Sum of Diracs in Frequency Domain I am attempting to compute the sum of dirac functions in the frequency domain via the following formulas:

$$ ReLu(f(x)) = f(x) * \sum_{i} \delta(x-x_i) $$
$$ x_i : f(x_i) &gt; 0 $$
$$ F(ReLu(f(x)))(k) = (\sum_{i} e^{2\pi jk x_i})F(f(x)) $$

From what I understand I need to find the values where x is larger than zero, then sum $e^{2\pi jk x_i}$ over these values. I made myself a trivial example to determine if I am doing this correctly and there is no difference to the image after the convolution by the sum of the exponentials. What am I interpreting wrong here?

Note this idea is from the paper: http://cs231n.stanford.edu/reports/2015/pdfs/tema8_final.pdf

import numpy as np
import matplotlib.pyplot as plt

# Make image array with a single negative value
img = np.array([[-1.0,2.3],[5,7.8]])

# Dirac values
N = 2
x = np.arange(0, N, 1)
y = np.arange(0, N, 1)
xm, ym = np.meshgrid(x, y)
x0 = xm
y0 = ym
shiftMat = np.exp(1j * ((2.0 * np.pi)) * (xm*x0 + ym*y0))

# Set equal to shift mat
# In this trival example I know that [0,0] is only negative position
# So set to 0 and compute sum of all positions in which f(x) &gt; 0 as detailed in paper
freqRelu = shiftMat
freqRelu[0,0] = 0
freqRelu = np.sum(freqRelu)

# Fourier Convolution and IFFT
imgFFT = np.fft.fft2(img)
freqR = np.multiply(imgFFT,freqRelu)
# freqR = np.outer(freqRelu,imgFFT)
reluedFreq = np.real(np.fft.ifft2(freqR))

# Spatial Relu For Comparision
reluedImg = img
reluedImg[0,0] = 0

plt.subplot(121)
plt.imshow(reluedImg)
plt.subplot(122)
plt.imshow(reluedFreq)
plt.show()

# print(np.allclose(reluedFreq,reluedImg))
print(reluedFreq)
print(reluedImg)
ReLu in the frequency domain becomes convolution, NOT multiplication. 
See chapter 4.5 of the paper: "...in frequency domain, ReLu() acts as a convolution with the function of a known form. "
But you don't need to calculate it at every point because of the pooling.