Clustering and Image Blurs

In this document, we demonstrate how zeroing out the high frequency Fourier coefficients of an image induces a Gaussian blur. This is similar to the effect of the heat equation, which attenuates high frequency components exponentially faster than lower frequency components.

from PIL import Image
import numpy as np
import matplotlib.pyplot as plt


def normalize_img(img):
    return (img - np.min(img))/np.ptp(img)

rishi_img = Image.open("images/rishi_fb.jpg").convert("L")
rishi_img.thumbnail((128, 128))
rishi = normalize_img(np.array(rishi_img))

plt.title("Test Image")
plt.axis('off')
plt.imshow(rishi, cmap="gray")
plt.show()

def two_d_fft(img, n=10):
    f = np.fft.fftshift(np.fft.fft2(img, norm="ortho"))
    half = int(len(f)/2)
    low_f = f[half-n:half+n + 1, half-n:half+n+1]
    return f, low_f

def two_d_ifft(f):
    return np.abs(np.fft.ifft2(np.fft.ifftshift(f), norm="ortho"))

rishi_waves, low_rishi_freqs = two_d_fft(rishi)

plt.subplot(1, 2, 1)
plt.title("Full Spectrum of Rishi")
plt.imshow(np.abs(rishi_waves))

plt.subplot(1, 2, 2)
plt.title("Rishi's Low Frequencies")
plt.imshow(np.abs(low_rishi_freqs))

plt.suptitle("Absolute Frequency Spectrum of Rishi")
plt.show()

plt.subplot(1, 2, 1)
plt.title("Full Log Spectrum of Rishi")
plt.imshow(np.log(np.abs(rishi_waves)))

plt.subplot(1, 2, 2)
plt.title("Rishi's Low Frequencies")
plt.imshow(np.log(np.abs(low_rishi_freqs)))

plt.suptitle("Log Frequency Spectrum of Rishi")
plt.show()

Smooth Basis Functions

Smooth Rishi Approximations

Can we approximate rishi with just a few of these smooth basis functions?

max_freq_radius = 40

for i, r in enumerate([5, 10, 50]):
    approx_r = np.fft.ifftshift(rishi_waves)
    #plt.subplot(1, 3, 1)
    plt.imshow(np.abs(np.fft.ifft2(approx_r, norm="ortho")), cmap="gray")
    approx_r[r : -r, :] = 0
    approx_r[:, r : -r] = 0
    approx_r = np.fft.fftshift(approx_r)
    #plt.subplot(1, 3, 2)
    #plt.imshow(np.nan_to_num(np.log(np.abs(approx_r)), neginf=0))
    plt.subplot(1, 3, i+1)
    plt.title(f"Frequency Cutoff Index: {r}")
    plt.imshow(np.abs(np.fft.ifft2(approx_r, norm="ortho")), cmap="gray")
    plt.axis('off')
    plt.gcf().set_size_inches((10, 10))
    plt.show()