concave | On-the-fly image resizing with AWS Lambda | Computer Vision library

There is a successful solution by switching to the APOPT solver m.options.SOLVER = 1. In this case the default solver, IPOPT, fails to find a solution but APOPT succeeds.

With Bilal's response I was able to come up with a suitable solution.

Basically because if f is concave, then -f is convex. So say we have:

Use names to access the names of the coefficients.

@fdireito Answer and response as follows:

@WesleyAllenWilliams That is surprising. Sorry for insisting, but are you sure you do not have some typo? Could you create a separate file with the following code line by line import math , import numpy as np , d = np.arange(.5,1000.5,.5), a_3 = (2/math.sqrt(6))*d, print(d), print(a_3) (copy paste from here, line by line). This works for me. Try that and see if you get d and a_3 with the values you want. – fdireito 13 hours ago

@WesleyAllenWilliams That should give d: [5.000e-01 1.000e+00 1.500e+00 ... 9.990e+02 9.995e+02 1.000e+03] and a_3 [4.08248290e-01 8.16496581e-01 1.22474487e+00 ... 8.15680084e+02 8.16088333e+02 8.16496581e+02] – fdireito 13 hours ago

@fdireito That worked really well! The issue was that Numpy doesn't like the way certain equations are structured with their parantheses and wanted simplification. Next up, I had to run through formula typos and I ultimately used a scatter plot for my million data points since it would NOT plot even with matching array lengths. Then I had to simplify the command for the legend since it was overworking my CPU...THEN I had to truncate the data set to 10,000 point as well as plot the domain as a logarithmic function to visualize the change. My hypothesis held true dude! Thanks!!! – Wesley Allen Williams 6 mins ago Delete

You can connect the hole (or holes) to the exterior perimeter so that you get a single "degenerate polygon" defined by a single point sequence that connects all points without self-intersection.

You go in and out through the same segment. If you follow the outer perimeter clockwise, you need to follow the hole perimeter counterclockwise or vice-versa. Otherwise, it would self-intersect.

You can use the fact (see the scipy.spatial.Delaunay documentation) that "for 2-D, the triangle points are oriented counterclockwise". Therefore, the outer edge points constructed in the Alpha shape will always be oriented counterclockwise i.e., the inner side of the shape will be to their left (if this was not certified in the documentation we could have checked it ourselves and flipped if necessary, but here there is no need).

This means that the points along the polygon between the leftmost point and the rightmost point are the lower hull, and the points between the rightmost and the leftmost are the upper hull. The code below implements this idea.

As you note, np.log(gamma(alpha.sum())) and -np.log(gamma(alpha)).sum() have different curvature, so you need to combine them as

This could be achieved like so:

1. Set the breaks for the axis to be unique x and y values from you line dfs
2. To get the you have to set axis.ticks.x/y = element_line(). Using axis.ticks will not do the job as in theme_ipsum axis.ticks.x/y are both set to element_blank()