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color-edge-qft.py
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color-edge-qft.py
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import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
from scipy import fftpack
import cv2
def img2uint8(img):
'''
Convert image to 8-bit format [0, 255].
'''
vmin = img.min()
vmax = img.max()
img = ((img - vmin) / (vmax - vmin)) * 255.0
return np.uint8(img)
def img_qft(img, mu):
"""
Obtain the Fourier Transform for Quaternions of an Image.
Parameters
----------
img : Color image [R, G, B].
mu : list
Pure quaternion unit.
e.g., (i + j + k) / sqrt(3) -> [1/sqrt(3), 1/sqrt(3), 1/sqrt(3)]
Return
------
Fuv : QFT in the frequency domain of the 4-D space image.
"""
if img.dtype != np.uint8:
img = img2uint8(img)
fr = img[:, :, 0]
fg = img[:, :, 1]
fb = img[:, :, 2]
DFTfr = np.fft.fft2(fr)
DFTfg = np.fft.fft2(fg)
DFTfb = np.fft.fft2(fb)
alpha = mu[0]
betha = mu[1]
gamma = mu[2]
Auv = - (alpha * DFTfr[:, :].imag) - (betha * DFTfg[:, :].imag) - (gamma * DFTfb[:, :].imag)
iBuv = DFTfr[:, :].real + (gamma * DFTfg[:, :].imag) - (betha * DFTfb[:, :].imag)
jCuv = DFTfg[:, :].real + (alpha * DFTfb[:, :].imag) - (gamma * DFTfr[:, :].imag)
kDuv = DFTfb[:, :].real + (betha * DFTfr[:, :].imag) - (alpha * DFTfg[:, :].imag)
Fuv = np.zeros((img.shape[0], img.shape[1], 4))
Fuv[:, :, 0] = Auv
Fuv[:, :, 1] = iBuv
Fuv[:, :, 2] = jCuv
Fuv[:, :, 3] = kDuv
return Fuv
def img_iqft(img_qft, mu):
"""
Inverse Fourier transform of the QFT of an image.
Parameters
----------
img_qft : QFT 4-D image [Auv, iBuv, jCuv, kDuv].
mu : list
Pure quaternion unit.
e.g., (i + j + k) / sqrt(3) -> [1/sqrt(3), 1/sqrt(3), 1/sqrt(3)]
Returns
-------
fmn : Color image.
"""
assert img_qft.shape[2] == 4, "Image is not in qft format"
A = img_qft[:, :, 0]
B = img_qft[:, :, 1]
C = img_qft[:, :, 2]
D = img_qft[:, :, 3]
IDFTA = np.fft.ifft2(A)
IDFTB = np.fft.ifft2(B)
IDFTC = np.fft.ifft2(C)
IDFTD = np.fft.ifft2(D)
alpha = mu[0]
betha = mu[1]
gamma = mu[2]
fa = (
IDFTA[:, :].real - (alpha * IDFTB[:, :].imag) -
(betha * IDFTC[:, :].imag) - (gamma * IDFTD[:, :].imag)
)
fr = (
IDFTB[:, :].real + (alpha * IDFTA[:, :].imag) +
(gamma * IDFTC[:, :].imag) - (betha * IDFTD[:, :].imag)
)
fg = (
IDFTC[:, :].real + (betha * IDFTA[:, :].imag) +
(alpha * IDFTD[:, :].imag) - (gamma * IDFTB[:, :].imag)
)
fb = (
IDFTD[:, :].real + (gamma * IDFTA[:, :].imag) +
(betha * IDFTB[:, :].imag) - (alpha * IDFTC[:, :].imag)
)
fmn = np.zeros((img_qft.shape[0], img_qft.shape[1], 4))
fmn[:, :, 0] = fa
fmn[:, :, 1] = fr
fmn[:, :, 2] = fg
fmn[:, :, 3] = fb
return fmn
def sobel_filter_qft(f):
"""
Vertical and horizontal sobel filter in the frequency domain applied to the
QFT of the image.
Parameters
----------
f : QFT of the image.
Returns
-------
Gx : complex
Sobel filter applied horizontally at qft in the frequency domain.
Gy : complex
Sobel filter applied vertically at qft in the frequency domain.
"""
# sobel in x direction
sobel_x = np.array([[-1, 0, 1],
[-2, 0, 2],
[-1, 0, 1]])
# sobel in y direction
sobel_y = np.flip(sobel_x.T, axis=0)
sz_x = (f.shape[0] - sobel_x.shape[0], f.shape[1] - sobel_x.shape[1])
sobel_x = np.pad(sobel_x, (((sz_x[0] + 1) // 2, sz_x[0] // 2),
((sz_x[1] + 1) // 2, sz_x[1] // 2)), 'constant')
sobel_x = fftpack.ifftshift(sobel_x)
sz_y = (f.shape[0] - sobel_y.shape[0], f.shape[1] - sobel_y.shape[1])
sobel_y = np.pad(sobel_y, (((sz_y[0] + 1) // 2, sz_y[0] // 2),
((sz_y[1] + 1) // 2, sz_y[1] // 2)), 'constant')
sobel_y = fftpack.ifftshift(sobel_y)
Gx = np.zeros((f.shape))
Gy = np.zeros((f.shape))
for i in range(f.shape[2]):
if i == 0:
Gx[:, :, i] = f[:, :, i] * fftpack.fft2(sobel_x).real
Gy[:, :, i] = f[:, :, i] * fftpack.fft2(sobel_y).real
else:
Gx[:, :, i] = f[:, :, i] * fftpack.fft2(sobel_x).imag
Gy[:, :, i] = f[:, :, i] * fftpack.fft2(sobel_y).imag
return Gx, Gy
def img_out(F, mu=[(1 / np.sqrt(3))] * 3):
"""
Retrieve color image using inverse Fourier transform for quaternions.
Parameters
----------
F : Quaternion Fourier transform image.
mu : list
Pure quaternion unit.
e.g., (i + j + k) / sqrt(3) -> [1/sqrt(3), 1/sqrt(3), 1/sqrt(3)]
Return
------
Normalized image in [0, 1] range.
"""
assert F.shape[2] == 4, "Image is not in qft format"
out = img_iqft(F, mu)
for d in range(out.shape[2]):
out[:, :, d] *= 255.0 / np.amax(out[:, :, d])
out[:, :, d] = np.clip(out[:, :, d], 0, 255)
return np.uint8(out)
# return np.uint8(cv2.normalize(out, None, alpha=0, beta=255, norm_type=cv2.NORM_MINMAX, dtype=cv2.CV_32F))
def color_xyedge_det(img, mu=[(1 / np.sqrt(3))] * 3):
"""
Color image recovered once the horizontal and vertical sobel filter is
applied.
"""
f = img_qft(img, mu)
Gx, Gy = sobel_filter_qft(f)
return img_out(Gx, mu), img_out(Gy, mu)
def correlate_qft(img, mu=[(1 / np.sqrt(3))] * 3):
"""
Retrieve Color Image from Horizontal and Vertical Sobel Filter Correlation.
"""
f = img_qft(img, mu)
Gx, Gy = sobel_filter_qft(f)
correlate = np.zeros(f.shape)
for j in range(f.shape[2]):
correlate[:, :, j] = Gx[:, :, j] - Gy[:, :, j]
return img_out(correlate, mu)
if __name__ == '__main__':
# Read image
img = plt.imread('./images/Lenna.png')
# Normalize image
img = img2uint8(img)
# Get a vertical and horizontal sobel filter apply in image
img_sobelx, img_sobely = color_xyedge_det(img)
# Show
fig, (ax1, ax2, ax3) = plt.subplots(ncols=3, nrows=1)
ax1.imshow(img, cmap='gray')
ax1.set_title('Input image'), ax1.set_xticks([]), ax1.set_yticks([])
ax2.imshow(img_sobelx[:, :, 1:], cmap='gray')
ax2.set_title('IQFT Sobel X'), ax2.set_xticks([]), ax2.set_yticks([])
ax3.imshow(img_sobely[:, :, 1:], cmap='gray')
ax3.set_title('IQFT Sobel Y'), ax3.set_xticks([]), ax3.set_yticks([])
# fig.savefig('sobel-hv.png', transparent=True
plt.show()
# Combine horizontal and vertical sobel filter using correlation
correlate = correlate_qft(img)
plt.figure(figsize=(8, 8))
plt.title('Sobel H-V Filter Correlation')
plt.xticks([])
plt.yticks([])
plt.imshow(correlate[:, :, 1:], cmap='gray')
# plt.savefig('sobel-correlate.png', transparent=True)
plt.show()