From I've understood, the transform order (rotate first or translate first) yield different [R|t].
So I want to know what's the order of the 4 possible poses you got from essential matrix SVD.
I've read a code implementation of Pose From Essential Matrix(Hartley and Zisserman's multiple view geometry (page 259)). And the author seems to interpret it as rotate first then translate, where he retrieve camera position by using p = -R^T * t.
Also, opencv seems to use trasnlate first then rotate rule. Because the t vector I got from calibrating camera is the position of camera.
Or maybe I have been wrong and the order doesn't matter?
You shouldn't use SVD to decompose a transformation into rotation and translation components. Viewed as x' = M*x = T*R*x, the translation is just the fourth column of M, and the rotation is in the upper-left 3x3 submatrix.
If you feed the whole 4x4 matrix into SVD, I'm not sure what you'll get out, but it won't be useful. (If nothing else, U and V will not be affine.)
Related
I'm trying to take a static list of vertices and, using a matrix, bend them about a circle as shown in the image below. I can find the new vertices using basic geometry, but I'm unsure how to implement it with a matrix, or if it is even possible. I believe, calling the center of the circle 'C' and calling the distance from the vertex to C 'd', that x' = Cx + d*cos(theta) and y' = Cy + d*sin(theta). (Cx and Cy are components of C). However, I don't think that that, in itself, can be converted to a matrix.
The Transformation
For context, I'm making a 3D Snake game. The Snake is made up of segments, and each segment should connect to the next. I have a direction and position for each connection between the segments, so I figured that I could build the segment by connecting each part as shown. However, it would be inefficient to calculate the vertices for each segment on each frame. If I could have the vertices statically listed and then transform it in the model-world matrix, that would be faster, and fit into the method I'm currently using. I included a simple drawing of the Snake (ignore the head) below.
The Snake
If it's not possible, are there any alternatives that can be done using a matrix while giving a similar result?
Yes, it is possible.
Skeleton Animation:
You can actually add a series of bones to the object and just apply rotation to the bones equally to get a perfect curve. This is the easiest way to achieve this. However, not necessarily the fastest way for your case.
Just Matrix & Weights:
If the bending is always in this proportion, you have to add weight information to each vertex and then multiply with a matrix with transformation/rotation/scale combination. This will give you better performance compared to Skeleton animation, as every vertex is multiplied using only one matrix and there is no hierarchy.
I have a 3d object which is free to rotate along x,y and z axis and it is then saved as a transform matrix. In a case where the sequence of rotation is not known and the object is rotated for more than 3 times (eg :-if i rotate the object x-60degress, y-30 degrees, z-45 degrees then again x->30 degrees), is it possible to extract the angles rotated from the transform matrix?.I know that it is possible to get angles if the sequence of rotation is known, but if I have only the final transform matrix with me and nothing else, is it possible to get the angles rotated(x,y,and z) from the transform matrix ?
Euler angle conversion is a pretty well known topic. Just normalize the matrix orientation vectors and then use something like this c source code.
The matrix is the current state of things it has no knowledge of what the transformation has been in the past. It does not know how the matrix was built. You can just take the matrix into and decompose it into any pieces you like, as long as:
The data do not overlap. For example:Two X turns after each other is indistinguishable form each other (no way to know if its 1 2 or three different rotations summed).
The sequence order is known
A decomposition can be built out of the data (for example scale can be measured)
Why hello thar. I have a 3D model whose starting angles along its 3 axis are known. Those angles are transformed into a Direction Cosine Matrix using this pattern (information found here) :
New angles values are obtained as time goes by, corresponding to an update of the model's orientation. To take those new values into consideration, I update the DCM with those in this way :
newDCM = oldDCM * newanglesDCM
What I want and how I do it : Now the tricky part. I actually only want the Y component of the rotation matrix. In order to apply a motion on my model, I need it to be flatten so the motion vector doesn't go either in the air or down on the ground. To do so I simply pull the 3 angles back from the rotation matrix and create a new one with the angles [0 Y 0].
The problem I have : When rotation are applied to the model, the DCM is updated. As a motion is detected, I multiply the motion vector [0 Yvalue 0] to the DCM that has been flatten (according to the previous explanation). The result is great when the instantaneous rotations have null or close to null values in X and Z components. But as soon as the model is in a situation where X and Z have significant values, the "orientation" of the model's motion is wrong. If I apply rotations that make it return into a "only Y" situation, it starts being good again.
What could go wrong : Either my Direction Cosine Matrix is wrong, or the technique I use to flatten the matrix is just completely stupid.
Thanks for your help, I'd really appreciate if you could give me a hand on this !
Greg.
EDIT : Example as requested
My model has 3 axis X,Y and Z. This defines the XYZ convention when the model is at rest. At starting point t0, I know the angles dAx, dAy and dAz that allow me to rotate the model from its original configuration to the one it is in at t0. If that bugs you, let's say that the model is at rest at t0, it doesn't matter.
I create the DCM just like explained in the image (let it be an identity matrix if it started at rest).
Every now and then rotations are applied to the model. Those rotations also are made of a dAx, dAy and dAz. Thus I update the rotation matrix (DCM) by multiplying the old one by the newly generated one : newDCM = oldDCM * newanglesDCM.
Now let's say I want the model to move, on the grid, from a point to another. Imagine the grid to be a street for example. Whether the model is oriented towards the sky, on a side or in front of it, I want the motion to be the same : alonside the road and not elevating in the air or diving into the ground.
If I kept the rotation matrix as it is, applying a [0 Y 0] rotation WOULD make it go somewhere I don't want it to. Thus I try to find my old original XZ frame, by flattening the DCM. Then I still have the Y component so I know where in the street the model is moving.
Imagine a character, whose head is the model and who is walking outside. If he looks to a building's window and walks, he won't walk in the air right up to the window - he'll walk to the feet of the building. That is exactly what I want to do :D
What you have to do is equate the elements two rotation matrices
E_1 = Rx(dθx)Ry(dθy)Rz(dθz)
and
E_2 = Rx(dφx)Rz(dφz)Ry(dφy)
in an element by element basis. Find two elemetns that contain only sin(dφy) and cos(dφy) and divide them to get tan(dφy)=... in terms of dθx, dθy and dθz.
I tried to do this with the DCM given but I cannot replicate the sequence of rotations to get what you have. My E_1 above is similar but some signs are different. In my example that I did I got the following expressions
dφy=atan( tan(dθy)/cos(dθz) )
dφz=atan( (cos(dθy)*sin(dθz))/(cos(dθy)*cos(dθz)*cos(dφy)+sin(dθy)*sin(dφy)) )
dφx=atan( (cos(dθx)*sin(dθy)*sin(dθz)+sin(dθx)*cos(dθz))/(cos(dθx)*cos(dθz)- ...
You have to work out your own relationships based on the sequences you use.
Note: that once dφy is known the above E_1 = E_2 equality becomes
Rx(dφx)Rz(dφz) = Rx(dθx)Ry(dθy)Rz(dθz)Ry(-dφy)
which is solved for dφz and then you have
Rx(dφx) = Rx(dθx)Ry(dθy)Rz(dθz)Ry(-dφy)Rz(-dφz)
for dφx.
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!
So first of all I have such image (and ofcourse I have all points coordinates in 2d so I can regenerate lines and check where they cross each other)
(source: narod.ru)
But hey, I have another Image of same lines (I know thay are same) and new coords of my points like on this image
(source: narod.ru)
So... now Having points (coords) on first image, How can I determin plane rotation and Z depth on second image (asuming first one's center was in point (0,0,0) with no rotation)?
What you're trying to find is called a projection matrix. Determining precise inverse projection usually requires that you have firmly established coordinates in both source and destination vectors, which the images above aren't going to give you. You can approximate using pixel positions, however.
This thread will give you a basic walkthrough of the techniques you need to use.
Let me say this up front: this problem is hard. There is a reason Dan Story's linked question has not been answered. Let provide an explanation for people who want to take a stab at it. I hope I'm wrong about how hard it is, though.
I will assume that the 2D screen coordinates and projection/perspective matrix is known to you. You need to know at least this much (if you don't know the projection matrix, essentially you are using a different camera to look at the world). Let's call each pair of 2D screen coordinates (a_i, b_i), and I will assume the projection matrix is of the form
P = [ px 0 0 0 ]
[ 0 py 0 0 ]
[ 0 0 pz pw]
[ 0 0 s 0 ], s = +/-1
Almost any reasonable projection has this form. Working through the rendering pipeline, you find that
a_i = px x_i / (s z_i)
b_i = py y_i / (s z_i)
where (x_i, y_i, z_i) are the original 3D coordinates of the point.
Now, let's assume you know your shape in a set of canonical coordinates (whatever you want), so that the vertices is (x0_i, y0_i, z0_i). We can arrange these as columns of a matrix C. The actual coordinates of the shape are a rigid transformation of these coordinates. Let's similarly organize the actual coordinates as columns of a matrix V. Then these are related by
V = R C + v 1^T (*)
where 1^T is a row vector of ones with the right length, R is an orthogonal rotation matrix of the rigid transformation, and v is the offset vector of the transformation.
Now, you have an expression for each column of V from above: the first column is { s a_1 z_1 / px, s b_1 z_1 / py, z_1 } and so on.
You must solve the set of equations (*) for the set of scalars z_i, and the rigid transformation defined R and v.
Difficulties
The equation is nonlinear in the unknowns, involving quotients of R and z_i
We have assumed up to now that you know which 2D coordinates correspond to which vertices of the original shape (if your shape is a square, this is slightly less of a problem).
We assume there is even a solution at all; if there are errors in the 2D data, then it's hard to say how well equation (*) will be satisfied; the transformation will be nonrigid or nonlinear.
It's called (digital) photogrammetry. Start Googling.
If you are really interested in this kind of problems (which are common in computer vision, tracking objects with cameras etc.), the following book contains a detailed treatment:
Ma, Soatto, Kosecka, Sastry, An Invitation to 3-D Vision, Springer 2004.
Beware: this is an advanced engineering text, and uses many techniques which are mathematical in nature. Skim through the sample chapters featured on the book's web page to get an idea.