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for my research project (98 % convergence as threshold for harmonics)

EFD_till_convergence.m

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+(* ::Package:: *)
+
+harmonicAnalysis[a_,b_,c_,d_]:=Block[{p=a,q=b,r=c,s=d,coeff,\[Theta],rotationMat,starMat,\[Psi],L,W,\[Phi]},
+\[Theta] = 0.5ArcTan[(p^2 +r^2 -q^2-s^2),2(p q + r s)];
+coeff = {p,q,r,s};
+rotationMat = {{Cos[\[Theta]],-Sin[\[Theta]]},{Sin[\[Theta]],Cos[\[Theta]]}};
+starMat=Partition[coeff,2,2].rotationMat;
+\[Psi]=ArcTan[starMat[[1,1]],starMat[[2,1]]]*(180 /Pi);
+\[Phi]=ArcTan[starMat[[1,2]],starMat[[2,2]]]*(180 /Pi);
+L = Norm[{starMat[[1,1]],starMat[[2,1]]}];
+W= Norm[{starMat[[1,2]],starMat[[2,2]]}];
+{L,W,\[Psi],\[Phi]}
+ ]
+
+
+ClearAll@EllipticalFourier;
+Options[EllipticalFourier] = {normalize-> True,sizeInvariance-> True,order-> 10,ptsToplot-> 300,locus-> {0.,0.},areathreshold-> 2};
+EllipticalFourier[img_,OptionsPattern[]]:=Module[{dXY,dt,t,T,\[Phi],contourpositions, contour,contourtour,contourregion,xi,A0,\[Delta],C0,dCval,coeffL,normalizeEFD,ellipticalCoefficients,plotEFD,dcCoefficients,dcVal},
+contourpositions = PixelValuePositions[MorphologicalPerimeter[img],1];
+contourtour=Last@FindShortestTour[contourpositions];
+contour = Part[contourpositions,contourtour];
+contourregion=BoundaryMeshRegion[contourpositions,Line@contourtour];
+dXY = Differences[contour];
+dt=(x\[Function]N@Sqrt[x])[Total/@(dXY^2)];
+t = Prepend[Accumulate[dt],0];
+T= Last@t;
+\[Phi] = 2 \[Pi] t/T;
+
+normalizeEFD[coeff_]:= With[
+{\[Theta]=0.5 ArcTan[(coeff[[1,1]]^2 -coeff[[1,2]]^2 + coeff[[1,3]]^2 -coeff[[1,4]]^2),
+2 (coeff[[1,1]] coeff[[1,2]] +coeff[[1,3]] coeff[[1,4]])]},
+Block[{coeffList = coeff,\[Psi],\[Psi]r,a,c,\[Theta]r},
+\[Theta]r[i_Integer]:={{Cos[i \[Theta]],-Sin[i \[Theta]]},{Sin[i \[Theta]],Cos[i \[Theta]]}};
+coeffList=Partition[#,2,2]&/@coeffList;
+coeffList=MapIndexed[(#1.\[Theta]r[First@#2])&,coeffList];
+a=First@Flatten@First@coeffList;
+c=(Flatten@First@coeffList)[[3]];
+\[Psi] = ArcTan[a,c];
+\[Psi]r= {{Cos[\[Psi]],Sin[\[Psi]]},{-Sin[\[Psi]],Cos[\[Psi]]}};
+coeffList=MapIndexed[Flatten[\[Psi]r.#]&,coeffList];
+If[OptionValue[sizeInvariance],a= First@First@coeffList;coeffList/=a,coeffList]]
+];
+
+ellipticalCoefficients[order_]:= Module[{coeffs,func},
+func[list:{{__}..},n_Integer]:=Block[{listtemp =list, const = T/(2 (n^2)( \[Pi]^2)),phiN= \[Phi] *n,
+dCosPhiN, dSinPhiN,an,bn,cn,dn},
+dCosPhiN = Cos[Rest@phiN]-Cos[Most@phiN];
+dSinPhiN = Sin[Rest@phiN]-Sin[Most@phiN];
+an = const*Total@(dXY[[All,1]]/dt *dCosPhiN);
+bn = const*Total@(dXY[[All,1]]/dt*dSinPhiN);
+cn = const*Total@(dXY[[All,2]]/dt*dCosPhiN);
+dn = const*Total@(dXY[[All,2]]/dt*dSinPhiN);
+listtemp[[n]]= {an,bn,cn,dn};
+listtemp
+];
+coeffs = ConstantArray[0,{order,4}];
+coeffL= If[OptionValue@normalize,normalizeEFD[#],#]&@Fold[func,coeffs,Range@order]
+];
+
+dcCoefficients:=( 
+xi = Accumulate[Part[dXY,All,1]]- (dXY[[All,1]]/dt)*Rest@t;
+A0= (1/T)*Total@(dXY[[All,1]]/(2 dt) * Differences[t^2] + xi*dt);
+\[Delta] = Accumulate[Part[dXY,All,2]]- (dXY[[All,2]]/dt)*Rest@t;
+C0= (1/T)*Total@(dXY[[All,2]]/(2 dt) * Differences[t^2] + \[Delta]*dt);
+{First@First@contour + A0,Last@First@contour+C0}
+);
+
+plotEFD[coeff_]:= Last@Reap@Block[{n = Length@coeff,nhalf,nrow=2,ti,xt,yt,func,\[Theta],
+i=OptionValue[ptsToplot],j,regionpts,shortesttour,regiondiff,regiondiffmetric},
+nhalf = Ceiling[n/2];
+ti = Array[#&,i,{0.,1}];
+If[Head@OptionValue[locus]===String,dcVal=dcCoefficients,dcVal=OptionValue@locus];
+xt = ConstantArray[dcVal[[1]],i];
+yt = ConstantArray[dcVal[[2]],i];
+regiondiffmetric =100;
+j=1;
+While[regiondiffmetric>OptionValue[areathreshold],
+xt+= coeff[[j,1]] Cos[2 (j) \[Pi] ti] + coeff[[j,2]] Sin[2 (j) \[Pi] ti];
+yt += coeff[[j,3]]Cos[2(j) \[Pi] ti] + coeff[[j,4]] Sin[2 (j)\[Pi] ti];
+
+regionpts=Most@Thread@{xt,yt};
+shortesttour = Line@Last@FindShortestTour@regionpts;
+regiondiff=BoundaryMeshRegion[regionpts,shortesttour]~RegionDifference~contourregion;
+regiondiffmetric = Area[regiondiff]/Area[contourregion] * 100;
+Sow@{harmonicAnalysis[Sequence@@coeff[[j]]],Graphics[{{Thick,XYZColor[0,0,1,0.6],Line@Thread[{xt,yt}]},{Thick,Dashed,XYZColor[1,0,0,0.6],Line@contour},Inset[Style[#,{Black,FontSize->12}]&@Text["harmonic: " <> ToString@j]]}]};
+j++
+] ;
+];
+ellipticalCoefficients[OptionValue[order]];
+plotEFD[coeffL]
+]