juliafftdct

Is this dct (FFTW.jl) behavior in julia normal?

I'm trying to do some exercises of Compressed Sensing on Julia, but i realize that the discrete cosine transformation (using FFTW.jl) of an identity matrix doesn't looks as the result of other programming languages (aka. Mathematica and Matlab).

For example in Julia

using Plots, FFTW, LinearAlgebra
n = 100
Psi = dct(Matrix(1.0I,n,n))
heatmap(Psi)

results in this matrix (which is essentially an identity matrix with some noise)

Julia's dct for an identity matrix

But in Matlab

imagesc(dct(eye(100,100),'Type',2))

this is the result (as expected)

Matlab dct type 2 of an identity matrix

Finally in Mathematica

MatrixPlot[N[FourierDCTMatrix[100, 2]], PlotLegends -> Automatic]

returns this

Mathematica dct of an identity matrix

Why Julia behaves so differently? And is this normal?

Solution

  • Matlab (and I guess Mathematica), does dct of each column in your matrix. FFTW performs a 2-dimensional dct when the input is two-dimensional. The same happens for fft.

    If you want column-wise transformation, you can specify the dimension:

    Psi1 = dct(Matrix(1.0I,n,n), 1);  # along first dimension
heatmap(Psi1)

    Notice that the direction of the y-axis is opposite for Plots.jl relative to Matlab.

    (BTW, you can also just write I(n) or 1.0I(n) instead of Matrix(1.0I,n,n))

    enter image description here

    This is something that sets Julia apart from some other languages. It tends to treat matrices as matrices, and not as just a collection of vectors or a bunch of scalars. For example exp(M) and log(M) for matrices not operate elementwise, but will calculate the matrix exponential and matrix logarithm according to their linear algebra definitions.