Fast Alpha Hulls (alpha shapes in 3d; parfor enabled)

Compute the alpha hulls (exterior and interior) of a set of points.

See also http://www.dylan-muir.com/articles/alpha_hulls/
Usage: [triHull, vbOutside, vbInside] = AlphaHull(mfPoints, fAlphaRadius , triDelaunay)

This function computes the alpha shape / alpha hulls of a set of points; both the external hull as well as interior voids. The Matlab parfor construct is used by the function, so that this code will run quickly on a machine running several instances of the Matlab parallel computing toolbox.

This algorithm is based on qhull and the delaunay tetrahedralisation of the set of points. It will return a hull triangulation, and ignore points connected only by a line.

'mfPoints' is an Nx3 ma­trix, where each row de­fines a point in 3-space. AlphaHull will find the hull of the set of points in 'mfPoints'.

'fAlphaRadius' is a scalar dis­tance which de­fines the pa­ra­me­ter alpha of the alpha hull. This dis­tance is in­ter­preted as the ra­dius of a sphere that will "roll around" the sur­face, with the bound­ary of the sphere touch­ing one to three of the points in 'mfPoints'. The tri­an­gles of the De­lau­nay tetra­he­dral­i­sa­tion where the spere can fit with­out in­ter­sect­ing any other points will form part of the alpha hull.

The op­tional pa­ra­me­ter 'triDelaunay' can be used to pro­vide the De­lau­nary tetra­he­dral­i­sa­tion of the set of points, if it is known in ad­vance.

'triHull' will be a tri­an­gu­la­tion con­tain­ing tri­an­gles that fall ei­ther on the alpha hull sur­face, or on the in­side sur­face of an alpha void (a hole) within the point set. The boolean vec­tors 'vbOutside' and 'vbInside' de­fine which rows of 'triHull' de­fine "out­side" and "in­side" hulls. The sur­faces re­turned by AlphaHull will be con­vex to the space pa­ra­me­ter alpha.

'triHull' will be a Tx3 ma­trix, where each row ['p1' 'p2' 'p3'] de­fines a tri­an­gle on an alpha sur­face. The in­dices 'pn' refer to rows in 'mfPoints', and so de­fine tri­an­gles in­clud­ing three of the orig­i­nal points.

* Caveats and room for improvement *
The method for la­belling "in­side" and "out­side" tri­an­gles is not ideal. It works by de­cid­ing whether the nor­mal of a tri­an­gle, in the di­rec­tion away from the rest of the point cloud, points in the di­rec­tion of the point set cen­troid. A bet­ter tech­nique might be to it­er­ate along the sur­face, la­belling tri­an­gles con­sis­tently as you go. If you im­prove on this, I'd love to hear about it.