If I have a A* function that supports finding the optimal path from a starting point to a target in a maze, how should I modify the heuristic function to be admissible so that if there are multiple targets the function still return the optimal result.
Assuming that the problem is to visit only one target:
The first solution that comes to mind is to loop over all possible targets, compute the admissible heuristic value for each of them, and then finally return the minimum of those values as the final heuristic value. That way you're sure that the heuristic is still admissible (does not overestimate the distance to any of the targets).
EDIT: Now assuming that the problem is to visit all targets:
In this case, A* may not even necessarily be the best algorithm to use. You can use A* with the heuristic as described above to first find the shortest path to the closest target. Then, upon reaching the first (closest) target, you can run it again in the same way to find the next closest target, etc.
But this may not give an optimal overall solution. It is possible that it is beneficial to first visit the second closest target (for example), because in some cases that may enable a cheaper follow-up path to the second target.
If you want an optimal overall solution, you'll want to have a look at a different overall approach probably. Your problem will be similar to the Travelling Salesman Problem (though not exactly the same, since I think you're allowed to visit the same point multiple times for example). This link may also be able to help you.
I'm looking for a solution to a Close-Enough Traveling Salesman Problem (CETSP) where I have a set of nodes that I need to visit all within a certain distance of optimally. I've found a couple of sources for some approaches towards this TSP variant but was unable to find a solver or a algorithm that I could easily use.
Do you have any suggestions for how I can go about getting a solution to my CETSP problem, whether it be running an implementation of it myself or using an existing solver.
You can try using UFFLP. They have an example where you can find the correct coordinates the salesman is supposed to pass given a predetermined sequence. So you can generate thousands of sequences and choose the best one (just a simple heuristic).
Have a look at http://www.gapso.com.br/en/ufflp-en/
You will find useful information.
The problem I am facing is following.
I have a number of 3D head scans, some of them are taken correctly (like attached example) but in many it is easy to see that the scanned person had his head not exactly aligned with the machine's front and thus one side of the texture (and depth map) seems to be "wider" (the exact reason is that one side was taken more from behind, it can be easily seen if you look at the ears).
Fortunately when I go from the cylindrical coordinates to carthesian ones and render the face with XNA, the face is symmetrical.
Now the thing is that I would like the texture and depth maps of all my heads by as nice and symmetrical as the correct one (because later i want to align them and perform PCA).
The idea I have at the moment is that I could interpolate the surfaces between all of the vertices and from those interpolations take new vertices that are equally distanced from each other.
This solutions seems a lot of work and maybe its an overkill.
Maybe there is some other way (like geting that interpolation data from DirectX/XNA that has to calculate it at some point anyway).
I will be most thankful for helpful answers.
The correct example:
http://i55.tinypic.com/332mio2.jpg
Incorrect example:
http://i54.tinypic.com/309ujvt.jpg
It's probably possible to salvage (some of) the bad scans to some degree using some coordinate transformations, but you would have to guess the "incorrectness" of the alignment and it's probably impossible to do automatically.
But, unless the original subject is dead (or otherwise unavailable); it's probably a lot easier to redo the scans.
Making another scan is very likely to be quicker, and you won't loose quality as transforming the bad scans probably will. The nose on the incorrect sample seems to be shadowing the side of the nose, and no fancy algorithm can ever fix the missing data.
I have a tile based map where several tiles are walls and others are walkable. the walkable tiles make up a graph I would like to use in path planning. My question is are their any good algorithms for finding a path which visits every node in the graph, minimising repeat visits?
For example:
map example http://img220.imageshack.us/img220/3488/mapq.png
If the bottom yellow tile is the starting point, the best path to visit all tiles with least repeats is:
path example http://img222.imageshack.us/img222/7773/mapd.png
There are two repeat visits in this path. A worse path would be to take a left at the first junction, then backtrack over three already visited tiles.
I don't care about the end node but the start node is important.
Thanks.
Edit:
I added pictures to my question but cannot see them when viewing it. here they are:
http://img220.imageshack.us/img220/3488/mapq.png
http://img222.imageshack.us/img222/7773/mapd.png
Additionally, in the graphs I need this for there will never be a situation where min repeats = 0. That is, to step on every tile in the map the player must cross his own path at least once.
Your wording is bad -- it allows a reduction to an NP-complete problem. If you could minimize repeat visits, then could you push them to 0 and then you would have a Hamiltonian Cycle. Which is solvable, but hard.
This sounds like it could be mapped onto the traveling salesman problem ... and so likely ends up being NP complete and no efficient deterministic algorithm is known.
Finding a path is fairly straight forward -- find a (or the minimum) spanning subtree and then do a depth/breadth-first traversal. Finding the optimal route is the really difficult bit.
You could use one of the dynamic optimization techniques to try and converge on a fairly good solution.
Unless there is some attribute of the minimum spanning subtree that could be used to generate the best path ... but I don't remember enough graph theory for that.
For instance:
An approach to compute efficiently the first intersection between a viewing ray and a set of three objects: one sphere, one cone and one cylinder (other 3D primitives).
What you're looking for is a spatial partitioning scheme. There are a lot of options for dealing with this, and lots of research spent in this area as well. A good read would be Christer Ericsson's Real-Time Collision Detection.
One easy approach covered in that book would be to define a grid, assign all objects to all cells it intersects, and walk along the grid cells intersecting the line, front to back, intersecting with each object associated with that grid cell. Keep in mind that an object might be associated with more grid-cells, so the intersection point computed might actually not be in the current cell, but actually later on.
The next question would be how you define that grid. Unfortunately, there's no one good answer, and you need to consider what approach might fit your scenario best.
Other partitioning schemes of interest are different tree structures, such as kd-, Oct- and BSP-trees. You could even consider using trees combined with a grid.
EDIT
As pointed out, if your set is actually these three objects, you're definately better of just intersecting each one, and just pick the earliest one. If you're looking for ray-sphere, ray-cylinder, etc, intersection tests, these are not really hard and a quick google should supply all the math you might possibly need. :)
"computationally efficient" depends on how large the set is.
For a trivial set of three, just test each of them in turn, it's really not worth trying to optimise.
For larger sets, look at data structures which divide space (e.g. KD-Trees). Whole chapters (and indeed whole books) are dedicated to this problem. My favourite reference book is An Introduction to Ray Tracing (ed. Andrew. S. Glassner)
Alternatively, if I've misread your question and you're actually asking for algorithms for ray-object intersections for specific types of object, see the same book!
Well, it depends on what you're really trying to do. If you'd like to produce a solution that is correct for almost every pixel in a simple scene, an extremely quick method is to pre-calculate "what's in front" for each pixel by pre-rendering all of the objects with a unique identifying color into a background item buffer using scan conversion (aka the z-buffer). This is sometimes referred to as an item buffer.
Using that pre-computation, you then know what will be visible for almost all rays that you'll be shooting into the scene. As a result, your ray-environment intersection problem is greatly simplified: each ray hits one specific object.
When I was doing this many years ago, I was producing real-time raytraced images of admittedly simple scenes. I haven't revisited that code in quite a while but I suspect that with modern compilers and graphics hardware, performance would be orders of magnitude better than I was seeing then.
PS: I first read about the item buffer idea when I was doing my literature search in the early 90s. I originally found it mentioned in (I believe) an ACM paper from the late 70s. Sadly, I don't have the source reference available but, in short, it's a very old idea and one that works really well on scan conversion hardware.
I assume you have a ray d = (dx,dy,dz), starting at o = (ox,oy,oz) and you are finding the parameter t such that the point of intersection p = o+d*t. (Like this page, which describes ray-plane intersection using P2-P1 for d, P1 for o and u for t)
The first question I would ask is "Do these objects intersect"?
If not then you can cheat a little and check for ray collisions in order. Since you have three objects that may or may not move per frame it pays to pre-calculate their distance from the camera (e.g. from their centre points). Test against each object in turn, by distance from the camera, from smallest to largest. Although the empty space is the most expensive part of the render now, this is more effective than just testing against all three and taking a minimum value. If your image is high res then this is especially efficient since you amortise the cost across the number of pixels.
Otherwise, test against all three and take a minimum value...
In other situations you may want to make a hybrid of the two methods. If you can test two of the objects in order then do so (e.g. a sphere and a cube moving down a cylindrical tunnel), but test the third and take a minimum value to find the final object.