# How can I extract data from an image to perform a discrete fourier transform?

I have the given image,

And I tried to extract the points on the wave patterns manually, using the point selection and COPY tool inn mathematica. However, the list of points I get do not present a piecewise function.

The upper plot of the image gives:

alwynpoints = {{60., 334.6579493799986}, {66.75,
246.90794937999863}, {68.25, 283.6579493799986}, {72.,
298.6579493799986}, {72.75, 286.6579493799986}, {79.5,
314.65794937999874}, {84., 244.90794937999874}, {85.5,
264.40794937999874}, {86.2, 284.65794937999874}, {86.24,
281.65794937999874}, {86.25, 280.90794937999874}, {89.26,
304.15794937999874}, {90., 319.15794937999874}, {93.75,
283.15794937999874}, {96.75, 268.15794937999874}, {99.,
280.90794937999874}, {99.1, 245.65794937999874}, {99.75,
325.15794937999874}, {102.75, 280.90794937999874}, {107.25,
372.40794937999874}, {109.5, 268.90794937999874}, {112.5,
172.90794937999874}, {115.5, 286.15794937999874}, {120.,
651.4079493799986}, {123.75, 247.65794937999874}, {125.25,
169.65794937999874}, {128.15, 282.15794937999874}, {128.25,
284.40794937999874}, {132.75, 401.40794937999874}, {137.15,
158.40794937999874}, {137.25, 283.65794937999874}, {142.5,
158.40794937999874}, {146.25, 282.90794937999874}, {148.5,
388.65794937999874}, {153., 287.40794937999874}, {154.5,
216.15794937999874}, {156., 286.65794937999874}, {158.25,
329.40794937999874}, {170.25, 282.90794937999874}};


However, when I try to plot these I get this incomplete plot

and from here I would like to do a Fourier transform of the plot, but I cannot find any information on Wolfram on how to do it on either an interpolation or a set of points.

Any ideas?

Thanks

• Try PlotRange->All? !Mathematica graphics since you have the points, may be what you need is Fourier? Which is FFT in Mathematica. Commented Nov 23, 2022 at 12:06
• There are lots of ways to extract the points from the image without having to do it manually. There is WebPlotDigitizer (be sure to pick the right color for the plot). There is the links provided by Szabolcs at the end here. The other answer given has some adjustments to make if I recall. I think the rescaling did not take into account the white space around the plot but I do not remember well. There is also this one Commented Nov 23, 2022 at 12:23
• You have way too few points to resolve all the wiggles in the picture. Commented Nov 23, 2022 at 12:32
• @Nasser, I tried Fourier, but that functionality rearranges the points in a different sequence, so the wavepattern is completely destroyed Commented Nov 23, 2022 at 13:16

Here is the coarsely digitized data of the first picture (using https://automeris.io/WebPlotDigitizer/ If you want more accurate results, improve on this):

dat=
{{5.48,1.25},{7.23,-2.03},{8.45,-1.83},{8.45,1.57},{8.97,0.113},{9.67,-1.04},{9.97,-1.47},{10.,2.01},{10.3,-0.697},{11.5,0.468},{12.1,-0.257},{12.1,0.783},{12.9,2.38},{21.3,1.16},{23.1,1.4},{24.3,0.0804},{25.8,1.84},{26.8,-1.72},{28.,-1.36},{28.,2.37},{28.,2.72},{28.,3.07},{28.,3.42},{29.6,-0.896},{30.8,-0.359},{32.9,-2.72},{32.9,-2.44},{32.9,-2.09},{33.7,0.688},{37.3,0.0908},{37.7,1.17},{38.2,1.61},{44.5,-1.15},{46.3,-0.776},{47.7,0.442},{49.,-0.391},{51.8,0.0232},{59.,-0.533},{59.7,0.903},{59.7,1.25},{61.3,1.7},{63.4,-0.87},{63.4,0.143},{63.4,0.603},{66.4,-1.38},{73.1,-2.34},{73.1,-2.},{74.,1.07},{74.,1.42},{74.6,-1.6},{75.6,1.85},{76.6,0.0798},{78.,2.32},{81.7,-1.06},{82.1,-2.64},{82.4,0.722},{82.9,-2.17},{82.9,-1.82},{85.4,-0.708},{87.8,-0.285},{87.8,1.2},{88.8,-1.4},{92.1,0.0181},{97.6,0.446},{99.,0.922},{101.,-1.03},{103.,-0.583},{106.,0.075},{106.,1.42},{106.,1.77},{106.,2.16},{106.,2.5},{106.,2.85},{107.,-1.69},{109.,-2.13},{115.,0.671},{115.,1.13},{115.,0.369},{116.,-1.41},{117.,-2.27},{120.,-1.04},{121.,-0.378},{121.,-0.735},{122.,0.0312},{124.,1.61},{125.,-1.88},{132.,-2.11},{132.,-1.41},{132.,0.413},{132.,0.76},{133.,-1.75},{133.,1.22},{133.,-0.611},{134.,-0.0938},{138.,-0.99},{144.,1.82},{150.,-1.03},{150.,0.048},{151.,-0.741},{151.,-0.411},{151.,0.683},{151.,1.03},{152.,0.351},{153.,1.44},{165.,-1.19},{166.,0.0782},{168.,0.469},{168.,0.821},{168.,1.17},{169.,-0.795},{170.,2.05},{170.,1.64},{176.,-0.242},{184.,-1.13},{184.,-0.834},{188.,0.0473},{188.,0.294},{188.,1.35},{188.,1.7},{188.,2.05},{189.,-2.64},{189.,-2.34},{189.,-1.99},{189.,0.617},{189.,0.998},{190.,2.46},{194.,-1.58},{196.,2.96},{198.,2.58},{200.,-1.26},{202.,-0.102},{202.,0.603},{202.,1.56},{202.,1.92},{202.,-0.821},{203.,-1.73},{204.,1.12},{213.,0.145},{216.,-0.487},{222.,-0.114},{223.,2.35},{223.,2.7},{225.,-1.49},{226.,-0.988},{226.,-0.485},{228.,0.0997},{228.,0.459},{228.,0.809},{228.,1.16},{228.,1.51},{228.,1.86},{234.,-1.8},{241.,0.571},{242.,0.165},{243.,0.92},{244.,-1.42},{244.,-1.07},{244.,-0.714},{244.,-0.218},{244.,1.38},{245.,-2.22},{246.,3.55},{246.,4.4},{247.,4.07},{248.,3.11},{248.,-1.75},{249.,2.08},{250.,2.57},{250.,1.78},{257.,0.122},{261.,-1.32},{261.,0.533},{261.,0.884},{262.,-0.903},{262.,1.38},{263.,-2.23},{266.,1.9},{266.,2.25},{267.,2.74},{271.,0.709},{271.,1.18},{271.,1.53},{271.,-1.69},{272.,-0.282},{273.,0.0812},{273.,-1.28},{275.,-0.932},{279.,0.427},{283.,-0.495},{288.,0.0485},{292.,0.402},{294.,0.769},{298.,-2.05},{299.,-1.68},{299.,-1.33},{299.,-0.974},{300.,-0.487},{304.,0.0895},{304.,-2.56},{305.,1.21},{307.,0.424},{308.,-1.75},{311.,1.74},{314.,-1.18},{317.,-0.734},{319.,0.794},{319.,0.0816},{323.,-0.382},{329.,1.12},{331.,0.601},{333.,-0.866},{334.,0.0928},{334.,-1.77},{337.,-1.45},{337.,-0.366},{338.,1.74},{339.,-3.1},{339.,-2.73},{340.,-2.25},{340.,1.38},{341.,-3.52},{343.,2.56},{343.,2.91},{343.,3.26},{343.,3.61},{343.,3.97},{345.,0.679},{345.,-0.799},{347.,2.12},{348.,0.0899},{350.,-0.33},{353.,1.12},{354.,1.59},{359.,-1.03},{361.,-1.43},{362.,0.521},{363.,-0.685},{364.,-0.4},{365.,0.00345},{371.,0.929},{371.,1.28},{371.,1.63},{371.,1.98},{371.,2.34},{373.,2.77},{375.,-2.4},{376.,-2.04},{376.,-1.34},{377.,-1.69},{377.,0.641},{377.,-0.819},{379.,0.0482},{380.,3.35},{381.,-0.34},{381.,2.65},{381.,3.},{384.,-2.93},{384.,1.54},{384.,1.89},{384.,2.24},{386.,-2.45},{388.,1.13},{392.,0.264},{393.,-1.8},{393.,-1.47},{393.,-1.12},{393.,0.672},{394.,-2.19},{394.,-0.72},{399.,0.00283},{403.,-0.339},{406.,1.05},{406.,1.4},{408.,1.77},{409.,2.18},{410.,0.694},{411.,-2.5},{411.,-2.12},{411.,-1.79},{411.,-1.44},{411.,-1.09},{412.,-0.69},{415.,0.0176},{420.,-2.14},{420.,-1.69},{420.,-1.33},{421.,1.54},{421.,1.89},{421.,2.24},{421.,2.59},{421.,2.94},{423.,0.366},{424.,-0.308},{425.,0.834},{425.,1.19},{426.,-0.842},{431.,0.161},{439.,0.63},{439.,-0.264},{440.,0.976},{440.,1.38},{441.,1.77},{441.,-1.46},{443.,-1.04},{443.,2.4},{444.,0.122},{444.,-0.6},{452.,-1.88},{453.,1.6},{457.,-0.59},{458.,0.529},{458.,0.895},{458.,1.25},{458.,0.2},{459.,-0.131},{461.,-1.92},{464.,-1.48},{466.,-1.02},{471.,-0.169},{471.,0.505},{471.,0.856},{473.,0.122},{473.,1.25},{476.,-0.473},{485.,-2.14},{485.,-1.8},{485.,1.56},{485.,2.03},{487.,0.564},{487.,0.198},{488.,1.03},{488.,-0.0119},{489.,-1.17},{489.,2.95},{489.,3.3},{490.,-0.791},{491.,-1.54},{493.,2.5},{499.,-0.256},{501.,0.266},{502.,-0.714},{502.,1.45},{502.,1.87},{503.,2.36},{503.,-1.5},{504.,-1.06},{505.,-1.71},{506.,1.03},{508.,0.702},{516.,0.203},{518.,-0.702},{520.,-1.04},{520.,0.786},{521.,-1.48},{522.,-0.173},{531.,0.415},{536.,-0.698},{536.,-0.0549},{538.,0.943},{540.,-0.463},{541.,1.45},{550.,0.529},{551.,-0.00119},{557.,-0.939},{557.,-0.536},{560.,-1.28},{561.,0.224},{562.,1.08},{562.,0.607},{568.,-0.0193},{573.,-0.452},{580.,-0.63},{580.,0.33},{581.,0.745},{582.,1.23},{585.,0.144},{586.,-1.21},{586.,1.78},{586.,2.13},{591.,-0.885},{594.,-0.416},{596.,-1.73},{596.,0.654},{596.,1.},{598.,1.48},{599.,-2.43},{599.,-2.09},{599.,3.07},{599.,3.42},{599.,0.0778},{601.,-1.38},{602.,-2.88},{602.,2.64},{604.,1.96},{604.,2.31},{606.,-0.491},{609.,-1.03},{609.,1.47},{611.,0.937},{612.,-1.79},{614.,-1.38},{614.,0.133},{616.,0.625},{621.,-0.38},{621.,-0.767},{629.,-2.28},{630.,1.04},{631.,0.0983},{632.,-2.07},{632.,-1.71},{633.,0.592},{633.,-1.25},{637.,-0.756},{639.,-0.279},{641.,1.67},{646.,0.273},{648.,-2.39},{649.,-2.01},{649.,-1.66},{649.,1.55},{650.,-1.18},{651.,2.16},{651.,0.773},{652.,-2.72},{652.,2.49},{652.,2.82},{653.,1.25},{653.,1.78},{653.,3.23},{654.,-0.721},{655.,-0.189},{657.,3.72},{661.,0.115},{666.,0.818},{666.,0.31},{668.,-1.65},{668.,-0.676},{668.,2.29},{668.,-1.2},{671.,-0.383},{671.,1.51},{671.,1.86},{672.,-2.21},{674.,1.16},{676.,-1.7},{677.,0.0666},{681.,0.684},{684.,-0.476},{687.,0.995},{687.,1.35},{688.,-1.15},{691.,0.0786},{697.,0.556},{700.,1.01},{701.,-0.753},{702.,-0.376},{705.,-1.09},{706.,0.111},{707.,1.44},{718.,-0.986},{718.,0.484},{718.,0.976},{719.,-0.533},{721.,1.5},{721.,1.85},{721.,2.2},{721.,2.55},{721.,2.9},{722.,0.0995},{723.,3.35},{726.,-1.42},{726.,-3.62},{726.,-3.27},{726.,-2.91},{726.,-2.56},{726.,-2.17},{726.,-1.84},{726.,4.95},{726.,5.3},{726.,5.66},{726.,6.01},{726.,6.36},{726.,6.71},{726.,7.06},{726.,7.41},{726.,7.77},{726.,8.12},{726.,8.47},{726.,8.82},{726.,9.17},{726.,9.52},{726.,9.88},{726.,10.2},{726.,10.6},{726.,10.9},{726.,11.3},{726.,11.6},{726.,12.},{726.,12.3},{726.,12.7},{726.,13.},{726.,13.4},{726.,13.7},{727.,-3.99},{727.,0.734},{730.,3.8},{730.,4.15},{730.,4.5},{731.,0.0109},{735.,1.06},{740.,3.61},{740.,4.31},{743.,-2.59},{743.,-4.39},{743.,-4.04},{743.,-3.69},{743.,-3.34},{743.,1.79},{743.,2.14},{743.,2.49},{743.,2.84},{744.,1.4},{745.,-3.},{745.,3.29},{745.,3.88},{746.,0.145},{748.,-1.89},{748.,-1.54},{748.,-1.1},{749.,0.862},{749.,-0.701},{749.,-0.349},{750.,-2.21},{751.,0.574},{760.,1.29},{761.,1.71},{761.,0.0773},{763.,0.892},{764.,0.566},{765.,-1.59},{765.,-1.26},{768.,-0.854},{770.,-0.429},{773.,0.326},{775.,0.97},{775.,1.32},{775.,1.67},{775.,2.03},{775.,2.38},{779.,-0.121},{781.,-0.729},{789.,-0.953},{792.,0.17},{794.,-1.98},{795.,0.898},{795.,1.37},{796.,0.581},{797.,-1.47},{799.,1.91},{800.,-0.502},{801.,-0.952},{807.,0.109},{813.,-0.742},{813.,0.914},{814.,0.66},{814.,1.38},{814.,-0.419},{819.,-2.8},{821.,0.102},{821.,-2.09},{821.,-1.74},{821.,-1.39},{821.,2.03},{821.,2.38},{821.,2.73},{821.,3.08},{821.,3.43},{822.,-2.48},{823.,-0.961},{828.,0.844},{830.,1.24},{831.,-0.292},{831.,1.62},{832.,-0.745},{836.,0.0996},{837.,0.534},{842.,-1.86},{842.,-0.696},{843.,-1.05},{843.,1.84},{844.,-1.52},{845.,2.26},{845.,2.62},{847.,-0.295},{849.,1.19},{851.,0.15},{851.,1.59},{855.,0.74},{857.,-2.07},{857.,-2.57},{858.,-3.24},{858.,-1.87},{858.,2.84},{858.,3.17},{860.,-0.329},{860.,2.07},{860.,2.42},{861.,-1.46},{863.,-1.08},{864.,-0.658},{864.,0.14},{867.,1.2},{869.,1.66},{871.,0.577},{873.,2.18},{876.,-1.28},{877.,-0.183},{878.,-0.758},{880.,0.83},{880.,1.18},{880.,1.53},{884.,-1.02},{885.,0.437},{886.,0.119},{897.,-0.172},{899.,-0.615},{903.,0.571},{903.,0.133},{909.,-0.954},{909.,1.39},{909.,-1.88},{910.,1.78},{911.,0.939},{912.,-1.46},{913.,-0.454},{915.,2.12},{916.,0.4},{917.,-0.00841},{917.,-0.966},{920.,1.62},{924.,1.07},{932.,-0.0653},{932.,-0.574},{936.,0.556},{939.,1.8},{939.,2.15},{940.,-1.5},{941.,-0.975},{942.,1.01},{942.,1.36},{943.,-1.95},{946.,-0.471},{946.,0.0401},{952.,0.575},{959.,-2.14},{961.,-0.323},{961.,0.163},{961.,-1.7},{961.,1.16},{962.,-1.04},{962.,-1.45},{963.,2.83},{963.,3.18},{964.,-0.727},{968.,0.732},{968.,1.79},{968.,2.14},{968.,2.49},{969.,1.45},{976.,0.0741},{978.,0.37},{978.,1.07},{980.,-1.63},{981.,-1.23},{981.,-0.874},{983.,-0.498},{985.,2.54},{986.,0.572},{989.,2.08},{990.,0.059},{990.,1.4},{990.,1.71},{991.,0.884},{992.,0.147},{995.,-2.17},{997.,-1.62},{997.,-1.26},{998.,-2.59},{998.,-0.541},{999.,-0.928},{1000.,3.03},{1000.,3.39},{1000.,3.74},{1000.,4.09},{1010.,0.0913},{1010.,0.475},{1010.,0.827},{1010.,1.18},{1010.,1.53},{1010.,1.88},{1010.,2.23},{1010.,2.58},{1010.,-2.14},{1010.,-1.12},{1010.,-0.496},{1010.,-1.65},{1020.,0.573},{1020.,0.127},{1030.,0.75},{1030.,1.1},{1030.,1.45},{1030.,1.8},{1030.,2.25},{1030.,-2.09},{1030.,-1.05},{1030.,-0.696},{1030.,-0.344},{1030.,-1.43},{1030.,-1.73},{1030.,2.71},{1030.,2.33},{1040.,0.219},{1040.,1.56},{1040.,1.9},{1040.,0.651},{1040.,1.16},{1040.,-1.09},{1040.,-0.715},{1050.,-0.509},{1050.,-0.0293},{1050.,0.282},{1050.,0.663},{1060.,-0.667},{1070.,-0.085},{1070.,0.547},{1070.,0.898},{1070.,0.232},{1070.,1.34},{1070.,2.57},{1070.,2.92},{1070.,3.37},{1070.,-0.933},{1080.,-1.47},{1080.,1.87},{1080.,2.22},{1080.,-0.339},{1080.,-1.74},{1090.,0.0641},{1090.,0.247},{1090.,0.636},{1090.,0.987},{1090.,1.47},{1090.,-0.873},{1090.,-1.34},{1100.,-1.75},{1100.,2.07},{1100.,-0.975},{1100.,-1.27},{1100.,0.244},{1100.,2.63},{1100.,2.98},{1100.,3.33},{1100.,-0.554},{1100.,1.93},{1100.,2.28},{1110.,0.852},{1110.,1.2},{1110.,1.55},{1110.,-0.0916},{1120.,1.37},{1120.,-0.68},{1120.,-0.994},{1120.,0.468},{1120.,0.982},{1120.,-0.432},{1120.,-0.0364},{1140.,0.404},{1140.,-1.05},{1140.,-0.0314},{1140.,-0.746},{1140.,-0.417},{1140.,1.04},{1140.,1.51},{1140.,0.756},{1140.,-1.5},{1150.,0.432},{1150.,1.6},{1150.,-1.},{1150.,1.11},{1150.,-0.101},{1150.,0.786},{1160.,-0.61},{1160.,-1.56},{1170.,-0.218},{1170.,0.21},{1170.,1.23},{1170.,1.58},{1170.,1.93},{1170.,0.816},{1170.,-1.87},{1170.,2.41},{1170.,-0.88},{1170.,2.96},{1170.,3.31},{1170.,-1.45},{1180.,-0.424},{1180.,0.435},{1180.,0.0839},{1190.,-2.04},{1190.,-1.7},{1190.,-2.48},{1190.,1.2},{1190.,-0.648},{1190.,0.809},{1190.,1.68},{1190.,2.03},{1190.,2.46},{1190.,-1.45},{1190.,2.75},{1190.,-1.04},{1190.,-0.343},{1200.,0.168},{1200.,0.602},{1200.,1.05},{1200.,1.4}}

ListLinePlot[dat,PlotRange->All]


And the power spectrum (squared magnitude of Fourier coefficients):

Periodogram[dat[[All, 2]], ScalingFunctions -> "dB", PlotRange -> All]


• this last plot is the magnitude. If you consider the first wave-pattern, the highest peak is a rogue wave in a recorded wave-train, why didn' t that rogue magnitude appearr? Commented Nov 23, 2022 at 18:04
• The second plot gives the square of the magnitudes of the frequencies. A peak in the original data will contribute to all the frequencies and not give a peak in the FFT. Commented Nov 23, 2022 at 19:10
• so does this mean that the source of the magnitude of a rogue wave is actually deriving from the entire wave train? if we consider it as a fourier series... Commented Nov 23, 2022 at 19:23
• I assume by "rogue wave" you mean the peak in the original data. Then note, the Fourie transform of a delta peak is a constant. meaning: a delta peak contains all possible frequencies. Commented Nov 23, 2022 at 19:45
• thanks I will read on this more. one thing, can your model give an analytic function too? Commented Nov 23, 2022 at 20:08