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freq_filters.py
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freq_filters.py
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import numpy as np
# Lowpass Filter
###############################################################################
# INPUT
# shape: P x Q shape of filter
# d0: Cutoff frequency
# ftype: Filter type ('ideal', 'butterworth' or 'gaussian')
# n: Order of filter (only applies for butterworth filters)
# u_k: u position of notch pair (only applies for notch filters)
# v_k: v position of notch pair (only applies for notch filters)
###############################################################################
# OUTPUT
# H: A lowpass filter with input parameters
###############################################################################
def lowpass_filter(shape, d0=160, ftype='butterworth', n=2, u_k=0, v_k=0):
P, Q = shape
# Initialize filter with zeros
H = np.zeros((P, Q))
# Traverse through filter
for u in range(0, P):
for v in range(0, Q):
# Get euclidean distance from point D(u,v) to the center
D_uv = np.sqrt((u - P/2 + u_k)**2 + (v - Q/2 + v_k)**2)
# Define lowpass transfer funtion according to filter type
if ftype == 'ideal':
if D_uv <= d0:
H[u, v] = 1.0
elif ftype == 'butterworth':
H[u, v] = 1/(1 + (D_uv/d0)**(2*n))
elif ftype == 'gaussian':
H[u, v] = np.exp(-D_uv**2 / (2*d0)**2)
return H
# Highpass Filter
###############################################################################
# INPUT
# shape: P x Q shape of filter
# d0: Cutoff frequency
# ftype: Filter type ('ideal', 'butterworth' or 'gaussian')
# n: Order of filter (only applies for butterworth filters)
# u_k: u position of notch pair (only applies for notch filters)
# v_k: v position of notch pair (only applies for notch filters)
###############################################################################
# OUTPUT
# H: A highpass filter with input parameters
###############################################################################
def highpass_filter(shape, d0=160, ftype='butterworth', n=2, u_k=0, v_k=0):
# Inverse of lowpass
H = 1.0 - lowpass_filter(shape, d0, ftype, n, u_k, v_k)
return H
# Bandreject Filter
###############################################################################
# INPUT
# shape: P x Q shape of filter
# d0: Cutoff frequency
# w: Width of the band
# ftype: Filter type ('ideal', 'butterworth' or 'gaussian')
# n: Order of filter (only applies for butterworth filters)
###############################################################################
# OUTPUT
# H: A bandreject filter with input parameters
###############################################################################
def bandreject_filter(shape, d0=160, w=20, ftype='butterworth', n=2):
P, Q = shape
# Initialize filter with ones
H = np.ones((P, Q))
# Traverse through filter
for u in range(0, P):
for v in range(0, Q):
# Get euclidean distance from point D(u,v) to the center
D_uv = np.sqrt((u - (P/2))**2 + (v - (Q/2))**2)
# Define bandreject transfer funtion for each filter type
if ftype == 'ideal':
if (d0 - (w/2)) <= D_uv <= (d0 + (w/2)):
H[u, v] = 0.0
elif ftype == 'butterworth':
if D_uv == d0: # To avoid dividing by zero
H[u, v] = 0
else:
H[u, v] = 1/(1 + ((D_uv*w)/(D_uv**2 - d0**2))**(2*n))
elif ftype == 'gaussian':
if D_uv == 0: # To avoid dividing by zero
H[u, v] = 1
else:
H[u, v] = 1.0 - np.exp(-((D_uv**2 - d0**2) / (D_uv * w))**2)
return H
# Bandpass Filter
###############################################################################
# INPUT
# shape: P x Q shape of filter
# d0: Cutoff frequency
# w: Width of the band
# ftype: Filter type ('ideal', 'butterworth' or 'gaussian')
# n: Order of filter (only applies for butterworth filters)
###############################################################################
# OUTPUT
# H: A bandpass filter with input parameters
###############################################################################
def bandpass_filter(shape, d0=160, w=20, ftype='butterworth', n=2):
# Inverse of bandreject
H = 1.0 - bandreject_filter(shape, d0, w, ftype, n)
return H
# Notch Reject Filter (several notch pairs not implemented)
###############################################################################
# INPUT
# shape: P x Q shape of filter
# d0: Cutoff frequency
# ftype: Filter type ('ideal', 'butterworth' or 'gaussian')
# n: Order of filter (only applies for butterworth filters)
# u_k: u position of notch pair
# v_k: v position of notch pair
###############################################################################
# OUTPUT
# H: A notch reject filter with input parameters
###############################################################################
def notch_reject_filter(shape, d0=160, ftype='butterworth', n=2, u_k=0, v_k=0):
# Form product of highpass filters at position (-u_k, -v_k) and (u_k, v_k)
H = highpass_filter(shape, d0, ftype, n, -u_k, -v_k) * highpass_filter(shape, d0, ftype, n, u_k, v_k)
return H
# Notch Pass Filter (several notch pairs not implemented)
###############################################################################
# INPUT
# shape: P x Q shape of filter
# d0: Cutoff frequency
# ftype: Filter type ('ideal', 'butterworth' or 'gaussian')
# n: Order of filter (only applies for butterworth filters)
# u_k: u position of notch pair
# v_k: v position of notch pair
###############################################################################
# OUTPUT
# H: A notch pass filter with input parameters
###############################################################################
def notch_pass_filter(shape, d0=160, ftype='butterworth', n=2, u_k=0, v_k=0):
# Inverse of notch reject
H = 1.0 - notch_reject_filter(shape, d0, ftype, n, u_k, v_k)
return H
# Filter image in frequency domain
###############################################################################
# INPUT
# img: Input image
# fclass: Filter class ('lowpass', 'highpass', 'bandreject' or 'bandpass')
# ftype: Filter type ('ideal', 'butterworth' or 'gaussian')
# d0: Cutoff frequency
# w: Width of the band (only applies for bandpass/bandreject filters)
# n: Order of filter (only applies for butterworth filters)
###############################################################################
# OUTPUT
# G: Output image
# H: Filter image
# P: Power spectrum of input image
###############################################################################
def filter_image_freq(img, fclass='lowpass', ftype='butterworth', d0=160, w=20, n=2, u_k=0, v_k=0):
# Get padding parameters
M, N = img.shape
P = 2*M
Q = 2*N
# Take the fourier transform of the image, with padding to shape P X Q
F = np.fft.fft2(img, s=(P,Q))
# Shift the low frequencies to the center.
F = np.fft.fftshift(F)
# Get power spectrum of the image
pow_spec = np.abs(F)**2
# Create a filter with input parameters
if fclass == 'lowpass':
H = lowpass_filter(F.shape, d0, ftype, n)
elif fclass == 'highpass':
H = highpass_filter(F.shape, d0, ftype, n)
elif fclass == 'bandreject':
H = bandreject_filter(F.shape, d0, w, ftype, n)
elif fclass == 'bandpass':
H = bandpass_filter(F.shape, d0, w, ftype, n)
elif fclass == 'notchreject':
H = notch_reject_filter(F.shape, d0, ftype, n, u_k, v_k)
elif fclass == 'notchpass':
H = notch_pass_filter(F.shape, d0, ftype, n, u_k, v_k)
# Form product of image with filter
G = F * H
# Shift frequencies back
G = np.fft.ifftshift(G)
# Inverse fourier transform to get output image in spatial domain
G = np.fft.ifft2(G)
# Get real values
G = np.abs(G)
# Extract M x N image from top left quadrant
G = G[0:M, 0:N]
# Return output image, the filter used and the power spectrum of input image
return G, np.abs(H), np.log(pow_spec)