How can I generate a set of image projections using MATLAB?
I have a 256x256 image. I want to get all 180 projections ranging from 0 degree to 180 degree projection angle.
Is there a way to do that in MATLAB?
You can use RADON
theta = 0:180; %# angle from 0 to 180 degrees
R = radon(yourImage,theta);
Each column of R corresponds to one angle from the list in theta
Related
I am trying to plot the displacement of a pixel from the original image to the fish eye image based on the radius from the center of the image.
I was successful in producing fish images in MATLAB using maketform
testImg = imread('ship.jpg');
optTra = maketform('custom',2,2,[],#radial,options);
newX = imtransform(testImg,optTra);
imshow(newX);
the radial function here helps me to get the fish eye distorted image.
I need to find the displacement of each pixel in the original image to that of the distorted image.
If the transformation applied (a.k.a "#radial") was angular, the inverse transformation is given by:
u = r cos(phi) + 0.5;
v = r sin(phi) + 0.5;
where
r = atan2(sqrt(x*x+y*y),p.z)/pi;
phi = atan2(y,x);
x,y are assumed to be normalized coordinates (centered and between -1 to 1).
I use camera calibration in matlab to detect some checkerboard patterns, after
figure; showExtrinsics(cameraParams, 'CameraCentric');
Now, I want to rotate the checkerboard patterns around the x-axis such that all of them have nearly the same y coordinates in the camera frame.
Method:
I get the positions of all patterns in the camera's frame. Then I do optimization,where the objective function is to minimize variance in y and the variable is rotation about x ranging from o to 360.
Problem:
But when I plot the transformed y-coordinates, they are even nearly in a line.
Code:
Get the checkerboad points:
%% Get rotation and translation matrices for each image;
T_cw=cell(num_imgs,1); % stores camera to world rotation and translation for each image
pixel_coordinates=zeros(num_imgs,2); % stores the pixel coordinates of each checkerboard origin
for ii=1:num_imgs,
% Calibrate the camera
im=imread(list_imgs_path{ii});
[imagePoints, boardSize] = detectCheckerboardPoints(im);
[r_wc, t_wc] = extrinsics(imagePoints, worldPoints, cameraParams);
T_wc=[r_wc,t_wc';0 0 0 1];
% World to camera matrix
T_cw{ii} = inv(T_wc);
t_cw{ii}=T_cw{ii}(1:3,4); % x,y,z coordinates in camera's frame
end
Data(num_imgs=10):
t_cw
[-1072.01388542262;1312.20387622761;-1853.34408157349]
[-1052.07856598756;1269.03455126794;-1826.73576892251]
[-1091.85978641218;1351.08261414473;-1668.88197803184]
[-1337.56358084648;1373.78548638383;-1396.87603554914]
[-1555.19509876309;1261.60428874489;-1174.63047408086]
[-1592.39596647158;1066.82210015055;-1165.34417772659]
[-1523.84307918660;963.781819272748;-1207.27444716506]
[-1614.00792252030;893.962075837621;-1114.73528985018]
[-1781.83112607964;708.973204727939;-797.185326205240]
[-1781.83112607964;708.973204727939;-797.185326205240]
Main code (Optimization and transformation):
%% Get theta for rotation
f_obj = #(x)var_ycors(x,t_cw);
opt_theta = fminbnd(f_obj,0,360);
%% Plotting (rotate ycor and check to fix theta)
y_rotated=zeros(1,num_imgs);
for ii=1:num_imgs,
y_rotated(ii)=rotate_cor(opt_theta,t_cw{ii});
end
plot(1:numel(y_rotated),y_rotated);
function var_computed=var_ycors(theta,t_cw)
ycor=zeros(1,numel(t_cw));
for ii =1:numel(t_cw),
ycor(ii)=rotate_cor(theta,t_cw{ii});
end
var_computed=var(ycor);
end
function ycor=rotate_cor(theta,mat)
r_x=[1 0 0; 0 cosd(theta) -sind(theta); 0 sind(theta) cosd(theta)];
rotate_mat=mat'*r_x;
ycor=rotate_mat(2);
end
This is a clear eigenvector problem!
Take your centroids:
t_cw=[-1072.01388542262;1312.20387622761;-1853.34408157349
-1052.07856598756;1269.03455126794;-1826.73576892251
-1091.85978641218;1351.08261414473;-1668.88197803184
-1337.56358084648;1373.78548638383;-1396.87603554914
-1555.19509876309;1261.60428874489;-1174.63047408086
-1592.39596647158;1066.82210015055;-1165.34417772659
-1523.84307918660;963.781819272748;-1207.27444716506
-1614.00792252030;893.962075837621;-1114.73528985018
-1781.83112607964;708.973204727939;-797.185326205240
-1781.83112607964;708.973204727939;-797.185326205240];
t_cw=reshape(t_cw,[3,10])';
compute PCA on them, so we know the principal conponents:
[R]=pca(t_cw);
And.... thats it! R is now the transformation matrix between your original points and the rotated coordinate system. As an example, I will draw in red the old points and in blue the new ones:
hold on
plot3(t_cw(:,1),t_cw(:,2),t_cw(:,3),'ro')
trans=t_cw*R;
plot3(trans(:,1),trans(:,2),trans(:,3),'bo')
You can see that now the blue ones are in a plane, with the best possible fit to the X direction. If you want them in Y direction, just rotate 90 degrees in Z (I am sure you can figure out how to do this with 2 minutes of Google ;) ).
Note: This is mathematically the best possible fit. I know they are not as "in a row" as one would like, but this is because of the data, this is honestly the best possible fit, as that is what the eigenvectors are!
I want to calculate the angle between 2 planes, Reference plane and Plane1. When I feed the X,Y,Z co-ordinates of pointCloud to the function plane_fit.m (by Kevin Mattheus Moerman), I get the coefficients:
reference_plane_coeff: [-0.13766204 -0.070385590 130.69409]
Plane1_coeff: [0.0044337390 -0.0013548643 95.890228]
Next, I find the intersection of both planes, separately on the XZ plane and get a line equation; ref_line_XZ and plane1_line_XZ respectively. For this, I make the second coefficient 0. (Is this right?)
Aref = reference_plane_coeff(1);
Cref = reference_plane_coeff(3);
ref_line_XZ = [Aref Cref];
Arun = Plane1_coeff(1);
Crun = Plane1_coeff(3);
plane1_line_XZ = [Arun Crun];
angle_XZ = acos( dot(ref_line_XZ,plane1_line_XZ ) / (norm(ref_line_XZ) * norm(plane1_line_XZ )) )
I get the angle_XZ value as 0.0012 rad. i.e. 0.0685 degrees
When I plot these planes on a graph and view it, the angle seems to be much more than 0.0012 degrees. I'm talking about the angle made by the two lines after intersection of both planes with the XZ plane.
What am I doing wrong?
Also, when I tried to find angle between its normals, using:
angle_beta_deg = acosd( dot(reference_plane_coeff,Plane1_coeff) / (norm(reference_plane_coeff) * norm(Plane1_coeff)) )
I got the angle as 0.0713.
On visual inspection of both planes' plots and manually calculating from the plot, angle_XZ should be around 9 degrees.
plane_fit.m (by Kevin Mattheus Moerman)
I have many 3D point clouds gathered by velodyne sensor. eg(x, y, z) in meter.
I'd like to convert 3D point clouds to range image.
Firstly, I've got transformtation from Catesian to spherical coordinate.
r = sqrt(x*x + y*y + z*z)
azimuth angle = atan2(x, z)
elevation angle = asin(y/r)
Now. How can I convert 3D point to Range image using these transformation in matlab?
Whole points are about 180,000 and I want 870*64 range image.
azimuth angle range(-180 ~ 180), elevation angle range(-15 ~ 15)
Divide up your azimuth and elevation into M and N ranges respectively. Now you have M*N "bins" (M = 870, N = 64).
Then (per bin) accumulate a histogram of points that project into that bin.
Finally, pick a representative value from each bin for the final range image. You could pick the average value (noisy, fast) or fit some distribution and then use that to pick the value (more precise, slow).
The pointcloud2image code available from Matlab File Exchange can help you to directly convert point cloud (in x,y,z format) to 2D raster image.
Imagine a dome with its centre in the +z direction. What I want to do is to move that dome's centre to a different axis (e.g. 20 degrees x axis, 20 degrees y axis, 20 degrees z axis). How can I do that ? Any hint/tip helps.
Add more info:
I've been dabbling with rotation matrices in wiki for a while. The problem is, it is not a commutative operation. RxRyRz is not same as RzRyRx. So based on the way I multiple it I get a different final results. For example, I want my final projection to have 20 degrees from the original X axis, 20 degrees from original Y axis and 20 degrees from original Z axis. Based on the matrix, giving alpha, beta, gamma values 20 (or its corresponding radian) does NOT result the intended rotation. Am I missing something? Is there a matrix that I can just put the intended angles and get it at the end ?
Using a rotation matrix is an easy way to rotate a collection of (x,y,z) points. You can calculate a rotation matrix for your case using the equations in the general rotation section. Note that figuring out the angle values to plug into those equations can be tricky. Think of it as rotating about one axis at a time and remember that the order of your rotations (order of multiplications) does matter.
An alternative to the general rotation equations is to calculate a rotation matrix from axis and angle. It may be easier for you to define correct parameters with this method.
Update: After perusing Wikipedia, I found a simple way to calculate rotation axis and angle between two vectors. Just fill in your starting and ending vectors for a and b here:
a = [0.0 0.0 1.0];
b = [0.5 0.5 0.0];
vectorMag = #(x) sqrt(sum(x.^2));
rotAngle = acos(dot(a,b) / (vectorMag(a) * vectorMag(b)))
rotAxis = cross(a,b)
rotAxis =
-0.5 0.5 0
rotAngle =
1.5708