I am looking for an efficient way to perform nearest neighbor searches within a specified radius in a two-dimensional plane. According to Wikipedia, space-partitioning data structures, such as :
k-d trees,
r-trees,
octrees,
quadtrees,
cover trees,
metric trees,
BBD trees
locality-sensitive hashing,
and bins,
are often used for organizing points in a multi-dimensional space and can provide O(log n) performance for search and insert operations. However, in my case, the points in the two-dimensional plane are moving at each iteration, so I need to update the tree accordingly. Rebuilding the tree from scratch at each iteration seems easier, but I would like to avoid it if possible because the points only move slightly between iterations.
I have read that k-d trees are not naturally balanced, which could be an issue in my case. R-trees, on the other hand, are better suited for storing rectangles. Bin algorithms, on the other hand, are easy to implement and provide near-linear search performance within local bins.
I am working on an autonomous agent simulation where 1,000,000 agents are rendered in the GPU, and the CPU is responsible for computing the next movement of each agent. Each agent is influenced by other agents within its line of sight, or in other words, other agents within a circular sector of angle θ and radius r. So here specific requirements for my use case:
Search space is a 2-d plane,
Each object is a point identified with the x,y coordinate.
All points are frequently updated by a small factor.
Cannot afford any O(n^2) algorithms.
Search within a radius (circular sector)
Search for all candidates within the search surface.
Given these considerations, what would be the best algorithms for my use case?
I think you could potentially solve this by doing a sort of scheduling approach. If you know that no object will move more than d distance in each iteration, and you want to know which objects are within X distance of each other on each iteration, then given the distances between all objects you know that on the next iteration the only potential pairs of objects that would change their neighbor status would be those with a distance between X-d and X+d. The iteration after that it would be X-2d and X+2d and so on.
So I'm thinking that you could do an initial distance calculation between all pairs of objects, and then based on each difference you can create an NxN matrix where the value in each cell is which iteration you will need to re-check their distance. Then when you re-check those during that iteration, you would update their values in this matrix for the next iteration that they need to be checked.
The only problem is whether calculating an initial NxN distance matrix is feasible.
Using the function convexityDefects takes the blob from a binary image and creates a vector with all the points surrounding my hand. This vector always begins at the Point at the highest Y location on my blob which is interfering with my ability to run K-Curvature on each finger to determine accurate finger position.
I have thought about reshuffling the vector so that the beginning of the vector starts away from a location away from the fingers on my hand. The problem is that this is difficult to implement because I don't have an effective way to pick a point on the hand to start the reordering of the points. Do you have any suggestions which could provide easy methods to fix this? My goal is to have none of the fingertips within ~30 points of the beginning of this array.
I'm looking for interpolating some contour lines to generating a 3D view. The contours are not stored in a picture, coordinates of each point of the contour are simply stored in a std::vector.
for convex contours :
, it seems (I didn't check by myself) that the height can be easily calculates (linear interpolation) by using the distance between the two closest points of the two closest contours.
my contours are not necessarily convex :
, so it's more tricky... actualy I don't have any idea what kind of algorithm I can use.
UPDATE : 26 Nov. 2013
I finished to write a Discrete Laplace example :
you can get the code here
What you have is basically the classical Dirichlet problem:
Given the values of a function on the boundary of a region of space, assign values to the function in the interior of the region so that it satisfies a specific equation (such as Laplace's equation, which essentially requires the function to have no arbitrary "bumps") everywhere in the interior.
There are many ways to calculate approximate solutions to the Dirichlet problem. A simple approach, which should be well suited to your problem, is to start by discretizing the system; that is, you take a finite grid of height values, assign fixed values to those points that lie on a contour line, and then solve a discretized version of Laplace's equation for the remaining points.
Now, what Laplace's equation actually specifies, in plain terms, is that every point should have a value equal to the average of its neighbors. In the mathematical formulation of the equation, we require this to hold true in the limit as the radius of the neighborhood tends towards zero, but since we're actually working on a finite lattice, we just need to pick a suitable fixed neighborhood. A few reasonable choices of neighborhoods include:
the four orthogonally adjacent points surrounding the center point (a.k.a. the von Neumann neighborhood),
the eight orthogonally and diagonally adjacent grid points (a.k.a. the Moore neigborhood), or
the eight orthogonally and diagonally adjacent grid points, weighted so that the orthogonally adjacent points are counted twice (essentially the sum or average of the above two choices).
(Out of the choices above, the last one generally produces the nicest results, since it most closely approximates a Gaussian kernel, but the first two are often almost as good, and may be faster to calculate.)
Once you've picked a neighborhood and defined the fixed boundary points, it's time to compute the solution. For this, you basically have two choices:
Define a system of linear equations, one per each (unconstrained) grid point, stating that the value at each point is the average of its neighbors, and solve it. This is generally the most efficient approach, if you have access to a good sparse linear system solver, but writing one from scratch may be challenging.
Use an iterative method, where you first assign an arbitrary initial guess to each unconstrained grid point (e.g. using linear interpolation, as you suggest) and then loop over the grid, replacing the value at each point with the average of its neighbors. Then keep repeating this until the values stop changing (much).
You can generate the Constrained Delaunay Triangulation of the vertices and line segments describing the contours, then use the height defined at each vertex as a Z coordinate.
The resulting triangulation can then be rendered like any other triangle soup.
Despite the name, you can use TetGen to generate the triangulations, though it takes a bit of work to set up.
I need to find for each point of the data set all its nearest neighbors. The data set contains approx. 10 million 2D points. The data are close to the grid, but do not form a precise grid...
This option excludes (in my opinion) the use of KD Trees, where the basic assumption is no points have same x coordinate and y coordinate.
I need a fast algorithm O(n) or better (but not too difficult for implementation :-)) ) to solve this problem ... Due to the fact that boost is not standardized, I do not want to use it ...
Thanks for your answers or code samples...
I would do the following:
Create a larger grid on top of the points.
Go through the points linearly, and for each one of them, figure out which large "cell" it belongs to (and add the points to a list associated with that cell).
(This can be done in constant time for each point, just do an integer division of the coordinates of the points.)
Now go through the points linearly again. To find the 10 nearest neighbors you only need to look at the points in the adjacent, larger, cells.
Since your points are fairly evenly scattered, you can do this in time proportional to the number of points in each (large) cell.
Here is an (ugly) pic describing the situation:
The cells must be large enough for (the center) and the adjacent cells to contain the closest 10 points, but small enough to speed up the computation. You could see it as a "hash-function" where you'll find the closest points in the same bucket.
(Note that strictly speaking it's not O(n) but by tweaking the size of the larger cells, you should get close enough. :-)
I have used a library called ANN (Approximate Nearest Neighbour) with great success. It does use a Kd-tree approach, although there was more than one algorithm to try. I used it for point location on a triangulated surface. You might have some luck with it. It is minimal and was easy to include in my library just by dropping in its source.
Good luck with this interesting task!
I'm looking for a data structure that would allow me to store an M-by-N 2D matrix of values contiguously in memory, such that the distance in memory between any two points approximates the Euclidean distance between those points in the matrix. That is, in a typical row-major representation as a one-dimensional array of M * N elements, the memory distance differs between adjacent cells in the same row (1) and adjacent cells in neighbouring rows (N).
I'd like a data structure that reduces or removes this difference. Really, the name of such a structure is sufficient—I can implement it myself. If answers happen to refer to libraries for this sort of thing, that's also acceptable, but they should be usable with C++.
I have an application that needs to perform fast image convolutions without hardware acceleration, and though I'm aware of the usual optimisation techniques for this sort of thing, I feel a specialised data structure or data ordering could improve performance.
Given the requirement that you want to store the values contiguously in memory, I'd strongly suggest you research space-filling curves, especially Hilbert curves.
To give a bit of context, such curves are sometimes used in database indexes to improve the locality of multidimensional range queries (e.g., "find all items with x/y coordinates in this rectangle"), thereby aiming to reduce the number of distinct pages accessed. A bit similar to the R-trees that have been suggested here already.
Either way, it looks that you're bound to an M*N array of values in memory, so the whole question is about how to arrange the values in that array, I figure. (Unless I misunderstood the question.)
So in fact, such orderings would probably still only change the characteristics of distance distribution.. average distance for any two randomly chosen points from the matrix should not change, so I have to agree with Oli there. Potential benefit depends largely on your specific use case, I suppose.
I would guess "no"! And if the answer happens to be "yes", then it's almost certainly so irregular that it'll be way slower for a convolution-type operation.
EDIT
To qualify my guess, take an example. Let's say we store a[0][0] first. We want a[k][0] and a[0][k] to be similar distances, and proportional to k, so we might choose to interleave the storage of first row and first column (i.e. a[0][0], a[1][0], a[0][1], a[2][0], a[0][2], etc.) But how do we now do the same for e.g. a[1][0]? All the locations near it in memory are now taken up by stuff that's near a[0][0].
Whilst there are other possibilities than my example, I'd wager that you always end up with this kind of problem.
EDIT
If your data is sparse, then there may be scope to do something clever (re Cubbi's suggestion of R-trees). However, it'll still require irregular access and pointer chasing, so will be significantly slower than straightforward convolution for any given number of points.
You might look at space-filling curves, in particular the Z-order curve, which (mostly) preserves spatial locality. It might be computationally expensive to look up indices, however.
If you are using this to try and improve cache performance, you might try a technique called "bricking", which is a little bit like one or two levels of the space filling curve. Essentially, you subdivide your matrix into nxn tiles, (where nxn fits neatly in your L1 cache). You can also store another level of tiles to fit into a higher level cache. The advantage this has over a space-filling curve is that indices can be fairly quick to compute. One reference is included in the paper here: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.8959
This sounds like something that could be helped by an R-tree. or one of its variants. There is nothing like that in the C++ Standard Library, but looks like there is an R-tree in the boost candidate library Boost.Geometry (not a part of boost yet). I'd take a look at that before writing my own.
It is not possible to "linearize" a 2D structure into an 1D structure and keep the relation of proximity unchanged in both directions. This is one of the fundamental topological properties of the world.
Having that that, it is true that the standard row-wise or column-wise storage order normally used for 2D array representation is not the best one when you need to preserve the proximity (as much as possible). You can get better result by using various discrete approximations of fractal curves (space-filling curves).
Z-order curve is a popular one for this application: http://en.wikipedia.org/wiki/Z-order_(curve)
Keep in mind though that regardless of which approach you use, there will always be elements that violate your distance requirement.
You could think of your 2D matrix as a big spiral, starting at the center and progressing to the outside. Unwind the spiral, and store the data in that order, and distance between addresses at least vaguely approximates Euclidean distance between the points they represent. While it won't be very exact, I'm pretty sure you can't do a whole lot better either. At the same time, I think even at very best, it's going to be of minimal help to your convolution code.
The answer is no. Think about it - memory is 1D. Your matrix is 2D. You want to squash that extra dimension in - with no loss? It's not going to happen.
What's more important is that once you get a certain distance away, it takes the same time to load into cache. If you have a cache miss, it doesn't matter if it's 100 away or 100000. Fundamentally, you cannot get more contiguous/better performance than a simple array, unless you want to get an LRU for your array.
I think you're forgetting that distance in computer memory is not accessed by a computer cpu operating on foot :) so the distance is pretty much irrelevant.
It's random access memory, so really you have to figure out what operations you need to do, and optimize the accesses for that.
You need to reconvert the addresses from memory space to the original array space to accomplish this. Also, you've stressed distance only, which may still cause you some problems (no direction)
If I have an array of R x C, and two cells at locations [r,c] and [c,r], the distance from some arbitrary point, say [0,0] is identical. And there's no way you're going to make one memory address hold two things, unless you've got one of those fancy new qubit machines.
However, you can take into account that in a row major array of R x C that each row is C * sizeof(yourdata) bytes long. Conversely, you can say that the original coordinates of any memory address within the bounds of the array are
r = (address / C)
c = (address % C)
so
r1 = (address1 / C)
r2 = (address2 / C)
c1 = (address1 % C)
c2 = (address2 % C)
dx = r1 - r2
dy = c1 - c2
dist = sqrt(dx^2 + dy^2)
(this is assuming you're using zero based arrays)
(crush all this together to make it run more optimally)
For a lot more ideas here, go look for any 2D image manipulation code that uses a calculated value called 'stride', which is basically an indicator that they're jumping back and forth between memory addresses and array addresses
This is not exactly related to closeness but might help. It certainly helps for minimation of disk accesses.
one way to get better "closness" is to tile the image. If your convolution kernel is less than the size of a tile you typical touch at most 4 tiles at worst. You can recursively tile in bigger sections so that localization improves. A Stokes-like (At least I thinks its Stokes) argument (or some calculus of variations ) can show that for rectangles the best (meaning for examination of arbitrary sub rectangles) shape is a smaller rectangle of the same aspect ratio.
Quick intuition - think about a square - if you tile the larger square with smaller squares the fact that a square encloses maximal area for a given perimeter means that square tiles have minimal boarder length. when you transform the large square I think you can show you should the transform the tile the same way. (might also be able to do a simple multivariate differentiation)
The classic example is zooming in on spy satellite data images and convolving it for enhancement. The extra computation to tile is really worth it if you keep the data around and you go back to it.
Its also really worth it for the different compression schemes such as cosine transforms. (That's why when you download an image it frequently comes up as it does in smaller and smaller squares until the final resolution is reached.
There are a lot of books on this area and they are helpful.