I've been playing around with both Matlab & Apples documentation in regards to CMRotationMatrix for weeks.
I've found that I could easily re-create CMRotationMatrix by calculating it with Roll, Yaw & Pitch.
However, I've found no resources/documentation on how to create a Rotation Matrix from XYZ rotations from either gravity or userAcceleration.
All I found was how they create a 4x4 matrix in their VideoSnake demo.
So my question is, does anyone have any input of how to create a 3x3 matrix from XYZ rotations?
To begin with rotation matrix has vast applications in Physics, Geometry and Computer Graphics according to Wikipedia. Now looking at it from this angle in relation to your question where you made mention of gravity and userAcceleration we are seeing a synergy between principles in relation to physics where we can make mention of spacecraft exploration which depends 100 percent on gravity.
Now getting to the meat of the matter on XYZ rotations in relation to Rotation Matrix there is an abstract figure which is denoted on the origin point of the XYZ axes without any specifics to a particular angle as a starting point.
Now this is the part you have to understand, since we are using abstract and arbitrary figures we need to convert this XYZ axis point into direction vectors which can then be understood in real life world coordinates.
Only then we will be able to synergistically relate Rotation Matrix and XYZ coordinate points
Now to conclude
The essence of using this direction vector is to convert the direction into equivalent direction in cognisance with the rotation matrix which can then be effectively utilised and expressed on the platform-local coordinates
Related
I'm working on an object tracking application using openCV. I want to convert my pixel coordinates to world coordinates to get more meaningful information. I have read a lot about computing the perspective transform matrix, and I know about cv2.solvePnP. But I feel like my case should be special, because I'm tracking a runner on a track and field runway with the runway orthogonal to the camera's z-axis. I will set up the camera to ensure this.
If I just pick two points on the runway edge, I can calculate a linear conversion from pixels to world coords at that specific height (ground level) and distance from the camera (i.e. along that line). Then I reason that the runner will run on a line parallel to the runway at a different height and slightly different distance from the camera, but the lines should still be parallel in the image, because they will both be orthogonal to the camera z-axis. With all those constraints, I feel like I shouldn't need the normal number of points to track the runner on that particular axis. My gut says that 2-3 should be enough. Can anyone help me nail down the method here? Am I completely off track? With both height and distance from camera essentially fixed, shouldn't I be able to work with a much smaller set of correspondences?
Thanks, Bill
So, I think I've answered this one myself. It's true that only two correspondence points are needed given the following assumptions.
Assume:
World coordinates are set up with X-axis and Y-axis parallel to the ground plane. X-axis is parallel to the runway.
Camera is translated and possibly rotated about X-axis (angled downward), but no rotation around Y-axis(camera plane parallel to runway and x-axis) or Z-axis (camera is level with respect to ground).
Camera intrinsic parameters are known from camera calibration.
Method:
Pick two points in the ground plane with known coordinates in world and image. For example, two points on the runway edge as mentioned in original post. The line connecting the poitns in world coordinates should not be parallel with either X or Z axis.
Since Y=0 for these points, ignore the second column of the rotation/translation matrix, reducing the projection to a planar homography transform (3x3 matrix). Now we have 9 degrees of freedom.
The rotation assumptions will enforce a certain form on the rotation/translation matrix. Namely, the first column and first row will be the identity (1,0,0). This further reduces the number of degrees of freedom in the matrix to 5.
Constrain the values of the second column of the matrix such that cos^2(theta)+sin^2(theta) = 1. This reduces the number of unknowns to only 4. Two correspondence points will give us the 4 equations we need to calculate the homography matrix for the ground plane.
Factor out the camera intrinsic parameter matrix from the homography matrix, leaving the rotation/translation matrix for the ground plane.
Due to the rotation assumptions made earlier, the ignored column of the rotation/translation matrix can be easily constructed from the third column of the same matrix, which is the second column in the ground plane homography matrix.
Multiply back out with the camera intrinsic parameters to arrive at the final universal projection matrix (from only 2 correspondence points!)
My test implentation has worked quite well. Of course, it's sensitive to the accuracy of the two correspondence points provided, but that's kind of a given.
I am have been using solvepnp() for the calculation of the rotation and translation matrix. But the euler angles calculated from the obtained rotation matrix gave very erratic values. Trying to find the problem, I had a set of 2D projection points for my marker and kept the other parameters of solvepnp() constant.
Eg values:
2D points
[219.67473, 242.78395; 363.4151, 238.61298; 503.04855, 234.56117; 501.70917, 628.16742; 500.58069, 959.78564; 383.1756, 972.02679; 262.8746, 984.56982; 243.17044, 646.22925]
The euler angle theta(x) calculated from the output rotation matrix of solvepnp() was -26.4877
Next, I incremented only the x value of the first point(i.e 219.67473) by 0.1 to check the variation of the theta(x) euler angle (keeping the remaining points and the other parameters constant) and ran the solvepnp() again .For that very small change,I had values which were decreasing from -19 degree, -18 degree (for x coord = 223.074) then suddenly jump to 27 degree for a while (for x coord = 223.174 to 226.974) then come down to 1.3 degree (for x coord = 227.074).
I cannot understand this behaviour at all.Could somebody please explain?
My euler angle calculation from the rotation matrix uses this procedure.
Try Rodrigues() for conversion between rotation matrix and rotation vector to make sure everything is clean and right. Non RANSAC version can be very sensitive to outliers that create a huge error in the parameters and thus bias a solution. Using RANSAC version of solvePnP may make it more stable to outliers. For example, adding too much to one of the points coordinates will eventually make it an outlier and it won’t influence a solution after that.
If everything fails, write a series unit tests: create an artificial set of points in 3D (possibly non planar), apply a simple translation first, in second variant apply rotation only, and in a third test apply both. Project using your camera matrix and then plug in your 2D, 3D points and projection matrix into your code to find the pose. If the result deviates from the inverse of the translations and rotations your applied to the points look for the bug in feeding parameters to PnP.
It seems the coordinate systems are different.OpenCV uses right-hand coordinate-system Y-pointing downwards. At nghiaho.com it says the calculations are based on this and if you look at the axis they don't seem to match. I guess you are using Rodrigues for matrix computation? Try comparing rotation vectors as well.
I'm trying to estimate the relative camera pose using OpenCV. Cameras in my case are calibrated (i know the intrinsic parameters of the camera).
Given the images captured at two positions, i need to find out the relative rotation and translation between two cameras. Typical translation is about 5 to 15 meters and yaw angle rotation between cameras range between 0 - 20 degrees.
For achieving this, following steps are adopted.
a. Finding point corresponding using SIFT/SURF
b. Fundamental Matrix Identification
c. Estimation of Essential Matrix by E = K'FK and modifying E for singularity constraint
d. Decomposition Essential Matrix to get the rotation, R = UWVt or R = UW'Vt (U and Vt are obtained SVD of E)
e. Obtaining the real rotation angles from rotation matrix
Experiment 1: Real Data
For real data experiment, I captured images by mounting a camera on a tripod. Images captured at Position 1, then moved to another aligned Position and changed yaw angles in steps of 5 degrees and captured images for Position 2.
Problems/Issues:
Sign of the estimated yaw angles are not matching with ground truth yaw angles. Sometimes 5 deg is estimated as 5deg, but 10 deg as -10 deg and again 15 deg as 15 deg.
In experiment only yaw angle is changed, however estimated Roll and Pitch angles are having nonzero values close to 180/-180 degrees.
Precision is very poor in some cases the error in estimated and ground truth angles are around 2-5 degrees.
How to find out the scale factor to get the translation in real world measurement units?
The behavior is same on simulated data also.
Have anybody experienced similar problems as me? Have any clue on how to resolve them.
Any help from anybody would be highly appreciated.
(I know there are already so many posts on similar problems, going trough all of them has not saved me. Hence posting one more time.)
In chapter 9.6 of Hartley and Zisserman, they point out that, for a particular essential matrix, if one camera is held in the canonical position/orientation, there are four possible solutions for the second camera matrix: [UWV' | u3], [UWV' | -u3], [UW'V' | u3], and [UW'V' | -u3].
The difference between the first and third (and second and fourth) solutions is that the orientation is rotated by 180 degrees about the line joining the two cameras, called a "twisted pair", which sounds like what you are describing.
The book says that in order to choose the correct combination of translation and orientation from the four options, you need to test a point in the scene and make sure that the point is in front of both cameras.
For problems 1 and 2,
Look for "Euler angles" in wikipedia or any good math site like Wolfram Mathworld. You would find out the different possibilities of Euler angles. I am sure you can figure out why you are getting sign changes in your results based on literature reading.
For problem 3,
It should mostly have to do with the accuracy of our individual camera calibration.
For problem 4,
Not sure. How about, measuring a point from camera using a tape and comparing it with the translation norm to get the scale factor.
Possible reasons for bad accuracy:
1) There is a difference between getting reasonable and precise accuracy in camera calibration. See this thread.
2) The accuracy with which you are moving the tripod. How are you ensuring that there is no rotation of tripod around an axis perpendicular to surface during change in position.
I did not get your simulation concept. But, I would suggest the below test.
Take images without moving the camera or object. Now if you calculate relative camera pose, rotation should be identity matrix and translation should be null vector. Due to numerical inaccuracies and noise, you might see rotation deviation in arc minutes.
I have a small cube with n (you can assume that n = 4) distinguished points on its surface. These points are numbered (1-n) and form a coordinate space, where point #1 is the origin.
Now I'm using a tracking camera to get the coordinates of those points, relative to the camera's coordinate space. That means that I now have n vectors p_i pointing from the origin of the camera to the cube's surface.
With that information, I'm trying to compute the affine transformation matrix (rotation + translation) that represents the transformation between those two coordinate spaces. The translation part is fairly trivial, but I'm struggling with the computation of the rotation matrix.
Is there any build-in functionality in OpenCV that might help me solve this problem?
Sounds like cvGetPerspectiveTransform is what you're looking for; cvFindHomograpy might also be helpful.
solvePnP should give you the rotation matrix and the translation vector. Try it with CV_EPNP or CV_ITERATIVE.
Edit: Or perhaps you're looking for RQ decomposition.
Look at the Stereo Camera tutorial for OpenCV. OpenCV uses a planar chessboard for all the computation and sets its Z-dimension to 0 to build its list of 3D points. You already have 3D points so change the code in the tutorial to reflect your list of 3D points. Then you can compute the transformation.
I am using the glMatrix to code Webgl and want to get the eye position, focal point and up direction from the existing projection and view matrix (kinda like the reverse of lookat function). Is there any way to do this?
I didn't implement one, no. I'm not even sure that you could decompose it into the original vectors, for that matter. The lookAt point could be anywhere along a ray from the origin, and how would you determine what the appropriate up vector was? I'm thinking this is a one-way algorithm (just too lazy to prove it!)
Beyond that, however, I question wether you would want to do this even if there was a method for it. I'll be willing to bet that it's almost always more beneficial to track the values you're using and manipulate them rather than to try and pull them back and forth from matrix to vectors and back.
Yes and No: Yes you can invert the model view transformation and no you will not get exactly all three vectors the same.
The model view transformation of lookAt is very similar to the connectTo operation as used in CSG models. It is mounting your scene in front of your camera. This is done by translation and three axis rotations. The eye point is translated to (0,0,0) and all further rotation is done around it. You can easily derive the eye point by transforming (0,0,0) with the inverse matrix.
But the center point is just used for adjusting the axis of view along the -Z axis. In openGL the eye is facing to -Z. The distance between center and eye is lost. So you can easy get a center point along your axis of view if you define the distance yourself. Let's say we want a distance of d. Then we just need to transform (0,0,-d) with the inverse matrix and we get a valid center point, but not exactly the same. The center point is defining only two rotation angles, the camera pan and tilt.
Even more worse is the reconstruction of the up vector. It is only used for the roll angle of the camera and thus only for one scalar value. Thus for the inverse transformation you can not only choose any positive value along the Y axis, you could choose any point in the YZ plane with a positive Y value. To get a up vector perfectly normal to the viewing axis and of size 1 we just transform (0,1,0) with the inverse matrix. Remember to transform as vector this time (not as point).
Now we have eye, center and up reconstructed in a way to get exactly the same result of lookAt next time. But since this matrix contains only 6 values of information (translation,pan,tilt,roll) we had to choose 3 values that were lost (distance center to eye, size and angle of up vector in YZ plane of camera).
The model view matrix can of course do other transformation (any affine) but the lookAt function is using this matrix only for translation and rotation. It is adjusting the scene in front of the camera without distorting it.