def _lowpassfilter(size: Tuple[int, int], cutoff: float,
n: int) -> torch.Tensor:
r"""
Constructs a low-pass Butterworth filter.
Args:
size: Tuple with heigth and width of filter to construct
cutoff: Cutoff frequency of the filter in (0, 0.5()
n: Filter order. Higher `n` means sharper transition.
Note that `n` is doubled so that it is always an even integer.
Returns:
f = 1 / (1 + w/cutoff) ^ 2n
Note:
The frequency origin of the returned filter is at the corners.
"""
assert 0 < cutoff <= 0.5, "Cutoff frequency must be between 0 and 0.5"
assert n > 1 and int(n) == n, "n must be an integer >= 1"
grid_x, grid_y = get_meshgrid(size)
# A matrix with every pixel = radius relative to centre.
radius = torch.sqrt(grid_x**2 + grid_y**2)
return ifftshift(1. / (1.0 + (radius / cutoff)**(2 * n)))
def _log_gabor(size: Tuple[int, int], omega_0: float,
sigma_f: float) -> torch.Tensor:
r"""Creates log Gabor filter
Args:
size: size of the requires log Gabor filter
omega_0: center frequency of the filter
sigma_f: bandwidth of the filter
Returns:
log Gabor filter
"""
xx, yy = get_meshgrid(size)
radius = (xx**2 + yy**2).sqrt()
mask = radius <= 0.5
r = radius * mask
r = ifftshift(r)
r[0, 0] = 1
lg = torch.exp((-(r / omega_0).log().pow(2)) / (2 * sigma_f**2))
lg[0, 0] = 0
return lg
def _construct_filters(x: torch.Tensor,
scales: int = 4,
orientations: int = 4,
min_length: int = 6,
mult: int = 2,
sigma_f: float = 0.55,
delta_theta: float = 1.2,
k: float = 2.0):
"""Creates a stack of filters used for computation of phase congruensy maps
Args:
x: Tensor with shape (N, 1, H, W).
scales: Number of wavelets
orientations: Number of filter orientations
min_length: Wavelength of smallest scale filter
mult: Scaling factor between successive filters
sigma_f: Ratio of the standard deviation of the Gaussian
describing the log Gabor filter's transfer function
in the frequency domain to the filter center frequency.
delta_theta: Ratio of angular interval between filter orientations
and the standard deviation of the angular Gaussian function
used to construct filters in the freq. plane.
k: No of standard deviations of the noise energy beyond the mean
at which we set the noise threshold point, below which phase
congruency values get penalized.
"""
N, _, H, W = x.shape
# Calculate the standard deviation of the angular Gaussian function
# used to construct filters in the freq. plane.
theta_sigma = math.pi / (orientations * delta_theta)
# Pre-compute some stuff to speed up filter construction
grid_x, grid_y = get_meshgrid((H, W))
radius = torch.sqrt(grid_x**2 + grid_y**2)
theta = torch.atan2(-grid_y, grid_x)
# Quadrant shift radius and theta so that filters are constructed with 0 frequency at the corners.
# Get rid of the 0 radius value at the 0 frequency point (now at top-left corner)
# so that taking the log of the radius will not cause trouble.
radius = ifftshift(radius)
theta = ifftshift(theta)
radius[0, 0] = 1
sintheta = torch.sin(theta)
costheta = torch.cos(theta)
# Filters are constructed in terms of two components.
# 1) The radial component, which controls the frequency band that the filter responds to
# 2) The angular component, which controls the orientation that the filter responds to.
# The two components are multiplied together to construct the overall filter.
# First construct a low-pass filter that is as large as possible, yet falls
# away to zero at the boundaries. All log Gabor filters are multiplied by
# this to ensure no extra frequencies at the 'corners' of the FFT are
# incorporated as this seems to upset the normalisation process when
lp = _lowpassfilter(size=(H, W), cutoff=.45, n=15)
# Construct the radial filter components...
log_gabor = []
for s in range(scales):
wavelength = min_length * mult**s
omega_0 = 1.0 / wavelength
gabor_filter = torch.exp(
(-torch.log(radius / omega_0)**2) / (2 * math.log(sigma_f)**2))
gabor_filter = gabor_filter * lp
gabor_filter[0, 0] = 0
log_gabor.append(gabor_filter)
# Then construct the angular filter components...
spread = []
for o in range(orientations):
angl = o * math.pi / orientations
# For each point in the filter matrix calculate the angular distance from
# the specified filter orientation. To overcome the angular wrap-around
# problem sine difference and cosine difference values are first computed
# and then the atan2 function is used to determine angular distance.
ds = sintheta * math.cos(angl) - costheta * math.sin(
angl) # Difference in sine.
dc = costheta * math.cos(angl) + sintheta * math.sin(
angl) # Difference in cosine.
dtheta = torch.abs(torch.atan2(ds, dc))
spread.append(torch.exp((-dtheta**2) / (2 * theta_sigma**2)))
spread = torch.stack(spread)
log_gabor = torch.stack(log_gabor)
# Multiply, add batch dimension and transfer to correct device.
filters = (spread.repeat_interleave(scales, dim=0) *
log_gabor.repeat(orientations, 1, 1)).unsqueeze(0).to(x)
return filters
