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# Tag Info

59

The main challenge to make something general is to create a list of points that, when joined, is as close as possible to the original image. The first example (elephant) focuses on the mathematical idea. In the second example (Bart Simpson), the code is wrapped in a function. The last example gives some ideas on how to apply the method to a photograph ...

56

It looks like you need some basic facts on numerical Fourier transforms. I am going to assume that you are going from the time domain to the frequency domain. The number of points in the time domain equals the number of points in the frequency domain. As the data is sampled in the time domain i.e. is a set of equally spaced points, then in the frequency ...

46

Here's what I did back in pre-MMA6 times. I believe you can easily adapt this method to work with all the fancy commands that have been added since then. Someone proficient in CDF programming could also make the mask building process interactive with disks and polygons: one window shows the FFT, the user places shapes to hide unwanted features, and another ...

44

I can explain this. The definite flavor of Integrate works with assumptions in a few ways. One is to use them in Simplify, Refine, and a few other places that accept Assumptions, to do what they will in the hope of attaining a better result (it also uses them to determine convergence and presence or absence of path singularities). Those places also get the ... 32 img = Import["ExampleData/lena.tif"]; 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@... 32 The Fourier transform is defined as: \begin{align*} H(f) &= \int_{-\infty}^\infty h(t) \mathrm e^{2\pi \mathrm i f t}\mathrm dt\\ h(t) &= \int_{-\infty}^\infty H(f) \mathrm e^{-2\pi \mathrm i f t}\mathrm df \end{align*} where h(t) $is the signal, and$ H(f) $is its Fourier transform; if$ t $is measured in seconds, then$ f $is measured ... 29 This is how I would do this. Define frequencies and sampling rate precisely. Then use Periodogram because it takes SampleRate as an option and rescales frequency axis automatically. Read up Docs on Periodogram - see examples there. data = Table[{t, Sin[2 Pi 697 t] + Sin[2 Pi 1209 t]}, {t, 0., 0.1, 1/8000.}]; ListLinePlot[data, AspectRatio -> 1/4, ... 26 I think there are at least three elements to consider here: FourierTransform and Fourier, by default, output results in different forms Plotting Sin[x] UnitStep[x] is not the same as Sin[x] and behaves differently when used in conjunction with Fourier and FourierTransform Plot does not handle DiracDelta elegantly The signal processing form of the Fourier ... 23 The problem here is that Mathematica doesn't recognize {x, y, z} as some kind of a vector object that should be treated as grouped together; instead, it substitutes in three independent variables, and probably starts integrating them one by one. The result is a very complicated integral. If you do the coordinate transformation yourself, you can reproduce ... 21 Here's a possible starting point for a solution. It splits the sample list into chunks and measures the Norm of the sample Differences in each chunk, and then does the FFT on that data. bpmplot[snd_, bpmmax_: 300] := Module[{samples, minfreq, signal, fft}, samples = snd[[1, 1, 1]]; minfreq = snd[[1, 2]]/Length[samples]; signal = (Norm[Differences[#]]) &... 21 I appreciate very much, that you wrote up such a nicely formatted question, although this is your first post. Therefore, let's put the comments into an answer. Your first issue was that you used ( ) where you should have used [ ]. That's maybe not obvious for starters and I have seen this mistake very often. There are different types of braces and it's ... 20 There is the function NFourierTransform[] (as well as NInverseFourierTransform[]) implemented in the package FourierSeries`. The function, as with the related kernel functions, takes a FourierParameters option so you can adjust computations to your preferred normalization as needed. For your specific normalization, you apparently want the setting ... 20 It always takes me a while to remember the best way to do a numerical Fourier transform in Mathematica (and I can't begin to figure out how to do that one analytically). So I like to first do a simple pulse so I can figure it out. I know the Fourier transform of a Gaussian pulse is a Gaussian, so pulse[t_] := Exp[-t^2] Cos[50 t] Now I set the timestep ... 19 Based on the MATLAB documentation, it would appear that this is accomplished by simple zero-filling. As such, you can obtain the same result in Mathematica using Fourier[PadRight[list, n, 0.], FourierParameters -> {1, -1}] where list is your signal and n is the desired length. For a multidimensional FFT, replace n with {n1, n2, ...}, where n1, n2, &... 18 If you take a look at the documentation, Mathematica's symbolic Fourier transform function, FourierTransform, computes $$\hat f(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{ikx}\mathrm{d}x$$ You can discretize some piece of this integral by limiting$x$and$k$to values$x_1 + (r-1)\Delta x$and$(s-1)\Delta k$respectively, where$\Delta x\...

18

update Just to clean things up a bit, we can use the discussion here to make a couple functions that help extract the frequency data from this dataset. I define two functions findPeriod and reconstruct: Clear[findPeriod]; findPeriod[data_, threshold_] := Module[{fs, s1, s = {}, i, a0f, af, pf, pos, fr, frpos, fdata, fdatac, n, per}, n = Length[...

18

Physicist chiming in - Hi!. I believe there has been some confusion here. It seems to me that OP is meaning to plot an Airy disk which was studied by G.B. Airy but is not given by the Airy function. It is given by the Fourier transform of the indicator function of the unit circle, which actually happens to be a Bessel function (see e.g. wikipedia). If I ...

17

Fourier[list] computes the discrete Fourier transform of list. I assume it uses the FFT when it can.

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CUDALink allows you to write custom kernels but unfortunately it doesn't allow to use its CUDAMemory in other functions. I found two methods to deal with it. Documented method You can use LibraryLink and call CUDA functions as in a regular CUDA program. For example, you can write something like fourier.cu. Then you can compile it and load "fourier_single" ...

16

I think perhaps you need codes like this: Func[x_] := Sin[x]; tmin = 0; tmax = 10; \[CapitalDelta]t = (tmax - tmin)/100; tgrid = Table[t, {t, tmin, tmax, \[CapitalDelta]t}]; wgrid = RotateRight[(2 \[Pi])/(tmax - tmin)* Range[-((Length@tgrid - 1)/2), (Length@tgrid - 1)/2], ( Length@tgrid - 1)/2]; ListLogLogPlot[{wgrid, (tmax - tmin)/Sqrt[2 \[Pi]*...

16

As mentioned by @Rahul, you have not sampled your sine wave often enough and have introduced artifacts due to aliasing. The frequency of Sin[500 x]=Sin[2 Pi f x] is $f=500/(2\pi)$, which is about 80 Hz. At least two samples per cycle are required to avoid aliasing, hence the default $x$ interval of 1 in {x,0,100} must be reduced to less than about $1/(2*80)=... 16 Finally I found the most promising algorithm proposed in this really good reference Manuel Guizar-Sicairos and Julio C. Gutiérrez-Vega, "Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields," J. Opt. Soc. Am. A 21, 53-58 (2004). The authors call the algorithm pth-order quasi-discrete Hankel Transform (pQDHT) ... 16 Looking at your plotted data you can see about 40 cycles of the dominant frequency, this tells you that the peak will appear somewhere around the 40th element of the DFT. That's in the region where your plot of the DFT is clipped, so it's no wonder you can't see the peak. Looking at the relevant part of the DFT you can see the peak quite clearly: ... 16 I like @Vitaliy's answer, but here's another approach using Fourierinstead of Periodogram. time = 2; tinc = 0.001; sampls = Table[Sin[n*(2 Pi) 4], {n, 0, 2, tinc}]; nyquist = 1/(2 tinc) len = Length@sampls; ListLinePlot[Sqrt[4/len] Abs@Fourier[sampls], PlotRange -> {{0, 10}, All}, DataRange -> {0, (len - 1)/time}] Briefly, I construct a sample set ... 16 So, you have a function$F(x,y) = f_x(x)g_y(y) + g_x(x)f_y(y)$, and you want to recover$f_x,g_y,g_x,f_y$. If you've tabulated the values of$F(x,y)$in a matrix$\mathbf F$with entries$f_{ij} = F(x_i,y_j)$, then this amounts to decomposing the matrix as $$\mathbf F \approx \mathbf f_x\mathbf g_y^T + \mathbf g_x\mathbf f_y^T,$$ where$\mathbf f_x,\mathbf ...

16

First, import the audio and extract usable data from it: audio = Sound[ SampledSoundList[ Flatten@ImageData@Import["https://i.stack.imgur.com/qHpp6.png"], 22050]] audioDuration = Duration[audio]; audioSampleRate = AudioSampleRate[audio]; data = AudioData[audio][[1]]; Second, use PeakDetect to see which points are peaks (= 1) and which points are ...

15

I will take the data as a time history. First I assume that the x-values are equally spaced and work out the time increment and frequency increment and then plot the data. tinc = data[[2, 1]] - data[[1, 1]]; finc = 1/(tinc Length[data]); ListPlot[data] This looks like almost two cycles of a sine wave with a frequency of about 0.1 and an amplitude of 17. ...

15

First, I have a few improvement suggestions for your Fourier code: The bright vertical and horizontal lines you see in your Fourier image are the sharp gradients at the borders of the image (because the Fourier transform assumes a periodic image). So you should get rid of the black border at the bottom: img = Import["http://i.stack.imgur.com/bIUkE.png"]; ...

14

Fourier uses FFT when possible

14

Here is an explicit way to calculate the frequency corresponding to each element of the output of the Fourier command. The frequencies will depend on two values: the sampling interval and the number of points in the data analysis. ssf = RotateRight[Range[-n/2, n/2 - 1]/(n sampInt), n/2]; where n is the number of points analyzed and sampInt is the time ...

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