Find the largest convex black area in an image

Question

I have an image of which this is a small cut-out:

As you can see it are white pixels on a black background. We can draw imaginary lines between these pixels (or better, points). With these lines we can enclose areas.

How can I find the largest convex black area in this image that doesn't contain a white pixel in it?

Here is a small hand-drawn example of what I mean by the largest convex black area:

P.S.: The image is not noise, it represents the primes below 10000000 ordered horizontally.

Answer 1

Trying to find maximum convex area is a difficult task to do. Wouldn't you just be fine with finding rectangles with maximum area? This problem is much easier and can be solved in O(n) - linear time in number of pixels. The algorithm follows.

Say you want to find largest rectangle of free (white) pixels (Sorry, I have images with different colors - white is equivalent to your black, grey is equivalent to your white).

You can do this very efficiently by two pass linear O(n) time algorithm (n being number of pixels):

1) in a first pass, go by columns, from bottom to top, and for each pixel, denote the number of consecutive pixels available up to this one:

repeat, until:

2) in a second pass, go by rows, read current_number. For each number k keep track of the sums of consecutive numbers that were >= k (i.e. potential rectangles of height k). Close the sums (potential rectangles) for k > current_number and look if the sum (~ rectangle area) is greater than the current maximum - if yes, update the maximum. At the end of each line, close all opened potential rectangles (for all k).

This way you will obtain all maximum rectangles. It is not the same as maximum convex area of course, but probably would give you some hints (some heuristics) on where to look for maximum convex areas.

Answer 2

I'll sketch a correct, poly-time algorithm. Undoubtedly there are data-structural improvements to be made, but I believe that a better understanding of this problem in particular will be required to search very large datasets (or, perhaps, an ad-hoc upper bound on the dimensions of the box containing the polygon).

The main loop consists of guessing the lowest point p in the largest convex polygon (breaking ties in favor of the leftmost point) and then computing the largest convex polygon that can be with p and points q such that (q.y > p.y) || (q.y == p.y && q.x > p.x).

The dynamic program relies on the same geometric facts as Graham's scan. Assume without loss of generality that p = (0, 0) and sort the points q in order of the counterclockwise angle they make with the x-axis (compare two points by considering the sign of their dot product). Let the points in sorted order be q₁, …, q_n. Let q₀ = p. For each 0 ≤ i < j ≤ n, we're going to compute the largest convex polygon on points q₀, a subset of q₁, …, q_{i - 1}, q_i, and q_j.

The base cases where i = 0 are easy, since the only “polygon” is the zero-area segment q₀q_j. Inductively, to compute the (i, j) entry, we're going to try, for all 0 ≤ k ≤ i, extending the (k, i) polygon with (i, j). When can we do this? In the first place, the triangle q₀q_iq_j must not contain other points. The other condition is that the angle q_kq_iq_j had better not be a right turn (once again, check the sign of the appropriate dot product).

At the end, return the largest polygon found. Why does this work? It's not hard to prove that convex polygons have the optimal substructure required by the dynamic program and that the program considers exactly those polygons satisfying Graham's characterization of convexity.

Answer 3

You could try treating the pixels as vertices and performing Delaunay triangulation of the pointset. Then you would need to find the largest set of connected triangles that does not create a concave shape and does not have any internal vertices.

Answer 4

If I understand your problem correctly, it's an instance of Connected Component Labeling. You can start for example at: http://en.wikipedia.org/wiki/Connected-component_labeling

Answer 5

I thought of an approach to solve this problem:

Out of the set of all points generate all possible 3-point-subsets. This is a set of all the triangles in your space. From this set remove all triangles that contain another point and you obtain the set of all empty triangles.

For each of the empty triangles you would then grow it to its maximum size. That is, for every point outside the rectangle you would insert it between the two closest points of the polygon and check if there are points within this new triangle. If not, you will remember that point and the area it adds. For every new point you want to add that one that maximizes the added area. When no more point can be added the maximum convex polygon has been constructed. Record the area for each polygon and remember the one with the largest area.

Crucial to the performance of this algorithm is your ability to determine a) whether a point lies within a triangle and b) whether the polygon remains convex after adding a certain point.

I think you can reduce b) to be a problem of a) and then you only need to find the most efficient method to determine whether a point is within a triangle. The reduction of the search space can be achieved as follows: Take a triangle and increase all edges to infinite length in both directions. This separates the area outside the triangle into 6 subregions. Good for us is that only 3 of those subregions can contain points that would adhere to the convexity constraint. Thus for each point that you test you need to determine if its in a convex-expanding subregion, which again is the question of whether it's in a certain triangle.

The whole polygon as it evolves and approaches the shape of a circle will have smaller and smaller regions that still allow convex expansion. A point once in a concave region will not become part of the convex-expanding region again so you can quickly reduce the number of points you'll have to consider for expansion. Additionally while testing points for expansion you can further cut down the list of possible points. If a point is tested false, then it is in the concave subregion of another point and thus all other points in the concave subregion of the tested points need not be considered as they're also in the concave subregion of the inner point. You should be able to cut down to a list of possible points very quickly.

Still you need to do this for every empty triangle of course.

Unfortunately I can't guarantee that by adding always the maximum new region your polygon becomes the maximum polygon possible.