I am currently having a part of code using euclidean distance to compute some likelihoods. I am trying to think of any ways to compute likelihoods with the same effectiveness, without using euclidean distance.
Is there anyone who could give me maybe a hint for a different way?
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I have a distribution of weighted 2D pose estimates (position + orientation) that are samples of an unknown PDF of a systems pose. All estimates and the underlying real position are constrained by a concave polygon.
The picture shows an exemplary distribution. The magenta colored circles are the estimates, the radius line indicates the estimated direction. The weights are indicated by the circles diameter. The red dot is the weighted mean, the yellow cirlce indicates the variance and the direction but is of no importance for the following problem:
From all estimates I want to derive the most likely position of the system.
Up to now I have evaluated the following approaches:
Using the estimate with the highest weight: Gives poor results since one estimate with a high weight outperforms several coinciding estimates with slightly lower weights.
Weighted Mean: Not applicable since the mean might lie outside the polygon as in the picture (red dot with yellow circle).
Weighted Median: Would work but does neglect potential clusters. E.g. in the image below two clusters are prominent of which one is more likely than the other.
Additionally I have looked into K-Means and K-Medoids. For K-Means I do not know the most efficient way to constrain the centers to the polygon. K-Medoids seems to work, but has poor performance (O(n^2)), which is important since I have a high number of estimates (contrary to explanatory picture)
What would be the ideal algorithm to solve this kind of problem ?
What complexity can be achieved ?
Are there readily available algorithms in c++ that solve this problem, or can be easily adapted to solve it?
k-means may also yield an estimate outside your polygons.
Such constraints are beyond the clustering use case. But nothing prevents you from devising a method to correct the estimates afterwards.
For non-convex data, DBSCAN may be worth a try. You could even incorporate line-of-sight into Generalized DBSCAN easily. But I'm not convinced that clustering will help for your overall objective.
I am trying to program a robot that detect obstacles and estimate distance.
I am using computer vision for this task . I calculated the disparity map but I do not know how to detect obstacles and estimate the distance.
what are the steps I need to follow to accomplish this goal?
what are the Open CV's functions I will need to use?
Is there any source codes?
Suppose we have an ellipse x^2/a^2 + y^2/b^2 .
Taking a point (a*cos(t),b*sint(t)) on the ellipse, what is the fastest way to find another point on the ellipse such that distance between them is a given d. [d is less than pi*a*b].
The problem was encountered when i have a corner [quarter ellipse] and need to find points along it seperated by some 'd'.
The length of a subsection of an ellipse is an elliptic integral, with no closed form solution.
In order to compute the distance along the ellipse, you will need a numerical integration routine. I recommend Romberg, or Gauss Quadrature (look up on Wikipedia). If you are doing this repeatedly, then precompute the distance across a bunch of points around the Ellipse so that you can rapidly get to the right region, then start integrating.
You will need to bisect (look up on Wikipedia) to find the desired length.
There is no analytical solution for the length of an elliptical arc. This means you won't be able to plug numbers into an equation to find a result, but instead use a method of numerical integration.
Simpsons rule is very easy to implement although most likely slower than the methods mentioned in other answers.
Now that you have a way to find the length of an elliptical arc, just measure different end points until you find one of length d to some acceptable tolerance
hey, I have been given a problem, I basically have been given a piece of grid paper of arbitary size and have to develop a distance matrix using only the coordinates for each of the grid points on the page.
I'm thinking the best approach would be something like the Floyd-Warshall or Djikstra algorithms for shortest path pair, but don't know how to adapt it to coordinate distances, as all the documentation uses a pre-determined distance matrix. so any help would be grand
the distance matrix contains simply the distances to all other points.
Basically, you just have to calculate the distances using an appropriate metric. If you want the "normal" distance, it's sqrt((x1-x2)^2+(y1-y2)^2) where (x/y) are the coordinates of a point in mm / inches. If you want the distance on the paper just following the lines its |x1-x2|+|y1-y2|.
Graph algorithms would be a overkill unless you have walls on the paper.
I'm trying to detect how well an input vector fits a given cluster centre. I can find the best match quite easily (the centre with the minimum euclidean distance to the input vector is the best), however, I now need to work how good a match that is.
To do this I need to find the spread (standard deviation?) of the vectors which build up the centroid, then see if the distance from my input vector to the centre is less than the spread. If it's more than the spread than I should be able to say that I have no clusters to fit it (given that the best doesn't fit the input vector well).
I'm not sure how to find the spread per cluster. I have all the centre vectors, and all the training vectors are labelled with their closest cluster, I just can't quite fathom exactly what I need to do to get the spread.
I hope that's clear? If not I'll try to reword it!
TIA
Ian
Use the distance function and calculate the distance from your center point to each labeled point, then figure out the mean of those distances. That should give you the standard deviation.
If you switch to using a different algorithm, such as Mixture of Gaussians, you get the spread (e.g., std. deviation) as part of the model (clustering result).
http://home.deib.polimi.it/matteucc/Clustering/tutorial_html/mixture.html
http://en.wikipedia.org/wiki/Mixture_model