Skip to main content

performance tuning - Radial distribution function


I have tried to rewrite my old IDL code to calculate the radial correlation function for a regular 2D crystal structure.


The theory behind this function is given here: https://en.wikipedia.org/wiki/Radial_distribution_function


Update 1:



The radial density distribution counts the number of points in a distance between $r$ and $r +\Delta r$ from each considered central point (below one marked as red). The area of such a "shell" is $2\pi r \Delta r$. The density distributions are averaged for all center points and then normalized by the total point density times the ring area for each radius.




enter image description here


Update 2:



It is important that the maximum radius (the maximum shell) for each center point does not cross the edges of the available point coordiantes (corresponding to the range defined by the smallest and largest x and y point value). That means that a point close to the edges has a maller maximum radius (smallest distance to the next edge) than a center point (for a square: maximum radius = half of the diagonal).



My question: is it possible to improve my "unreadable" and slow code, which does not use any special mathematica functions?.


Also I must have made a normalization error, since the function does not converge at g(r)=1 (see plot below).


As input I have taken a crystal image recorded with a high resolution camera:


enter image description here


Dr. belisarius has detected all coordinates by the following one-liner:



pts = ComponentMeasurements[Binarize@ImageSubtract
[image, BilateralFilter[image, 4, 1]], "Centroid"];

These points pts I have used to determine the radial correlation function.


The resulting plot is:


enter image description here


The full code is given here:


Clear[radialDensityDistribution];
radialDensityDistribution [listData_, mrr_: 0, mrc_: dDiag,
subdivision_: 50] :=

(
(*listData: list of 2D data points*)
(*mrr: distance from edge*)
(*mrc: calculation radius from central point*)
(*subdivision: number of in size's from mean point distance*)

n = Length[listData];

x = listData[[All, 1]];
y = listData[[All, 2]];


minCorner = {Min[x], Min[y]};
maxCorner = {Max[x], Max[y]};

diag = maxCorner - minCorner;
dDiag = Sqrt[diag.diag];

area = diag[[1]]*diag[[2]];

pointDensity = n/area;


deltaR = (area/n)^(1.0/2);
dr = deltaR/subdivision;

maxShell = Floor[mrc/dr];

g = Array[0 &, maxShell];
centralPoint = Array[0 &, n];

com = {Mean[x], Mean[y]};

radii = Sqrt[(x - com[[1]])^2 + (y - com[[2]])^2];
maxrad = Max[radii] // N;

centralIndex = Flatten@Position[radii + mrr, n_ /; n <= maxrad];
nCentral = Length[centralIndex];

p = {x[[centralIndex]], y[[centralIndex]]};

g = 0;


Table[
dist = {p[[1, 2 ;; All]] - p[[1, i]],
p[[2, 2 ;; All]] - p[[2, i]]}[[All, i ;; All]];
shell =
Floor[Sqrt[
dist[[1, All]]*dist[[1, All]] + dist[[2, All]]*dist[[2, All]]]/
dr];
h = HistogramList[shell, {0, maxShell - 1, 1}][[2, All]];
g = h + g,
{i, 1, nCentral - 1}

];

Table[
areaShell = Pi*(((shell + 1.0)*dr)^2 - (shell*dr)^2);
g[[shell]] = g[[shell]]/(1.0*nCentral*areaShell*pointDensity),
{shell, 1, maxShell - 1}
];

rn = (Range[maxShell - 1] - 0.5)*dr;
{g, rn, deltaR}

)

image = Import["http://i.stack.imgur.com/czhuI.png"];

pts = ComponentMeasurements[
Binarize@ImageSubtract[image, BilateralFilter[image, 4, 1]],
"Centroid"][[All, 2]];

extx = Max[pts[[All, 1]]] - Min[pts[[All, 1]]];
exty = Max[pts[[All, 2]]] - Min[pts[[All, 2]]];

ext = Min[extx, exty];

{g, rn, deltaR} = radialDensityDistribution [pts, ext/4, ext/4, 20];

ListLinePlot[Transpose[{rn, g}], PlotRange -> Full, Frame -> True,
FrameLabel -> {{"g(r)", ""}, {"r (pixels)", ""}}, ImageSize -> Large]

Answer



I will show a simple and fast approach to computing the pair correlation function (radial distribution function) for a 2D system of point particles.:


radialDistributionFunction2D[pts_?MatrixQ, boxLength_Real, nBins_: 350] :=
Module[{gr, r, binWidth = boxLength/(2 nBins), npts = Length@pts, rho},

rho = npts/boxLength^2; (* area number density *)
{r, gr} = HistogramList[(*compute and bin the distances between points of interest*)
Flatten @ DistanceMatrix @ pts, {0.005, boxLength/4., binWidth}];
r = MovingMedian[r, 2]; (* take center of each bin as r *)
gr = gr/(2 Pi r rho binWidth npts); (* normaliza g(r) *)
Transpose[{r, gr}] (* combine r and g(r) *)
]

Here is how you use it:


rdf = radialDistributionFunction2D[pts, 1023.];

ListLinePlot[rdf, PlotRange ->{{0, 150}, All}, Mesh -> 80]

Mathematica graphics


Notice that you get the correct normalization for free. This took about 1.2 seconds on my machine. I have restricted the plot range to show the interesting features.


Comments

Popular posts from this blog

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...