I have the following equations of motion of a point:
x:3*sin(4*t);
y:2*cos(4*t);
I already created a vector:
r:[x,y];
and did some operations but I don't know how to plot the path (the result should be an ellipse).
I tried various commands and only this gave me something similar to what I want to achieve:
load(draw);
draw2d(parametric(x,y,t,0,10));
But the result is strange:
Is there a way to get a proper plot in form of an ellipse in wxMaxima in this case?
Looks like draw2d(parametric(...)) is not applying adaptive subdivision to get a smoother plot; it's just sampling from a fixed grid. You can say draw2d(nticks = <large number>, parametric(...)) to get a somewhat smoother plot, but the problem doesn't go away.
Try plot2d([parametric, x, y, [t, 0, 10]]) -- I find that gives a nice smooth plot.
Related
I am trying to do 3D reconstruction using SFM (Structure From Motion). I am pretty new to computer vision and doing this as a hobby, so if you use acronyms please also let me know what it stands for so I can look it up.
Learning wise, I have been following this information :
https://www.youtube.com/watch?v=SyB7Wg1e62A&list=PLgnQpQtFTOGRYjqjdZxTEQPZuFHQa7O7Y&ab_channel=CyrillStachniss
https://imkaywu.github.io/tutorials/sfm/#triangulation
Plus links below from quick question.
My end goal is to use this on persons face, to create a 3D face reconstruction. If people have advice on this topic specifically please let me know as well.
I do the following steps :
IO using OpenCV. A video taken using a single camera.
Find intrinsic parameters and distortion coefficients of the camera using Zhangs method.
Use SIFT to find features from frame 1 and frame 2.
Feature matching is done using cv2.FlannBasedMatcher().
Compute essential matrix using cv2.findEssentialMat().
Projection matrix of frame 1 is set to numpy.hstack((numpy.eye(3), numpy.zeros((3, 1))))
Rotation and Translation are obtained using cv2.recoverPose().
Using Rotation and Translation we get the Projection Matrix of frame 2
curr_proj_matrix = cv2.hconcat([curr_rotation_matrix, curr_translation_matrix]).
I use cv2.undistortPoints() on feature pts for frame 1 and 2, using information from step 2.
Lastly, I do triangulation points_4d = triangulation.triangulate(prev_projection_matrix, curr_proj_matrix, prev_pts_u, curr_pts_u)
Then I reassign prev values to be equal curr values and continue through the video.
I use matplotlib to display the scatter plot.
Quick Question :
Why do some articles do E = (K^-1)T * F * K and some E = (K)T * F * K.
First way : What do I do with the fundamental matrix?
Second way : https://harish-vnkt.github.io/blog/sfm/
Issue :
As you can see the scatter plot looks a bit warped, I am unsure why, or if I am missing a step, or doing something wrong. Hence looking for advise.
Also the Z axis, is all negative.
One of the guesses I had, was that the video is in 60 FPS and even though I am moving the camera relatively quickly, it might not be enough of the rotation + translation to determine the triangulation. However, removing frames in between, did not make much difference.
Please let me know if you would like me to provide some of the code.
I believe I have an answer but I am not sure why it works. Hence if someone could expand, plus mention what the 3rd column of the 4D points is, then I will approve that answer and delete this.
Doing this on 4D points after triangulation : points_4d /= points_4d[3] (1)
The documentation does not mention it : https://docs.opencv.org/4.5.3/d9/d0c/group__calib3d.html#gad3fc9a0c82b08df034234979960b778c
My best guess, is that doing (1) is similar to doing this : cv2.convertPointsFromHomogeneous(). Converting from homogeneous space to euclidean space.
Edit 20211003 : Please see a comment for further explanation.
Say I have the lines shown in the image below, represented in polar coordinate format (rho and theta). These lines are the output of OpenCV's HoughLines function after some post processing. (Sorry I'm not allowed to embed images yet.)
What I want to do is, given any one line, find all of the lines that are perpendicular to that line, as shown in the second image below.
I understand how to do this with Cartesian lines, but I'm having trouble wrapping my mind around what properties of rho and theta the two lines would have to have to be perpendicular, although I understand how polar lines work at least fundamentally. Sorry if this is elementary stuff, but I'm having trouble finding any explanation of this online anywhere. Do I need to first convert the lines to Cartesian coordinates, or is there some simpler way to do this? Any help would be much appreciated, thanks!
To get perpendicular lines in polar coordinates, you simply take the theta for the first line, and find all lines whose theta = +/- 90° of the first theta.
You have to normalize the angles to be within 0°-360° or some other range, when comparing them.
So if line 1 has a theta line1.Theta
Then the angle to another line is a = (line2.Theta - line1.Theta)
and you want all lines where a is close to -90°, 90°, 270°, -270°, ...
depending on how you normalize your angles
I have UIBezierPathes like this:
And I want to calculate the area size of this kind of custom shapes. I don't have any idea to manage this calculation for this complex shapes in Objective c. I found here something for Android but didn't found something similar for iOS: Click
I dont need a working solution, just need an idea or some pseudo code to solve this problem if there is an way.
Thanks!
If you regions are surrounded by Bezier curves (including straight lines), then you can exploit Green's theorem for parametric curves (formula 10 here) to find area of region. Just calculate value of integral for every curve of contour:
A = Integral[t=0..1] (y(t)*x'(t)*dt)
for cubic Bezier curve, defined by control points P[]:
A = Integral[0..1](y(t)*x'(t)*dt)=
Integral[0..1](
(P[0].Y*(1-t)^3+3*P[1].Y*t*(1-t)^2+3*P[2].Y*t^2*(1-t)+P[3].Y*t^3)*
(P[0].X*(1-t)^3+3*P[1].X*t*(1-t)^2+3*P[2].X*t^2*(1-t)+P[3].X*t^3)' * dt)
We have to expand the brackets, differentiate the second line expression, multiply expressions, and integrate the result. Resulting formulas are rather large, but easy to implement. Maple work to derive formulas:
There a is an ellipse on the picture,just as following.
I have got the points of the contour by using opencv. But you can see the pictrue,because the resolution is low, there is a straight line on the contour.How can i fit it into curve like the blue line?
One Of the method to solve your problem is to vectorize your shape (moving from simple intensity space to vectors space).
I am not aware of the state-of-art in this field. However, from school information, I can suggest this solution.
Bezier curves, you can try to model your shape using simple bezier curve.This is not a hard operation you can google for dozen of them. Then, you can resizing it as much as you want after that you may render it to simple image.
Be aware that you may also Splines instead of Bezier.
Another method would be more simple but less efficient. Since you mentioned OpenCV, you can apply the cv::fitEllipse on the points. Be aware that this will return a RotatedRect which contains the ellipse. You can infer your ellipse simply like this:
Center = Center of RotatedRect.
Longest Radius = The Line which pass from the center and intersect with the two small sides of the RotatedRect.
Smallest Radius = The Line which pass from the center and intersect with the two long sides of the RotatedRect.
After you got your Ellipse Parameters, You can resize it as you want then just repaint it in the size you want using cv::ellipse.
I know that this is a pseudo answer. However, I think every thing is easy to apply. If you faced any problem implementing it, just give me a comment.
If instead of getting all edges, I only want edges that make 45 degree angles. What is a method to detect these?
Would it be possible to detect all edges, then somehow run a constrained hough transform to detect which edges form 45 degrees?
What is wrong with using an diagonal structure element and simply convolve the image??
Details
Please read here and it should become clear how to build the structuring element. If you are familiar with convolution than you can build a simple structure matrix which amplifies diagonals without theory
{ 0, 1, 2},
{-1, 0, 1},
{-2, -1, 0}
The idea is: You want to amplify pixel in the image, where 45deg below it is something different than 45deg above it. Thats the case when you are at a 45deg edge.
Taking an example. Following picture
convolved by the above matrix gives a graylevel image where the highest pixel values have those lines which are exactly 45deg.
Now the approach is to simply binarize the image. Et voila
First of all, it is possible to do this as post processing.
The result of Hough is in the parameter space of (angle,radius).
So you can simply take a slice in say angle=(45-5,45+5) and all radiuses.
An alternative method is that the output of edge detection will contain only 45/135 angle edges.
If you use a kernel but want line equations, then you'll still have to perform a line fit after the edge pixels are found. If you're certain the lines are exactly 45 degrees, then knowing the (x,y) point on any discovered line or line segment is sufficient to find the line equation.
Hough (rho, theta) parameter space can use whatever ranges of rho and theta that you'd like. You might preprocess the image to favor neighbor pixels at the proper angle. For example, give a "bonus point" to an edge pixel if it has 8-neighbors at the appropriate angle. You can certainly mix a kernel-based method (such as halirutan suggested) with a parametric or parameterless Hough algorithm.
A recent implementation of Hough runs at blazing fast speeds, so if you're looking for a quick solution you might download the open source code and then simply filter the output.
"Real-time line detection through an improved Hough transform voting scheme"
by Fernandes and Oliveira
http://www.ic.uff.br/~laffernandes/projects/kht/index.html