Splines extraction on Kopf-Lischinski algorithm, part 2


This is a sequel of a series of posts. The last one was “part 0” and this order is a bit insane, but nothing to worry about. Check the first one to understand what all this is about. I hope this will be the final post in this series.

Once upon a time… a polygon with holes appeared

The polygon-union described in the previous posts works correctly and there are no better answers than the correct answer, but these polygons were meant to be B-Splines and when I described them, I already knew that the mentioned algorithm would be insufficient.

Consider the following image.

Normal polygon (left) and bordered polygon (right)

Normal polygon (left) and bordered polygon (right)

The left polygon is the polygon that you see before and/or after the polygon-union. They are visually indistinguishable. The right polygon has an outline to allow you understand its internal representation. You can understand how the polygon-union generates a polygon with a “hole” seeing the right polygon.

If the polygon-union don’t generate visually identifiable pixels, then there shouldn’t exist any problem, but when we convert the polygon to a B-Spline, some areas of the image won’t be filled. The polygons won’t fit correctly, like show in the image below.

After B-Spline conversion

After B-Spline conversion

The solution is to create some holes. With the representation of polygons used in libdepixelize, this task is very simple and key operation to accomplish the goal is to encounter the “edges that the polygon share with itself”. I’ll explain this task further in the following paragraphs.

A holed polygon with labelled points represented with a single path

The above image has 2 pairs of hidden points, pair <5, 6>, that is equal to pair <11, 10>, and pair <12, 13>, that is equal to pair <18, 17>. Well, to extract internal polygons that represent the holes, you just iterate over the points and, for each point, try to find another point that is equal to it, then get the internal sequence and use it to construct the internal polygon.

  • Remark #1: libdepixelize will do these steps in backward order, to avoid too much moves of elements in the internal vector.
  • Remark #2: The polygon is represented in clockwise order, but the holes will be represented in counter-clockwise order, but there is no reason to normalize. You can safely ignore this (de)normalization feature.


I forgot to mention that edges not always are two points only and you should use the same “largest common edge” from the algorithm of previous posts.


Things became more complicated than I predicted and now involve some recursive functions and I needed to extend (extend != change) the algorithm. To see the whole extension, check out the Splines::_fill_holes function in libdepixelize source code and where this function is used.


Things became even more complicated than the complicated things that I didn’t anticipate. Complicated code to fix the position of T-junction nodes created a pattern where the polygon itself shared a point with itself and this pattern propagated wrongly in the “holes-detection” algorithm and afterwards.

The algorithm “3rd ed.” check if the common edge has “lenth” 1 to solve the problem.

And then, the polygon were meant to be a B-Spline

It’s a series about splines extraction and it’s fair to end with this step.

The representation of the data in three different moments

It’s evolution, baby

The algorithm is very simple. The points you get through the polygon-union algorithm in the previous posts will be the control points of quadratic Bézier curves and the interpolating/ending points of each Bézier curve will be the middle points between two adjacent control points. This is the special case where all points are smooth. The image below can help you get the idea.

Locations of the B-Spline points

Locations of the B-Spline points

If you have non-smooth points, then all you need to do is interpret them as interpolating points, instead of control points. You’ll have three interpolating points that are part of two straight lines. There is a bit more of theory to understand why the generated B-Spline is smooth, but we’re ALMOST done! In next post (a small one, I hope), I’ll detail how to get correctly positioned points for T-junctions.


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