Image derivative in the Fourier space

Question

It can be shown that the image derivative in the $x$ direction is given by: $$ \frac{\partial f}{\partial x}=\frac{2\pi i}{N} \mathcal F^{-1}\left(u\cdot \mathcal F(f(x,y)\right) $$ where $N$ is the width of the image in the $x$ direction and $u$ denotes a variable in the fourier space ($F(u,v)$).

Now, in the image space, derivation is a convolution with this matrix: $$\partial_x =\left(\begin{array}{cc} -1 & 1\end{array}\right)$$

this means (from to convolution theorem) that the fourier transform of $\partial_x$ is a matrix $D(u,v)$ such that $D(u,v)=u$.

so I tried to see this in python, but something seems to be wrong here:

d = np.zeros((21,21))
d[10,10] = 1
d[10,9] = -1
D = np.fft.fft2(d)
plt.matshow(np.fft.fftshift(D))

Can anyonw see where I am wrong?

Answer 1

The Fourier transform property in your first equation is for continuous time but the image is a discrete time signal. $\partial_x$ isn't exactly differentiation, it's a discrete time filter which does finite differencing.

Since you are only taking derivatives in one direction, let's think about a 1-D signal for simplicity (or, equivalently, let's assume your image has only one row). Let the signal be $f(x)$ and its continuous time Fourier transform $F(u)$. Then we have the transform pair $\frac{df(x)}{dx} \leftrightarrow j 2\pi u F(u)$. In discrete time, if $x[n] \leftrightarrow X[k]$, you can't really say $\{x[n]-x[n-1]\} \leftrightarrow j2\pi k X[k]$.

Instead, we can think of $\partial_x$ as a high pass filter. It then agrees with the intuitive meaning of the continuous time Fourier transform property too (where multiplying by $u$ attenuates lower frequencies more than higher frequencies). Try this code, to see the high pass response:

d = np.zeros(10000)
d[0] = -1.0
d[1] = 1.0
D = np.fft.fft(d)
plt.plot( np.abs(D) )

Answer 2

(picks up the chalk)

I believe fervently that, in the 2D image space, this rotation to grad is useful:

• $M = M$: (no rotation)
• $a = 0, 90, 180, 270, 360$ degrees, or $\left[ 0 , 360 \right]$ degress
  • 1. N = 2dcfft(M); The elements are cos and sin, recall that?
  • 2. O = e^-ia(M); Element by element component rotation, ok?
  • 3. P = 2dicfft(O); Image reformation.

(offers chalk, no takers)

Note for $a = 0$, or $360$ degrees, $P = M$ to computational accuracy. So, WTF happens in between? Well, it's all about pencil sketches and modeling for art classes. I do that. And in the studio, I hold still, shut TF up, and think about the brain dynamics.

In between:

At $a = 90$, $O = \left| \nabla \left( M \right) \right|$. The other angles are left as an exercise for the interested reader!

Cheers from Dana at Replikon Dot Net Mathcad 6.0 Plus!

(chalk returned)