I am using matlab's built in function called Procrustes to see the rotation translation and scale between two images. But, I am just using coordinates of the brightest points in the image and rotating these coordinates about the center of the image. Procrustes compares two matrices and gives you the rotation, translation, and scale. However, procrustes only works correctly if the matrices are in the same order for comparison.
I am given an image and a separate comparison coordinate matrix. The end goal is to find how much the image has been rotated, translated, and scaled compared to the coordinate matrix. I can just use Procrustes for this, but I need to correctly order the coordinates found from the image to match the order in the comparison coordinate matrix. My thought was to compare the distance between every possible combination of points in the coordinate matrix and compare it to the coordinates that I find in the picture. I just do not know how to write this code due to the fact if there is n coordinates, there will be n! possible combinations.
Just searching for the shortest distance is not so hard.
A = rand(1E4,2);
B = rand(1E4,2);
tic
idx = nan(1,1E4);
for ct = 1:size(A,1)
d = sum((A(ct,:)-B).^2,2);
idx(ct) = find(d==min(d));
end
toc
plot(A(1:10,1),A(1:10,2),'.r',B(idx(1:10),1),B(idx(1:10),2),'.b')
takes half a second on my PC.
The problems can start when two points in set A are matched to the same location in set B.
length(unique(idx))==length(idx)
This can be solved in several ways. The best (imho) is to determine a probability that point B matches with point A based on the distance (usually something that decreases exponentially), and solve for the most probable situation.
A simpler method (but more error prone) is to remove the matched point from set B.
Related
I want to generate a series of points as building in simulation.
Points density is 1/1000m^3
The points have its shape just like the real buildings(circle or rectangle or something else)
In order to reach the reality, these shapes should not be overlapped.
The question is how to generate the center point of these 'buildings'?
I tried this
clusterNumber = round((pi*areaRadius^2)/1000);
radius = unifrnd (0,areaRadius,clusterNumber,1);
angle = unifrnd (-pi,pi,clusterNumber,1);
for i=1:clusterNumber
Coordinate(i,1) = cos(angle(i))*radius(i); % x
Coordinate(i,2) = sin(angle(i))*radius(i); % y
and the result showed as what I expected... it did'nt work
When I used scatter it showed
So, my question is how to generate non-uniform and non-overlap circles or rectangles in a specific circle.
If you want your buildings not to intersect, you must check for intersections with already created buildings before creating one at your random position.
Of course, if you create many buildings, collision detection will be costly. You can speed it up with an efficient nearest neighbour search, for example with kd-trees or by creating a fine grid in the building space so that you have only a few neighbour cells to check.
Imposing the condition that buildings must not intersect will also alter your distribution. You will no longer have the marked clustering in the circle's centre. You still generate morre random positionsthere, but as your area gets more populated, most of them will be rejected.
Here's an example distribution:
Enforcing the criterion may also affect your algorithm: It might be a good idea to limit the number of randomly gerenated positions, so that you don't run into an infinite loop when no more buildings can be placed or when the probability to find a suitable space is very low.
I've got question about algorithms to compare if two sets of points are in a similar place on the image.
They don't create similar shapes likes circles, rectangles etc, but they are something like irregular clouds.
For example:
The first cloud of points is learning set of desired area on image and we are checking if second cloud is in similar position.
I was thinking of drawing simple shapes to form points (like rectangles which will accumulate all points) and checking if one is in another or distance between centers of figures, but this method doesn't seem to be very accurate.
Are there better algorithms to solve this problem?
Image Moments
Don't worry about the fancy name, it's just a standard method in image processing to do exactly what you require.
Image moment of power n w.r.t. x and m w.r.t. y is actually the
integration of (pixel value * xPosition^n * xPosition^m) over the
entire image.
So (0, 0)th order moment i.e moment(0, 0) is actually area of the cloud.
Similarly, moment(1, 0)/moment(0, 0) is X coordinate of centroid of the cloud.
And, moment(0, 1)/moment(0, 0) is Y coordinate of centroid of the cloud.
Higher order moments give additional features/information peculiar to shape of the clouds.
Now you can easily compare the arbitrary shapes.
These functions are available in opencv and matlab.
Hope this helps.
Good luck.
Sets will have quite similar shapes (it will be set of points of human skeleton from kinect > sensor) and I want to check if person is sitting in the same place as it was learned in the > first place
Then you will probably be able to derive a correspondence between two points (i.e. you will know that a given point is SHOULDER_RIGHT or ELBOW_LEFT or...). If that is the case you can simply calculate the SUM(SQRT((Xi1-Xi2)^2+(Yi1-Yi2)^2) for each i-th pair of points (X1,Y1) and (X2,Y2) (same goes if you can obtain the third dimension Z).
The value thus obtained will have a minimum of zero when the two sets of points are perfectly coinciding.
If I have 2 sets of points I can rotate one around with Procrustes analysis to align one with the other.
But suppose these 2 sets of points each are attached to images and I would like to rotate the images as well. Is there any way I can also rotate the image, instead of rotating just the points? The tutorial there uses a dot product for rotation (solve u, s, v = svd(p1', p2) and then do p2 . v . u', p' is transposed p)
However that doesn't tell me what the angle between the images is.
The page on wikipedia calculates an angle between each pair of points I think.
Maybe what I'm asking is impossible? If I rotate the first set of points to align it with the first, can't I also rotate the respective images by an angle as well? Point being, which angle is that?
I noticed that v . u' gives me a 2 x 2 matrix which seems to be the rotation matrix (there's a wikipedia page but I can't link there due to posting priviledges). I got the sin and cos of the third and first elements and then used arctan2, but the results I'm getting are kind of weird. I know they have to be transformed from radians but I'm not convinced what I'm doing is right. Trying the rotation it gives me on gimp makes it seem like it's not what I want, but I'll test some more.
It seems like your approach is mostly correct. Two things which come to mind:
1) The paper you linked to (Procrustes analysis) involves a translation and a scaling in addition to rotation. Depending on whether or not those operations are also performed on your images, you may end up with strange results that don't appear to match.
2) I think you may be overcomplicating your angle calculation. v * u' appears to be the correct rotation matrix, but I believe the correct angle only requires one of the matrix entries in the 2x2 matrix. For instance, just use acos() of the first matrix entry. As you've noticed, this will (depending on the program) give you an answer in radians which you'll have to convert to degrees if you want to try out the rotation in gimp, etc.
Hope this helps!
Problem
Given a set of known cartesian points (set A), and a 2d transformation (rotation, translation, scale) of some subset of those points (set B), find the orientation of the subset (rotation, translation, scale) relative to the original set of points.
I.E. Suppose I take a "picture" of a known set of 2d points on a wall. I want to know what position the camera was in relative to "upright and centered" when the picture was taken. Some of the points may not be visible in the picture (they may be occluded). (in this analogy, assume the camera is orthoganal and always pointed directly at the plane of the wall, so you don't need to take distortion or perspective into account)
Proposed approach:
Step 1: Scale B to the same "range" as A
Don't know how; open to suggestions. Maybe take the area of a convex hull around all the points in B, and scale it to nearly that of the convex hull around A. This is tricky, because points may be missing from B.
Step 2: Match some arbitrary point in "B" to its twin in "A"
Pick some random point in set B. Call this point K. Somehow take a "fingerprint" of K relative to all the other points in B (using distance only). Find its match in A by fingerprinting all points in A and taking the point with the most similar fingerprint of K.
Step 3: Rotate B (around K) until all points in B are aligned with a point in A
Multiple solutions are possible, so keep rotating though 360d looking for solutions.
That's just shooting from the hip, I may be way off base. Anyone have any ideas?
Assuming you don't actually know the correspondence between the points in the two clouds, you could try a statistical approach.
First, compute the mean x0 of the original cloud, then compute the mean x1 of the subset cloud. The difference of the mean vectors, x1-x0, is a good estimate of the required translation.
Now, subtract the relevant mean vector from each set to give two clouds centered at the origin. Compute the covariance matrix for each cloud and find its eigenvalues and eigenvectors. The required rotation can be found from the eigenvectors, while the scaling corresponds to the eigenvalues.
Compose all of this and you should have a good statistical estimate of the desired transform. Obviously, its quality will be a function of how well the subset spans the original set.
"Give me a place to stand on, and I will move the Earth" Archimede
I think we should follow the steps of Archimede
Arpi's algoritm:
We must choose a point (X1) of set A with coordinates (0, 0). (this will be the place to stand on)
Choose another point (X2) and put it on the OX vector (to simplify things)
All the other points' coordinates from set A will be calculated based on the coordinates of X1(0, 0) and X2(some_Coordinate, 0).
Now, choose a point from set B (Y1) and that will be the center of the B set. Choose another point from set B (Y2) and put it to OX of the B set. Now, we have a scale scalar and a rotation angle. If this will be a solution, than Y1 in the B set represents X1 from the A set and Y2 from the B set represents X2 from the A set. If we can find a map between the B set and A set based on this, using all the points of the B set and Yi <> Yj if i <> j, where i and j are the indexes of the points in our representation than we have a potential solution and we store that.
End of Arpi's algoritm
To find all the potential solutions you must do the following:
foreach point in A as X1 do
foreach point in A as X2 do
arpi's algoritm(X1, X2)
Of course, you can optimize this, but for the sake of simplicity I described it without optimizations (complications), it will be your job to optimize this and only if you need that.
I would attempt to minimize the deviation between the target points and the found points. Meaning I would pair each target point with a found point, and apply any transformation (rotation, scale or skew) to all the target points which decreases the sum of the deviations. I would repeat this for all potential pairs, eventually taking the match to be the set of pairs and the necessary transformations with the smallest total deviation.
The real question is how you optimize this so the performance to be better than O(n^2). I suppose some sort of heuristic matching, perhaps caching the intermediary results, or finding a method of eliminating some pairs earlier in the process.
As a followup to my previous question about determining camera parameters I have formulated a new problem.
I have two pictures of the same rectangle:
The first is an image without any transformations and shows the rectangle as it is.
The second image shows the rectangle after some 3d transformation (XYZ-rotation, scaling, XY-translation) is applied. This has caused the rectangle to look a trapezoid.
I hope the following picture describes my problem:
alt text http://wilco.menge.nl/application.data/cms/upload/transformation%20matrix.png
How do determine what transformations (more specifically: what transformation matrix) have caused this tranformation?
I know the pixel locations of the corners in both images, hence i also know the distances between the corners.
I'm confused. Is this a 2d or a 3d problem?
The way I understand it, you have a flat rectangle embedded in 3d space, and you're looking at two 2d "pictures" of it - one of the original version and one based on the transformed version. Is this correct?
If this is correct, then there is not enough information to solve the problem. For example, suppose the two pictures look exactly the same. This could be because the translation is the identity, or it could be because the translation moves the rectangle twice as far away from the camera and doubles its size (thus making it look exactly the same).
This is a math problem, not programming ..
you need to define a set of equations (your transformation matrix, my guess is 3 equations) and then solve it for the 4 transformations of the corner-points.
I've only ever described this using German words ... so the above will sound strange ..
Based on the information you have, this is not that easy. I will give you some ideas to play with, however. If you had the 3D coordinates of the corners, you'd have an easier time. Here's the basic idea.
Move a corner to the origin. Thereafter, rotations will take place about the origin.
Determine vectors of the axes. Do this by subtracting the adjacent corners from the origin point. These will be a local x and y axis for your world.
Determine angles using the vectors. You can use the dot and cross products to determine the angle between the local x axis and the global x axis (1, 0, 0).
Rotate by the angle in step 3. This will give you a new x axis which should match the global x axis and a new local y axis. You can then determine another rotation about the x axis which will bring the y axis into alignment with the global y axis.
Without the z coordinates, you can see that this will be difficult, but this is the general process. I hope this helps.
The solution will not be unique, as Alex319 points out.
If the second image is really a trapezoid as you say, then this won't be too hard. It is a trapezoid (not a parallelogram) because of perspective, so it must be an isosceles trapezoid.
Draw the two diagonals. They intersect at the center of the rectangle, so that takes care of the translation.
Rotate the trapezoid until its parallel sides are parallel to two sides of the original rectangle. (Which two? It doesn't matter.)
Draw a third parallel through the center. Scale this to the sides of the rectangle you chose.
Now for the rotation out of the plane. Measure the distance from the center to one of the parallel sides and use the law of sines.
If it's not a trapezoid, just a quadralateral, then it'll be harder, you'll have to use the angles between the diagonals to find the axis of rotation.