How to find likely path of connection between dots

I have a set of data (2x2000) consisting of X & Y coordinates for a series of points. These datapoints are generated as all the possible solutions to a function.

As you can see in the plot (left side) the datapoints can be grouped together into 'chains' of smooth continuous lines (I highlighted some examples in red on the Right side) and I would like to find an automated process for determining the lines which can be fitted to these datapoints.

Any help would be appreciated. Many thanks!

[Bjorn Gustavsson]

QD suggestion: Start by chaining up a random point with its two nearest neighbors, with some sensibly selected threshold on how far between they can be before giving up on a chain (and perhaps on direction - how far from anti-parallel the two connections can be). If you plot those it would "surely" create a number of rather long chains. If you put neighboring points connected like that into cell-array element for each chain maybe you can chain up some 70 - 80 % of the points and the easy part (10-25%?) of your problem is solved.

[Jeffrey Clark]

@jon gorecki, be aware that reading too much into patterns can lead you away from the correct solution - keep an open mind and try looking at the data in the order given rather than the x or y values. As in this example the blue circles seem to have a nice pattern with one such line drawn in red. However, the actual points connected in sequence as a line whould show another pattern in green, which may be closer to the truth but itself could just be an undersampling of the actual function which in my case was a sine function.

[Bjorn Gustavsson]

Another suggestion is to have a look at chebfun - it might be a better approach to this type of "half-analytical-half-numeric" type of problems.

[jon gorecki]

Thank you Bjorn for your suggestion it was very helpful

[Bjorn Gustavsson]

@jon gorecki, good! If you have some half-final results or a working function at least I would be interested in looking at your results or downloading your function from the File Exchange, if you can share. This was a nice problem in the sense that it is "obvious with naked eye" but tricky to get to work, and I think it is not a very uncommon problem either.

[jon gorecki]

unfortunately I have not come to a good end solution but I was able to chip away a few layers of the problem. In my case I am only concerned with a 'chain' of solutions that can stretch fully from the left (x=300) to the right (x=400) in a smooth continuous path.

step 1: remove any dot with a Y value of 1.1 or 2 (this removes the extreme values as I can say they are going outside of the physical domain of the problem)

Step 2: go through the points one by one and measure if they have a neighbouring point on either side (horizontally) within a Y value tolerance. If not I remove this point. I keep cycling through this loop until the solution stops changing. In this way I am able to remove many of the unwanted datapoints.

Step 3 was to do a 2D convolution with a bowtie shape mask (example below) where the rationale was that this should preferentially highlight pixels with a smooth continuous chain.

step 4: take the convoluted image and pick the point of maximum intensity at each X value. Finished

Overall it works OK, it gets quite close to the true answer but im not happy with at as its not a very precice method. I would much prefer a method to isolate individual 'chains' of datapoints as you may do by eye.

I have also tried a method to of gradient minimisation whereby I start at a datapoint and then look at the points on the right and left and find which combination of 3 points results in the minimum gradients. This method did not work very well, but I think if I increased it to 5 points (+/- 2 on each side) then it may be much improved.

Finally, I looked at functions of graydist and bwdist. I feel they could be very useful in this problem but its not immediately obvious how I would utilise them to solve this problem.