I am trying to find a way to determine whether two frusta intersect and, if so, how big of an intersection that is (example 100% if the two frusta are in exact same location, 0% if they don't touch).
I have the position, volume and all sort of data about the two frusta, I just have no idea how to use it. I took a look at the Separating Axis Theorem for collision detection but I can't figure out exactly whether it's what I'm looking for.
Does anybody have any suggestion on the direction to go?
The SAT will only tell you if they are touching. It won't be able to give you a percentage overlap. To calculate the frusta overlap percentage, I think you will need to calculate the volume of the poyhedron created by intersecting the frusta and divide by the volume of the "main" frustum.
Calculating the intersection of the frusta will tell you if they are overlapping. One way to do it is to build a bsp out of each one, and do a CSG Intersection operation.
Once you have the interesection polyhedron, you can calculate its volume by splitting it up into tetrahedrons and adding up all the volumes of the tetrahedrons. There are academic papers out there that do tetrahedralization directly from the BSP representation.
Related
I have a markered robot with circular markers and two images from different perspective as shown: (Circular white rings are the markers)
I want to match the markers in the two images, by matching I mean the bottommost marker of 1st image should be treated as correspondence point of bottom most marker of 2nd image and so on.
The finger-like robot given in the image can bend in any direction given in space (can also bend in a U-like manner).
If it helps, the camera geometry is fixed and known beforehand.
I am lost, as simple correspondence algorithm would not work, since the perspectives are very different. How should I go about matching the two images?
You can start like this:
You know the position of the mounting point on the base panel for each perspective.
You know the positions of the white rings for each perspective as discussed here.
You can derive the direction of the arm at each ring by its tilt.
So you can easily determine the sequence of the positions starting with the mounting point stepping from ring to ring even if the arm is bent. With this you can match the rings from both images. If you have any situation where this fails, please add an according example to your question!
Unfortunately, you don't have matching points but matching curves. You might try to fit ellipses on the rings and take the ellipse centers for points to be matched.
This is an approximation, as the center of a circle does not exactly project as the center of the ellipse, but I don't think that this will be the major source of error: as you only see half circles, the fitting will not be that accurate.
If all nine circles remain visible and are ordered vertically, the matching of the centers is trivial. If they are not ordered but don't form a loop, you can probably start from the lowest and follow the chain of nearest neighbors.
I'm working on an IPhone robot that would be moving around. One of the challenges is estimating distance to objects- I don't want the robot to run into things. I saw some very expensive (~1000$) laser rangefinders, and would like to emulate one using iPhone.
I got one or two camera feeds and two laser pointers. The laser pointers are mounted about 6 inches apart, at an angle The angle of lasers in relation to the cameras is known. The Angle of cameras to each other is known.
The lasers are pointing ahead of cameras, creating 2 dots on a camera feed. Is it possible to estimate the distance to the dots by looking at the distance between the dots in a camera image?
The lasers form a trapezoid from the
/wall \
/ \
/laser mount \
As the laser mount gets closer to the wall, the points should be moving further away from each other.
Is what I'm talking about feasible? Has anyone done something like that?
Would I need one or two cameras for such calculation?
If you just don't want to run into things, rather than have an accurate idea of the distance to them, then you could go "dambusters" on it and just detect when the two points become one - this would be at a known distance from the object.
For calculation, it is probaby cheaper to have four lasers instead, in two pairs, each pair at a different angle, one pair above the other. Then a comparison between the relative differences of the dots would probably let you work out a reasonably accurate distance. Math overflow for that one, though.
In theory, yes, something like this can work. Google "light striping" or "structured light depth measurement" for some good discussions of using this sort of idea on a larger scale.
In practice, your measurements are likely to be crude. There are a number of factors to consider: the camera intrinsic parameters (focal length, etc) and extrinsic parameters will affect how the dots appear in the image frame.
With only two sample points (note that structured light methods use lines, etc), the environment will present difficulties for distance measurement. Surfaces that are directly perpendicular to the floor (and direction of travel) can be handled reasonably well. Slopes and off-angle walls may be detectable, but you will find many situations that will give ambiguous or incorrect distance measures.
The following problem:
Given is an arbitrary polygon. It shall be covered 100% with the minimum number of circles of a given radius.
Note:
1) Naturally the circles have to overlap.
2) I try to solve the problem for ARBITRARY polygons. But also solutions for CONVEX polygons are appreciated.
3) As far as Im informed, this problem is NP-hard ( an algorithm to find the minimum size set cover for the Set-cover problem )
Choose: U = polygon and S1...Sk = circles with arbitrary centers.
My solution:
Ive already read some papers and tried a few things on my own. The most promising idea that I came up with was in fact one already indicated in Covering an arbitrary area with circles of equal radius.
So I guess it’s best I quickly try to describe my own idea and then refine my questions.
The picture gives you already a pretty good idea of what I do
IDEA and Problem Formulation
1. I approximate the circles with their corresponding hexagons and tessellate the whole R2, i.e. an sufficiently large area; keyword hexagonally closest packaging. (cyan … tessellation, red dotted, centers of the cyan hexagons)
2. I put the polygon somewhere in the middle of this tessellated area and compute the number of hexagons that are needed to cover the polygon.
In the following Im trying to minimize N, which is number ofhexagons needed to cover the polygon, by moving the polygon around step by step, after each step “counting” N.
Solving the problem:
So that’s when it gets difficult (for me). I don’t know any optimizers that solve this problem properly, since they all terminate after moving the polygon around a bit and not observing any change.
My solution is the following:
First note that this is a periodic problem:
1. The polygon can be moved in horizontal direction x with a period of 3*r (side length = radius r) of the hexagon.
2. The polygon can be moved in vertical direction y with a period of r^2+r^2-2*rrcos(2/3*pi) of the hexagon.
3. The polygon can be rotated phi with a period of 2/3*pi.
That means, one has to search a finite area of possible solutions to find the optimal solution.
So what I do is, I choose a stepsize for (x,y,phi) and simply brute force compute all possible solutions, picking out the optimum.
Refining my questions
1) Is the problem formulated intelligently? Right now im working on an algorithm that only tessellates a very small area, so that as little hexagons as possible have to be computed.
2) Is there a more intelligent optimizer to solve the problem?
3) FINALLY: I also have difficulties finding appropriate literature, since I don’t guess I don’t know the right keywords to look for. So if anybody can provide me with literature, it would also be appreciated a lot.
Actually I could go on about other things ive tried but I think no one of u guys wants to spend the whole afternoon just reading my question.
Thx in advance to everybody who takes the time to think about it.
mat
PS i implement my algorithms in matlab
I like your approach! When you mention your optimization I think a good way to go about it is by rotating the hexagonal grid and translating it till you find the least amount of circles that cover the region. You don't need to rotate 360 since the pattern is symmetric so just 360/6.
I've been working on this problem for a while and have just published a paper that contains code to solve this problem! It uses genetic algorithms and BFGS optimization. You can find a link to the paper here: https://arxiv.org/abs/2003.04839
Edit: Answer rewritten (there's no limitation that circles couldn't go outside the polygon).
You might be interested in Covering a simple polygon with circles. I think the algorithm works or is extendable also to complex polygons.
1.Inscribe the given polygon in a minimum sized rectangle
2.Cover the rectangle optimally by circles (algorithm is available)
I have a series of nature reserves that need to be plotted, as polygon overlays, on a map using the coordinates contained within KML data. I’ve found a tutorial on the Apple website for displaying KML overlays on map instances.
The problem is that the reserves vary in size greatly - from a small pond right up to several hundred kilometers in size. As a result I can’t use the coordinates of the center point to find the nearest reserves. Instead I need to calculate the nearest point of the reserves polygon to find the nearest one. With the data in KML - how would I go about trying to achieve this?
I've only managed to find one other person ask this and no one had replied :(
Well, there are a couple different solutions depending on your needs. The higher the accuracy required, the more work required. I like Phil's meanRadius parameter idea. That would give you a rough idea of which polygon is closest and would be pretty easy to calculate. This idea works best if the polygons are "circlish". If the polygon are very irregular in shape, this idea loses it's accuracy.
From a math standpoint, here is what you want to do. Loop through all points of all polygons. Calculate the distance from those points to your current coordinate. Then just keep track of which one is closest. There is one final wrinkle. Imagine a two points making a line segment that is very long. You are located one meter away from the midpoint of the line. Well, the distance to these two points is very large, while, in fact you are very close to the polygon. You will need to calculate the distance from your coordinate to every possible line segment which you can do in a variety of manners which are outlined here:
http://www.worsleyschool.net/science/files/linepoint/distance.html
Finally, you need to ask yourself, am I in any polygons? If you're 10 meters away from a point on a polygon, but are, in fact, inside the polygon, obviously, you need to consider that. The best way to do that is to use a ray casting algorithm:
http://en.wikipedia.org/wiki/Point_in_polygon#Ray_casting_algorithm
I need to compare two or more images to calculate how much a point shifted in the x and y direction. How do I go about doing this in MATLAB?
What you are looking for is an "Optical Flow" algorithm. There are many around, some faster but less accurate, some slower and more accurate.
Click here to find a MATLAB optical flow implementation (Lucas Kanade).
Gilads suggestion about a Lucas-Kanade tracker/optical flow calculator is really good, and is what I would use. It does however have the drawback of not working very well if the scene has changed too much.
If the scenes are indeed very different (say you moved and rotated the camera quite a lot) you would have to find your corresponding points in some other way. One example could be to use a SIFT descriptor to find image features in the two images and then determine which points correspond to each other. If you know the camera matrices of the two images then it becomes quite easy.