx, Gy);
figure, imshow(Gmag, [ ]), title('Gradient magnitude')
I = Gmag;
[x,y] = ginputc(N, 'Color' , 'r');
matchedpoint1 = [x y];
J = imread('2.png');
[Gx, Gy] = imgradientxy(J, 'Sobel');
[Gmag, ~] = imgradient(Gx, Gy);
figure, imshow(Gmag, [ ]), title('Gradient magnitude')
J = Gmag;
[x, y] = ginputc(N, 'Color' , 'r');
matchedpoint2 = [x y];
[tform,inlierPtsDistorted,inlierPtsOriginal] = estimateGeometricTransform(matchedpoint2,matchedpoint1,'similarity');
figure; showMatchedFeatures(J,I,inlierPtsOriginal,inlierPtsDistorted);
title('Matched inlier points');
I = imread('2.jpg'); J = imread('2.png');
I = rgb2gray(I);
outputView = imref2d(size(I));
Ir = imwarp(J,tform,'OutputView',outputView);
figure; imshow(Ir, []);
title('Recovered image');
figure,imshowpair(I,J,'diff'),title('Difference with original');
figure,imshowpair(I,Ir,'diff'),title('Difference with restored');
Step 1
I used the sobel edge detector to extract the edges for both the depth and rgb images and then used a thresholding values to get the edge map. I will be primarily working with the gradient magnitude only. This gives me two images as this :
Step 2
Next I use the ginput or ginputc function to mark keypoints on both the images. The correspondence between the points are established by me beforehand. I tried using SURF features but they do not work well on depth images.
Step 3
Use the estimategeometrictransform to get the transformation matrix tform and then use this matrix to recover the original position of the moved image. The next set of images tells this story.
Granted I still believe the results can be further improved if the keypoint selections in either of the images are more judiciously done. I also think #Specktre method is better. I just noticed that I used a separate image-pair in my answer compared to that of the question. Both images come from the same dataset to be found here vap rgb-d-t dataset.
I am trying to allign two images - one rgb and another depth using MATLAB. Please note that I have checked several places for this - like here , here which requires a kinect device, and here here which says that camera parameters are required for calibration. I was also suggested to use EPIPOLAR GEOMETRY to match the two images though I do not know how. The dataset I am referring to is given in rgb-d-t face dataset. One such example is illustrated below :
The ground truth which basically means the bounding boxes which specify the face region of interest are already provided and I use them to crop the face regions only. The matlab code is illustrated below :
I = imread('1.jpg');
I1 = imcrop(I,[218,198,158,122]);
I2 = imcrop(I,[243,209,140,108]);
figure, subplot(1,2,1),imshow(I1);
subplot(1,2,2),imshow(I2);
The two cropped images rgb and depth are shown below :
Is there any way by which we can register/allign the images. I took the hint from
here where basic sobel operator has been used on both the rgb and depth images to generate an edge map and then keypoints will need to be generated for matching purposes. The edge maps for both the images are generated here.
.
However they are so noisy that I do not think we will be able to do keypoint matching for this images.
Can anybody suggest some algorithms in matlab to do the same ?
prologue
This answer is based on mine previous answer:
Does Kinect Infrared View Have an offset with the Kinect Depth View
I manually crop your input image so I separate colors and depth images (as my program need them separated. This could cause minor offset change by few pixels. Also as I do not have the depths (depth image is 8bit only due to grayscale RGB) then the depth accuracy I work with is very poor see:
So my results are affected by all this negatively. Anyway here is what you need to do:
determine FOV for both images
So find some measurable feature visible on both images. The bigger in size the more accurate the result. For example I choose these:
form a point cloud or mesh
I use depth image as reference so my point cloud is in its FOV. As I do not have the distances but 8bit values instead I converted that to some distance by multiplying by constant. So I scan whole depth image and for every pixel I create point in my point cloud array. Then convert the dept pixel coordinate to color image FOV and copy its color too. something like this (in C++):
picture rgb,zed; // your input images
struct pnt3d { float pos[3]; DWORD rgb; pnt3d(){}; pnt3d(pnt3d& a){ *this=a; }; ~pnt3d(){}; pnt3d* operator = (const pnt3d *a) { *this=*a; return this; }; /*pnt3d* operator = (const pnt3d &a) { ...copy... return this; };*/ };
pnt3d **xyz=NULL; int xs,ys,ofsx=0,ofsy=0;
void copy_images()
{
int x,y,x0,y0;
float xx,yy;
pnt3d *p;
for (y=0;y<ys;y++)
for (x=0;x<xs;x++)
{
p=&xyz[y][x];
// copy point from depth image
p->pos[0]=2.000*((float(x)/float(xs))-0.5);
p->pos[1]=2.000*((float(y)/float(ys))-0.5)*(float(ys)/float(xs));
p->pos[2]=10.0*float(DWORD(zed.p[y][x].db[0]))/255.0;
// convert dept image x,y to color image space (FOV correction)
xx=float(x)-(0.5*float(xs));
yy=float(y)-(0.5*float(ys));
xx*=98.0/108.0;
yy*=106.0/119.0;
xx+=0.5*float(rgb.xs);
yy+=0.5*float(rgb.ys);
x0=xx; x0+=ofsx;
y0=yy; y0+=ofsy;
// copy color from rgb image if in range
p->rgb=0x00000000; // black
if ((x0>=0)&&(x0<rgb.xs))
if ((y0>=0)&&(y0<rgb.ys))
p->rgb=rgb2bgr(rgb.p[y0][x0].dd); // OpenGL has reverse RGBorder then my image
}
}
where **xyz is my point cloud 2D array allocated t depth image resolution. The picture is my image class for DIP so here some relevant members:
xs,ys is the image resolution in pixels
p[ys][xs] is the image direct pixel access as union of DWORD dd; BYTE db[4]; so I can access color as single 32 bit variable or each color channel separately.
rgb2bgr(DWORD col) just reorder color channels from RGB to BGR.
render it
I use OpenGL for this so here the code:
glBegin(GL_QUADS);
for (int y0=0,y1=1;y1<ys;y0++,y1++)
for (int x0=0,x1=1;x1<xs;x0++,x1++)
{
float z,z0,z1;
z=xyz[y0][x0].pos[2]; z0=z; z1=z0;
z=xyz[y0][x1].pos[2]; if (z0>z) z0=z; if (z1<z) z1=z;
z=xyz[y1][x0].pos[2]; if (z0>z) z0=z; if (z1<z) z1=z;
z=xyz[y1][x1].pos[2]; if (z0>z) z0=z; if (z1<z) z1=z;
if (z0 <=0.01) continue;
if (z1 >=3.90) continue; // 3.972 pre vsetko nad .=3.95m a 4.000 ak nechyti vobec nic
if (z1-z0>=0.10) continue;
glColor4ubv((BYTE* )&xyz[y0][x0].rgb);
glVertex3fv((float*)&xyz[y0][x0].pos);
glColor4ubv((BYTE* )&xyz[y0][x1].rgb);
glVertex3fv((float*)&xyz[y0][x1].pos);
glColor4ubv((BYTE* )&xyz[y1][x1].rgb);
glVertex3fv((float*)&xyz[y1][x1].pos);
glColor4ubv((BYTE* )&xyz[y1][x0].rgb);
glVertex3fv((float*)&xyz[y1][x0].pos);
}
glEnd();
You need to add the OpenGL initialization and camera settings etc of coarse. Here the unaligned result:
align it
If you notice I added ofsx,ofsy variables to copy_images(). This is the offset between cameras. I change them on arrows keystrokes by 1 pixel and then call copy_images and render the result. This way I manually found the offset very quickly:
As you can see the offset is +17 pixels in x axis and +4 pixels in y axis. Here side view to better see the depths:
Hope It helps a bit
Well I have tried doing it after reading lots of blogs and all. I am still not sure whether I am doing it correct or not. Please feel free to give comments if something is found amiss. For this I used a mathworks fex submission that can be found here : ginputc function.
The matlab code is as follows :
clc; clear all; close all;
% no of keypoint
N = 7;
I = imread('2.jpg');
I = rgb2gray(I);
[Gx, Gy] = imgradientxy(I, 'Sobel');
[Gmag, ~] = imgradient(Gx, Gy);
figure, imshow(Gmag, [ ]), title('Gradient magnitude')
I = Gmag;
[x,y] = ginputc(N, 'Color' , 'r');
matchedpoint1 = [x y];
J = imread('2.png');
[Gx, Gy] = imgradientxy(J, 'Sobel');
[Gmag, ~] = imgradient(Gx, Gy);
figure, imshow(Gmag, [ ]), title('Gradient magnitude')
J = Gmag;
[x, y] = ginputc(N, 'Color' , 'r');
matchedpoint2 = [x y];
[tform,inlierPtsDistorted,inlierPtsOriginal] = estimateGeometricTransform(matchedpoint2,matchedpoint1,'similarity');
figure; showMatchedFeatures(J,I,inlierPtsOriginal,inlierPtsDistorted);
title('Matched inlier points');
I = imread('2.jpg'); J = imread('2.png');
I = rgb2gray(I);
outputView = imref2d(size(I));
Ir = imwarp(J,tform,'OutputView',outputView);
figure; imshow(Ir, []);
title('Recovered image');
figure,imshowpair(I,J,'diff'),title('Difference with original');
figure,imshowpair(I,Ir,'diff'),title('Difference with restored');
Step 1
I used the sobel edge detector to extract the edges for both the depth and rgb images and then used a thresholding values to get the edge map. I will be primarily working with the gradient magnitude only. This gives me two images as this :
Step 2
Next I use the ginput or ginputc function to mark keypoints on both the images. The correspondence between the points are established by me beforehand. I tried using SURF features but they do not work well on depth images.
Step 3
Use the estimategeometrictransform to get the transformation matrix tform and then use this matrix to recover the original position of the moved image. The next set of images tells this story.
Granted I still believe the results can be further improved if the keypoint selections in either of the images are more judiciously done. I also think #Specktre method is better. I just noticed that I used a separate image-pair in my answer compared to that of the question. Both images come from the same dataset to be found here vap rgb-d-t dataset.
I have a list of coordinates, which are generated from another program, and I have an image.
I'd like to load those coordinates (making circular regions of interest (ROIs) with a diameter of 3 pixels) onto my image, and extract the intensity of those pixels.
I can load/impose the coordinates on to the image by using;
imshow(file);
hold on
scatter(xCoords, yCoords, 'g')
But can not extract the intensity.
Can you guys point me in the right direction?
I am not sure what you mean by a circle with 3 pixels diameter since you are in a square grid (as mentioned by Ander Biguri). But you could use fspecial to create a disk filter and then normalize. Something like this:
r = 1.5; % for diameter = 3
h = fspecial('disk', r);
h = h/h(ceil(r),ceil(r));
You can use it as a mask to get the intensities at the given region of the image.
im = imread(file);
ROI = im(xCoord-1:xCoord+1; yCoord-1:yCoord+1);
I = ROI.*h;
I have two images. One is the original, and the another is rotated.
Now, I need to discover the angle that the image was rotated. Until now, I thought about discovering the centroids of each color (as every image I will use has squares with colors in it) and use it to discover how much the image was rotated, but I failed.
I'm using this to discover the centroids and the color in the higher square in the image:
i = rgb2gray(img);
bw = im2bw(i,0.01);
s = regionprops(bw,'Centroid');
centroids = cat(1, s.Centroid);
colors = impixel(img,centroids(1),centroids(2));
top = max(centroids);
topcolor = impixel(img,top(1),top(2));
You can detect the corners of one of the colored rectangles in both the image and the rotated version, and use these as control points to infer the transformation between the two images (like in image registration) using the CP2TFORM function. We can then compute the angle of rotation from the affine transformation matrix:
Here is an example code:
%# read first image (indexed color image)
[I1 map1] = imread('http://i.stack.imgur.com/LwuW3.png');
%# constructed rotated image
deg = -15;
I2 = imrotate(I1, deg, 'bilinear', 'crop');
%# find blue rectangle
BW1 = (I1==2);
BW2 = imrotate(BW1, deg, 'bilinear', 'crop');
%# detect corners in both
p1 = corner(BW1, 'QualityLevel',0.5);
p2 = corner(BW2, 'QualityLevel',0.5);
%# sort corners coordinates in a consistent way (counter-clockwise)
p1 = sortrows(p1,[2 1]);
p2 = sortrows(p2,[2 1]);
idx = convhull(p1(:,1), p1(:,2)); p1 = p1(idx(1:end-1),:);
idx = convhull(p2(:,1), p2(:,2)); p2 = p2(idx(1:end-1),:);
%# make sure we have the same number of corner points
sz = min(size(p1,1),size(p2,1));
p1 = p1(1:sz,:); p2 = p2(1:sz,:);
%# infer transformation from corner points
t = cp2tform(p2,p1,'nonreflective similarity'); %# 'affine'
%# rotate image to match the other
II2 = imtransform(I2, t, 'XData',[1 size(I1,2)], 'YData',[1 size(I1,1)]);
%# recover affine transformation params (translation, rotation, scale)
ss = t.tdata.Tinv(2,1);
sc = t.tdata.Tinv(1,1);
tx = t.tdata.Tinv(3,1);
ty = t.tdata.Tinv(3,2);
translation = [tx ty];
scale = sqrt(ss*ss + sc*sc);
rotation = atan2(ss,sc)*180/pi;
%# plot the results
subplot(311), imshow(I1,map1), title('I1')
hold on, plot(p1(:,1),p1(:,2),'go')
subplot(312), imshow(I2,map1), title('I2')
hold on, plot(p2(:,1),p2(:,2),'go')
subplot(313), imshow(II2,map1)
title(sprintf('recovered angle = %g',rotation))
If you can identify a color corresponding to only one component it is easier to:
Calculate the centroids for each image
Calculate the mean of the centroids (in x and y) for each image. This is the "center" of each image
Get the red component color centroid (in your example) for each image
Subtract the mean of the centroids for each image from the red component color centroid for each image
Calculate the ArcTan2 for each of the vectors calculated in 4), and subtract the angles. That is your result.
If you have more than one figure of each color, you need to calculate all possible combinations for the rotation and then select the one that is compatible with the other possible rotations.
I could post the code in Mathematica, if you think it is useful.
I would take a variant to the above mentioned approach:
% Crude binarization method to knock out background and retain foreground
% features. Note one looses the cube in the middle
im = im > 1
Then I would get the 2D autocorrelation:
acf = normxcorr2(im, im);
From this result, one can easily detect the peaks, and as rotation carries into the autocorrelation function (ACF) domain, one can ascertain the rotation by matching the peaks between the original ACF and the ACF from the rotated image, for example using the so-called Hungarian algorithm.