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Find the curvature of an object from an image

Posted 14 days ago
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curvatureMeasure.m is a Mathematica script for calculating curvature along the boundary of an image object. It might be useful to people working in the computer vision community. This simple script can be easily extended to even track the curvature of an object as it deforms (will add soon) across several images.

Here is our input image:

enter image description here

boundary is discretized into equidistant points

enter image description here

circle fits to a given point point Pi, Pi + N-left, Pi + N-right, where N is the neighbour to the point Pi.

enter image description here

curvatures (found as 1/radius) is the colormap:

enter image description here

code

(* ::Package:: *)

BeginPackage["curvatureMeasure`"];


curvatureMeasure::usage = "measures the curvature along the image object";


Begin["`Private`"];


shiftPairs[perimeter_,shift_]:=Module[{newls},
newls=perimeter[[-shift;;]]~Join~perimeter~Join~perimeter[[;;shift]];
Table[{newls[[i-shift]],newls[[i]],newls[[i+shift]]},{i,1+shift,Length[newls]-shift}]
];

(* we can either use the suppressed code below to fit circles or the Built-In Circumsphere to find the fits 
(* from Mathematica StackExchange: courtesy ubpdqn *)
circfit[pts_]:=Module[{reg,lm,bf,exp,center,rad},
reg={2 #1,2 #2,#2^2+#1^2}&@@@pts;
lm=LinearModelFit[reg,{1,x,y},{x,y}];
bf=lm["BestFitParameters"];
exp=(x-#2)^2+(y-#3)^2-#1-#2^2-#3^2&@@bf;
{center,rad}={{#2,#3},Sqrt[#2^2+#3^2+#1]}&@@bf;
circlefit[{"expression"->exp,"center"->center,"radius"->rad}]
];
circlefit[list_][field_]:=field/.list;
circlefit[list_]["Properties"]:=list/.Rule[field_,_]:>field;
circlefit/:ReplaceAll[fields_,circlefit[list_]]:=fields/.list;
Format[circlefit[list_],StandardForm]:=HoldForm[circlefit]["<"<>ToString@Length@list<>">"]
*)

curvatureMeasure[img_Image,div_Integer,shift_Integer]:=Module[{\[ScriptCapitalR],polygon,t,interp,sub,sampledPts,
pairedPts,circles,\[Kappa],midpts,regMem,col,g,fn},
\[ScriptCapitalR] = ImageMesh[img, Method -> "Exact"];
polygon = Append[#,#[[1]]]&@MeshCoordinates[\[ScriptCapitalR]][[ MeshCells[\[ScriptCapitalR],2][[1,1]] ]];
t = Prepend[Accumulate[Norm/@Differences[polygon]],0.];
interp = Interpolation[Transpose[{t,polygon}],InterpolationOrder -> 1,
PeriodicInterpolation->True];
sub = Subdivide[interp[[1,1,1]],interp[[1,1,2]],div];
sampledPts = interp[sub];
Print[Show[\[ScriptCapitalR],Graphics@Point@sampledPts,ImageSize-> 250]];
pairedPts = shiftPairs[sampledPts, shift];

circles = (Circumsphere/@pairedPts)/. Sphere -> Circle;
(*circles = (fn=circfit[#]; Circle[fn["center"],fn["radius"]])&/@pairedPts;*)

Print[Graphics[{{Red,Point@sampledPts},{XYZColor[0,0,0,0.1],circles}}]];
\[Kappa] = 1/Cases[circles,x_Circle:> Last@x];
midpts = Midpoint/@pairedPts[[All,{1,-1}]];
regMem = RegionMember[\[ScriptCapitalR],midpts]/.{True-> 1,False-> -1};
\[Kappa] *= regMem;
col = ColorData["Rainbow"]/@Rescale[\[Kappa], MinMax[\[Kappa]],{0,1}];
g = Graphics[{PointSize[0.018],MapThread[Point[#1,VertexColors->#2]&,{sampledPts,col}]}];
Print[Show[HighlightMesh[\[ScriptCapitalR],{Style[1, Black],Style[2,White]}],g,ImageSize->Medium]];
\[Kappa]
]


End[];
EndPackage[];
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Interesting! An alternative way to do this is using convolution with Gaussian functions to take the derivatives and second derivatives around the curve and use the formula for signed curvature:

k = (xu*yuu - xuu*yu)/((xu^2 + yu^2)^(3/2))

Where xu is the derivative of the x coordinates, xuu is the second derivative of the x coordinates, and yu and yuu are the same for the y coordinates. Computing the derivatives like this is the basis of the Curvature Scale Space image introduced in the mid-1980s, which is a method for characterizing the curvature of a shape at different scales (where the scale corresponds to the width, sigma, of the Gaussian).

We can define the Gaussian and its derivatives as:

gaussian[x_, \[Sigma]_] := 1/(\[Sigma]*Sqrt[2*\[Pi]]) * Exp[-(x^2)/(2*\[Sigma]^2)]
dgaussian[x_, \[Sigma]_] := D[gaussian[y, \[Sigma]], y] /. y -> x
ddgaussian[x_, \[Sigma]_] := D[dgaussian[y, \[Sigma]], y] /. y -> x

These can be used to generate the convolution kernel for a sequence of points pts and a given sigma value:

makeKernel[f_, pts_, \[Sigma]_] := Table[f[x, \[Sigma]], {x, -Length[pts]/2, Length[pts]/2}]

The curvature then can be defined as:

k[pts_, \[Sigma]_] := Module[
   {xu, xuu, yu, yuu},
   xu = ListConvolve[makeKernel[dgaussian, pts, \[Sigma]], pts[[;; , 1]], 1];
   xuu = ListConvolve[makeKernel[ddgaussian, pts, \[Sigma]], pts[[;; , 1]], 1];
   yu = ListConvolve[makeKernel[dgaussian, pts, \[Sigma]], pts[[;; , 2]], 1];
   yuu = ListConvolve[makeKernel[ddgaussian, pts, \[Sigma]], pts[[;; , 2]], 1];
   (xu*yuu - xuu*yu)/(((xu^2) + (yu^2))^(3/2))];

Using your image and the points derived from the boundary of the shape with a sigma value of 20, we get this:

\[Kappa] = k[pts, 20];

col = ColorData["Rainbow"] /@ Rescale[\[Kappa], MinMax[\[Kappa]], {0, 1}];

g = Graphics[{PointSize[0.018], MapThread[Point[#1, VertexColors -> #2] &, {pts, col}]}]

The result doesn't look exactly the same as yours, but is roughly similar.

Image with curvature coloring

An alternative method to this that I've used in the past is to use a spectral method to compute the derivatives. Unfortunately I only have that code in Matlab and haven't ported it to Mathematica.

Thanks. This is an awesome method. Do you mind if I add your code to my GitHub repository with your name?

Sure. It’s worth noting that the curvature given by that method is the curvature of the shape smoothed by the Gaussian of a given width. It’s sometimes easier to interpret in terms of the smoothed curve instead of the original points.

Some comments:

  • A spectral method to compute derivatives is built-in
  • The Gaussian is built-in as the PDF of the NormalDistribution
  • GaussianMatrix gives a matrix that corresponds to a Gaussian kernel of radius $r$
  • Since the package uses Interpolation, after smoothing, one could use this to compute derivatives and curvature directly
  • The work of Bart ter Haar Romeny’s group is relevant to this topic
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