Using 2D Fourier transform of an image to detect typical wavelengths

I am new to Mathematica, and I have read this post to understand how to perform Fourier transform on an image. My mission is to extract information on the typical distance between the black patches in the image I have attached here. The code that I attach here gives me the Fourier transform, but I don't know how to take out from the Fourier transform the values of the wavenumbers.

I have used this piece of code

img = Import["example.jpg"];
Image[img, ImageSize -> 300]

data = ImageData[img];(*get data*)
{nRow, nCol, nChannel} = Dimensions[data];
d = data[[All, All, 2]];
d = d*(-1)^Table[i + j, {i, nRow}, {j, nCol}];
fw = Fourier[d, FourierParameters -> {1, 1}];

(*adjust for better viewing as needed*)
fudgeFactor = 100;
abs = fudgeFactor*Log[1 + Abs@fw];
Labeled[Image[abs/Max[abs], ImageSize -> 300],Style["Magnitude spectrum", 18]]


I have the following image on which I would like to perform this analysis -

• fw contains the values. I.e. fw[[1,1]] is the value for frequency 0/0, fw[[1,2]] is the value for frequency 0/1... – Niki Estner Feb 11 '15 at 14:05
• What nikie says. You'd probably want to leave out the line d = d*(-1)^Table[i + j, {i, nRow}, {j, nCol}]; which seems to apply some checkerboard pattern to the white background pixels. – Sjoerd C. de Vries Feb 11 '15 at 14:58
• @SjoerdC.deVries is right, I didn't notice that. IIRC, this is done to "rotate" the 0 frequencies to the center of the FFT output. If you want to interpret the values numerically (instead of displaying the FT as a picture), you'd want to remove that line. – Niki Estner Feb 11 '15 at 15:05

This is not an answer more of an extended comment...

My mission is to extract information on the typical distance between the black patches in the image I have attached here.

Do we have to use Fourier transform for this?

For example we can get the required estimate with these commands:

img = Import["http://i.stack.imgur.com/hALsH.jpg"];
Row[{"Image dimensions:", ImageDimensions[img]}]
imgB = ColorNegate@Binarize[img, 0.4];
Block[{img = imgB},
Show[img,
Graphics[{Red, Thick,
Circle[#[[1]], #[[2]]] & /@
ComponentMeasurements[
]


comps = ComponentMeasurements[

dists = Flatten@
Map[Take[Sort[#], UpTo[5]] &,
Outer[EuclideanDistance[#1[[1]], #2[[1]]] - (#1[[2]] + #2[[2]]) &,
comps, comps, 1]];

qs = Range[0, 1, 0.25];
TableForm[{qs, Quantile[dists, qs]}]

Histogram[dists]


It looks like random blobs, and that's what the FFT suggests...

img = Import["http://i.stack.imgur.com/hALsH.jpg"];
imgBW = ImageData@ColorConvert[img, "Grayscale"];
imgZ = imgBW - Mean@Mean[imgBW];
xf = Abs[Fourier[imgZ, FourierParameters -> {1, -1}]];
{d1, d2} = Ceiling[Dimensions[xf]/2];
xCentered = RotateLeft[xf, {d1, d2}];
ArrayPlot[xCentered]


Here we zoom into the center where we can see it looks a lot like a randomized Gaussian blob:

zoom = 50;
ArrayPlot[xCentered, PlotRange -> {{d1 - zoom, d1 + zoom}, {d2 - zoom, d2 + zoom}}]


I think this means you are going to have to approach this a different way.

• If I squint I see the letter S in the zoomed center – shrx May 14 '16 at 18:28

One approach is to locate the black components and then measure some properties of them. Here we locate them using MorphologicalComponents, find the centroids using ComponentMeasurements and then calculate the distance between the centroids using Nearest.

img = Import["http://i.stack.imgur.com/hALsH.jpg"];
imgBW = Binarize@ColorConvert[img, "Grayscale"];
comp = MorphologicalComponents[ColorNegate[Dilation[imgBW, 5]]] // Colorize
cent = ComponentMeasurements[comp, "Centroid"];
allNears = Nearest[cent[[All, 2]], #, 2] & /@ cent[[All, 2]];
Total[Norm[allNears[[#, 1]] - allNears[[#, 2]]] & /@
Range[Length[allNears]]]/Length[allNears]


As you can see, the average distance form the center of each blob to the nearest adjacent blob is about 33 pixels.

• Basically very similar to my answer (+1, btw). I do like the application of Nearest and Dilation. (A minor point, I think using Mean/Median might be better than using Total[_]/Length[_]. – Anton Antonov May 14 '16 at 15:28
• @Anton -- after I finished the Fourier answer, I worked out the morphological approach - by the time I finished, you had already posted... so +1 to you for beating me! – bill s May 14 '16 at 15:54
• I think you should combine your FFT answer with this one. From my perspective, after applying FFT and concluding non-regularity the next natural thing to do is to apply the morphological components approach. – Anton Antonov May 14 '16 at 15:59