Commit a63d2044 authored by André Anjos's avatar André Anjos 💬

Fixes to Maximum Curvature implementation; Improved references & documentation

parent ee00ee5d
......@@ -3,14 +3,10 @@
import math
import numpy
import bob.core
import scipy.ndimage
import bob.io.base
from bob.bio.base.extractor import Extractor
from .. import utils
class MaximumCurvature (Extractor):
"""
......@@ -18,12 +14,15 @@ class MaximumCurvature (Extractor):
Based on N. Miura, A. Nagasaka, and T. Miyatake, Extraction of Finger-Vein
Pattern Using Maximum Curvature Points in Image Profiles. Proceedings on IAPR
conference on machine vision applications, 9 (2005), pp. 347--350
conference on machine vision applications, 9 (2005), pp. 347--350.
**Parameters:**
Parameters:
sigma (:py:class:`int`, optional): standard deviation for the gaussian
smoothing kernel used to denoise the input image. The width of the
gaussian kernel will be set automatically to 4x this value (in pixels).
sigma : :py:class:`int`
Optional: Sigma used for determining derivatives.
"""
......@@ -32,227 +31,477 @@ class MaximumCurvature (Extractor):
self.sigma = sigma
def maximum_curvature(self, image, mask):
"""Computes and returns the Maximum Curvature features for the given input
fingervein image"""
def detect_valleys(self, image, mask):
"""Detects valleys on the image respecting the mask
This step corresponds to Step 1-1 in the original paper. The objective is,
for all 4 cross-sections (z) of the image (horizontal, vertical, 45 and -45
diagonals), to compute the following proposed valley detector as defined in
Equation 1, page 348:
.. math::
\kappa(z) = \\frac{d^2P_f(z)/dz^2}{(1 + (dP_f(z)/dz)^2)^\\frac{3}{2}}
We start the algorithm by smoothing the image with a 2-dimensional gaussian
filter. The equation that defines the kernel for the filter is:
.. math::
\mathcal{N}(x,y)=\\frac{1}{2\pi\sigma^2}e^\\frac{-(x^2+y^2)}{2\sigma^2}
This is done to avoid noise from the raw data (from the sensor). The
maximum curvature method then requires we compute the first and second
derivative of the image for all cross-sections, as per the equation above.
We instead take the following equivalent approach:
1. construct a gaussian filter
2. take the first (dh/dx) and second (d^2/dh^2) deritivatives of the filter
3. calculate the first and second derivatives of the smoothed signal using
the results from 3. This is done for all directions we're interested in:
horizontal, vertical and 2 diagonals. First and second derivatives of a
convolved signal
.. note::
Item 3 above is only possible thanks to the steerable filter property of
the gaussian kernel. See "The Design and Use of Steerable Filters" from
Freeman and Adelson, IEEE Transactions on Pattern Analysis and Machine
Intelligence, Vol. 13, No. 9, September 1991.
Parameters:
finger_mask = numpy.zeros(mask.shape)
finger_mask[mask == True] = 1
image (numpy.ndarray): an array of 64-bit floats containing the input
image
mask (numpy.ndarray): an array, of the same size as ``image``, containing
a mask (booleans) indicating where the finger is on ``image``.
winsize = numpy.ceil(4*self.sigma)
x = numpy.arange(-winsize, winsize+1)
y = numpy.arange(-winsize, winsize+1)
X, Y = numpy.meshgrid(x, y)
Returns:
h = (1/(2*math.pi*self.sigma**2))*numpy.exp(-(X**2 + Y**2)/(2*self.sigma**2))
hx = (-X/(self.sigma**2))*h
hxx = ((X**2 - self.sigma**2)/(self.sigma**4))*h
hy = hx.T
hyy = hxx.T
hxy = ((X*Y)/(self.sigma**4))*h
numpy.ndarray: a 3-dimensional array of 64-bits containing $\kappa$ for
all considered directions. $\kappa$ has the same shape as ``image``,
except for the 3rd. dimension, which provides planes for the
cross-section valley detections for each of the contemplated directions,
in this order: horizontal, vertical, +45 degrees, -45 degrees.
# Do the actual filtering
"""
fx = utils.imfilter(image, hx)
fxx = utils.imfilter(image, hxx)
fy = utils.imfilter(image, hy)
fyy = utils.imfilter(image, hyy)
fxy = utils.imfilter(image, hxy)
# 1. constructs the 2D gaussian filter "h" given the window size,
# extrapolated from the "sigma" parameter (4x)
# N.B.: This is a text-book gaussian filter definition
winsize = numpy.ceil(4*self.sigma) #enough space for the filter
window = numpy.arange(-winsize, winsize+1)
X, Y = numpy.meshgrid(window, window)
G = 1.0 / (2*math.pi*self.sigma**2)
G *= numpy.exp(-(X**2 + Y**2) / (2*self.sigma**2))
f1 = 0.5*numpy.sqrt(2)*(fx + fy) # \ #
f2 = 0.5*numpy.sqrt(2)*(fx - fy) # / #
f11 = 0.5*fxx + fxy + 0.5*fyy # \\ #
f22 = 0.5*fxx - fxy + 0.5*fyy # // #
# 2. calculates first and second derivatives of "G" with respect to "X"
# (0), "Y" (90 degrees) and 45 degrees (?)
G1_0 = (-X/(self.sigma**2))*G
G2_0 = ((X**2 - self.sigma**2)/(self.sigma**4))*G
G1_90 = G1_0.T
G2_90 = G2_0.T
hxy = ((X*Y)/(self.sigma**4))*G
# 3. calculates derivatives w.r.t. to all directions of interest
# stores results in the variable "k". The entries (last dimension) in k
# correspond to curvature detectors in the following directions:
#
# [0] horizontal
# [1] vertical
# [2] diagonal \ (45 degrees rotation)
# [3] diagonal / (-45 degrees rotation)
image_g1_0 = scipy.ndimage.convolve(image, G1_0, mode='nearest')
image_g2_0 = scipy.ndimage.convolve(image, G2_0, mode='nearest')
image_g1_90 = scipy.ndimage.convolve(image, G1_90, mode='nearest')
image_g2_90 = scipy.ndimage.convolve(image, G2_90, mode='nearest')
fxy = scipy.ndimage.convolve(image, hxy, mode='nearest')
# support calculation for diagonals, given the gaussian kernel is
# steerable. To calculate the derivatives for the "\" diagonal, we first
# **would** have to rotate the image 45 degrees counter-clockwise (so the
# diagonal lies on the horizontal axis). Using the steerable property, we
# can evaluate the first derivative like this:
#
# image_g1_45 = cos(45)*image_g1_0 + sin(45)*image_g1_90
# = sqrt(2)/2*fx + sqrt(2)/2*fx
#
# to calculate the first derivative for the "/" diagonal, we first
# **would** have to rotate the image -45 degrees "counter"-clockwise.
# Therefore, we can calculate it like this:
#
# image_g1_m45 = cos(-45)*image_g1_0 + sin(-45)*image_g1_90
# = sqrt(2)/2*image_g1_0 - sqrt(2)/2*image_g1_90
#
image_g1_45 = 0.5*numpy.sqrt(2)*(image_g1_0 + image_g1_90)
image_g1_m45 = 0.5*numpy.sqrt(2)*(image_g1_0 - image_g1_90)
# NOTE: You can't really get image_g2_45 and image_g2_m45 from the theory
# of steerable filters. In contact with B.Ton, he suggested the following
# material, where that is explained: Chapter 5.2.3 of van der Heijden, F.
# (1994) Image based measurement systems: object recognition and parameter
# estimation. John Wiley & Sons Ltd, Chichester. ISBN 978-0-471-95062-2
# This also shows the same result:
# http://www.mif.vu.lt/atpazinimas/dip/FIP/fip-Derivati.html (look for
# SDGD)
# He also suggested to look at slide 75 of the following presentation
# indicating it is self-explanatory: http://slideplayer.com/slide/5084635/
image_g2_45 = 0.5*image_g2_0 + fxy + 0.5*image_g2_90
image_g2_m45 = 0.5*image_g2_0 - fxy + 0.5*image_g2_90
img_h, img_w = image.shape #Image height and width
# Calculate curvatures
k = numpy.zeros((img_h, img_w, 4))
k[:,:,0] = (fxx/((1 + fx**2)**(3/2)))*finger_mask # hor #
k[:,:,1] = (fyy/((1 + fy**2)**(3/2)))*finger_mask # ver #
k[:,:,2] = (f11/((1 + f1**2)**(3/2)))*finger_mask # \ #
k[:,:,3] = (f22/((1 + f2**2)**(3/2)))*finger_mask # / #
# ######################################################################
# [Step 1-1] Calculation of curvature profiles
# ######################################################################
# Peak detection (k or kappa) calculation as per equation (1) page 348 on
# Miura's paper
finger_mask = mask.astype('float64')
return numpy.dstack([
(image_g2_0 / ((1 + image_g1_0**2)**(1.5)) ) * finger_mask,
(image_g2_90 / ((1 + image_g1_90**2)**(1.5)) ) * finger_mask,
(image_g2_45 / ((1 + image_g1_45**2)**(1.5)) ) * finger_mask,
(image_g2_m45 / ((1 + image_g1_m45**2)**(1.5))) * finger_mask,
])
def eval_vein_probabilities(self, k):
'''Evaluates joint vein centre probabilities from cross-sections
This function will take $\kappa$ and will calculate the vein centre
probabilities taking into consideration valley widths and depths. It
aggregates the following steps from the paper:
* [Step 1-2] Detection of the centres of veins
* [Step 1-3] Assignment of scores to the centre positions
* [Step 1-4] Calculation of all the profiles
Once the arrays of curvatures (concavities) are calculated, here is how
detection works: The code scans the image in a precise direction (vertical,
horizontal, diagonal, etc). It tries to find a concavity on that direction
and measure its width (see Wr on Figure 3 on the original paper). It then
identifies the centers of the concavity and assign a value to it, which
depends on its width (Wr) and maximum depth (where the peak of darkness
occurs) in such a concavity. This value is accumulated on a variable (Vt),
which is re-used for all directions. Vt represents the vein probabilites
from the paper.
Parameters:
k (numpy.ndarray): a 3-dimensional array of 64-bits containing $\kappa$
for all considered directions. $\kappa$ has the same shape as
``image``, except for the 3rd. dimension, which provides planes for the
cross-section valley detections for each of the contemplated
directions, in this order: horizontal, vertical, +45 degrees, -45
degrees.
Returns:
numpy.ndarray: The un-accumulated vein centre probabilities ``V``. This
is a 3D array with 64-bit floats with the same dimensions of the input
array ``k``. You must accumulate (sum) over the last dimension to
retrieve the variable ``V`` from the paper.
'''
V = numpy.zeros_like(k)
def _prob_1d(a):
'''Finds "vein probabilities" in a 1-D signal
This function efficiently counts the width and height of concavities in
the cross-section (1-D) curvature signal ``s``.
It works like this:
1. We create a 1-shift difference between the thresholded signal and
itself
2. We compensate for starting and ending regions
3. For each sequence of start/ends, we compute the maximum in the
original signal
Example (mixed with pseudo-code):
a = 0 1 2 3 2 1 0 -1 0 0 1 2 5 2 2 2 1
b = a > 0 (as type int)
b = 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1
0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1
0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 (-)
-------------------------------------------
X 1 0 0 0 0 -1 0 0 0 1 0 0 0 0 0 0 X (length is smaller than orig.)
starts = numpy.where(diff > 0)
ends = numpy.where(diff < 0)
-> now the number of starts and ends should match, otherwise, we must
compensate
-> case 1: b starts with 1: add one start in begin of "starts"
-> case 2: b ends with 1: add one end in the end of "ends"
-> iterate over the sequence of starts/ends and find maximums
Parameters:
a (numpy.ndarray): 1D signal with curvature to explore
Returns:
numpy.ndarray: 1D container with the vein centre probabilities
'''
b = (a > 0).astype(int)
diff = b[1:] - b[:-1]
starts = numpy.argwhere(diff > 0)
starts += 1 #compensates for shifted different
ends = numpy.argwhere(diff < 0)
ends += 1 #compensates for shifted different
if b[0]: starts = numpy.insert(starts, 0, 0)
if b[-1]: ends = numpy.append(ends, len(a))
z = numpy.zeros_like(a)
if starts.size == 0 and ends.size == 0: return z
for start, end in zip(starts, ends):
maximum = numpy.argmax(a[int(start):int(end)])
z[start+maximum] = a[start+maximum] * (end-start)
return z
# Scores
Vt = numpy.zeros(image.shape)
Wr = 0
# Horizontal direction
bla = k[:,:,0] > 0
for y in range(0,img_h):
for x in range(0,img_w):
if (bla[y,x]):
Wr = Wr + 1
if ( Wr > 0 and (x == (img_w-1) or not bla[y,x]) ):
if (x == (img_w-1)):
# Reached edge of image
pos_end = x
else:
pos_end = x - 1
pos_start = pos_end - Wr + 1 # Start pos of concave
if (pos_start == pos_end):
I=numpy.argmax(k[y,pos_start,0])
else:
I=numpy.argmax(k[y,pos_start:pos_end+1,0])
pos_max = pos_start + I
Scr = k[y,pos_max,0]*Wr
Vt[y,pos_max] = Vt[y,pos_max] + Scr
Wr = 0
for index in range(k.shape[0]):
V[index,:,0] += _prob_1d(k[index,:,0])
# Vertical direction
for index in range(k.shape[1]):
V[:,index,1] += _prob_1d(k[:,index,1])
# Direction: 45 degrees (\)
curv = k[:,:,2]
i,j = numpy.indices(curv.shape)
for index in range(-curv.shape[0]+1, curv.shape[1]):
V[i==(j-index),2] += _prob_1d(curv.diagonal(index))
# Direction: -45 degrees (/)
# NOTE: due to the way the access to the diagonals are implemented, in this
# loop, we operate bottom-up. To match this behaviour, we also address V
# through Vud.
curv = numpy.flipud(k[:,:,3]) #required so we get "/" diagonals correctly
Vud = numpy.flipud(V) #match above inversion
for index in reversed(range(curv.shape[1]-1, -curv.shape[0], -1)):
Vud[i==(j-index),3] += _prob_1d(curv.diagonal(index))
return V
def connect_centres(self, V):
"""Connects vein centres by filtering vein probabilities ``V``
This function does the equivalent of Step 2 / Equation 4 at Miura's paper.
The operation is applied on a row from the ``V`` matrix, which may be
acquired horizontally, vertically or on a diagonal direction. The pixel
value is then reset in the center of a windowing operation (width = 5) with
the following value:
.. math::
b[w] = min(max(a[w+1], a[w+2]) + max(a[w-1], a[w-2]))
Parameters:
V (numpy.ndarray): The accumulated vein centre probabilities ``V``. This
is a 2D array with 64-bit floats and is defined by Equation (3) on the
paper.
Returns:
numpy.ndarray: A 3-dimensional 64-bit array ``Cd`` containing the result
of the filtering operation for each of the directions. ``Cd`` has the
dimensions of $\kappa$ and $V_i$. Each of the planes correspond to the
horizontal, vertical, +45 and -45 directions.
"""
def _connect_1d(a):
'''Connects centres in the given vector
The strategy we use to vectorize this is to shift a twice to the left and
twice to the right and apply a vectorized operation to compute the above.
Parameters:
a (numpy.ndarray): Input 1D array which will be window scanned
Returns:
numpy.ndarray: Output 1D array (must be writeable), in which we will
set the corrected pixel values after the filtering above. Notice that,
given the windowing operation, the returned array size would be 4 short
of the input array.
'''
return numpy.amin([numpy.amax([a[3:-1], a[4:]], axis=0),
numpy.amax([a[1:-3], a[:-4]], axis=0)], axis=0)
Cd = numpy.zeros(V.shape + (4,), dtype='float64')
# Horizontal direction
for index in range(V.shape[0]):
Cd[index, 2:-2, 0] = _connect_1d(V[index,:])
# Vertical direction
bla = k[:,:,1] > 0
for x in range(0,img_w):
for y in range(0,img_h):
if (bla[y,x]):
Wr = Wr + 1
if ( Wr > 0 and (y == (img_h-1) or not bla[y,x]) ):
if (y == (img_h-1)):
# Reached edge of image
pos_end = y
else:
pos_end = y - 1
pos_start = pos_end - Wr + 1 # Start pos of concave
if (pos_start == pos_end):
I=numpy.argmax(k[pos_start,x,1])
else:
I=numpy.argmax(k[pos_start:pos_end+1,x,1])
pos_max = pos_start + I
Scr = k[pos_max,x,1]*Wr
Vt[pos_max,x] = Vt[pos_max,x] + Scr
Wr = 0
# Direction: \ #
bla = k[:,:,2] > 0
for start in range(0,img_w+img_h-1):
# Initial values
if (start <= img_w-1):
x = start
y = 0
else:
x = 0
y = start - img_w + 1
done = False
while (not done):
if(bla[y,x]):
Wr = Wr + 1
if ( Wr > 0 and (y == img_h-1 or x == img_w-1 or not bla[y,x]) ):
if (y == img_h-1 or x == img_w-1):
# Reached edge of image
pos_x_end = x
pos_y_end = y
else:
pos_x_end = x - 1
pos_y_end = y - 1
pos_x_start = pos_x_end - Wr + 1
pos_y_start = pos_y_end - Wr + 1
if (pos_y_start == pos_y_end and pos_x_start == pos_x_end):
d = k[pos_y_start, pos_x_start, 2]
elif (pos_y_start == pos_y_end):
d = numpy.diag(k[pos_y_start, pos_x_start:pos_x_end+1, 2])
elif (pos_x_start == pos_x_end):
d = numpy.diag(k[pos_y_start:pos_y_end+1, pos_x_start, 2])
else:
d = numpy.diag(k[pos_y_start:pos_y_end+1, pos_x_start:pos_x_end+1, 2])
I = numpy.argmax(d)
pos_x_max = pos_x_start + I
pos_y_max = pos_y_start + I
Scr = k[pos_y_max,pos_x_max,2]*Wr
Vt[pos_y_max,pos_x_max] = Vt[pos_y_max,pos_x_max] + Scr
Wr = 0
if((x == img_w-1) or (y == img_h-1)):
done = True
else:
x = x + 1
y = y + 1
# Direction: /
bla = k[:,:,3] > 0
for start in range(0,img_w+img_h-1):
# Initial values
if (start <= (img_w-1)):
x = start
y = img_h-1
else:
x = 0
y = img_w+img_h-start-1
done = False
while (not done):
if(bla[y,x]):
Wr = Wr + 1
if ( Wr > 0 and (y == 0 or x == img_w-1 or not bla[y,x]) ):
if (y == 0 or x == img_w-1):
# Reached edge of image
pos_x_end = x
pos_y_end = y
else:
pos_x_end = x - 1
pos_y_end = y + 1
pos_x_start = pos_x_end - Wr + 1
pos_y_start = pos_y_end + Wr - 1
if (pos_y_start == pos_y_end and pos_x_start == pos_x_end):
d = k[pos_y_end, pos_x_start, 3]
elif (pos_y_start == pos_y_end):
d = numpy.diag(numpy.flipud(k[pos_y_end, pos_x_start:pos_x_end+1, 3]))
elif (pos_x_start == pos_x_end):
d = numpy.diag(numpy.flipud(k[pos_y_end:pos_y_start+1, pos_x_start, 3]))
else:
d = numpy.diag(numpy.flipud(k[pos_y_end:pos_y_start+1, pos_x_start:pos_x_end+1, 3]))
I = numpy.argmax(d)
pos_x_max = pos_x_start + I
pos_y_max = pos_y_start - I
Scr = k[pos_y_max,pos_x_max,3]*Wr
Vt[pos_y_max,pos_x_max] = Vt[pos_y_max,pos_x_max] + Scr
Wr = 0
if((x == img_w-1) or (y == 0)):
done = True
else:
x = x + 1
y = y - 1
## Connection of vein centres
Cd = numpy.zeros((img_h, img_w, 4))
for x in range(2,img_w-3):
for y in range(2,img_h-3):
Cd[y,x,0] = min(numpy.amax(Vt[y,x+1:x+3]), numpy.amax(Vt[y,x-2:x])) # Hor #
Cd[y,x,1] = min(numpy.amax(Vt[y+1:y+3,x]), numpy.amax(Vt[y-2:y,x])) # Vert #
Cd[y,x,2] = min(numpy.amax(Vt[y-2:y,x-2:x]), numpy.amax(Vt[y+1:y+3,x+1:x+3])) # \ #
Cd[y,x,3] = min(numpy.amax(Vt[y+1:y+3,x-2:x]), numpy.amax(Vt[y-2:y,x+1:x+3])) # / #
#Veins
img_veins = numpy.amax(Cd,axis=2)
# Binarise the vein image
md = numpy.median(img_veins[img_veins>0])
img_veins_bin = img_veins > md
return img_veins_bin.astype(numpy.float64)
for index in range(V.shape[1]):
Cd[2:-2, index, 1] = _connect_1d(V[:,index])
# Direction: 45 degrees (\)
i,j = numpy.indices(V.shape)
border = numpy.zeros((2,), dtype='float64')
for index in range(-V.shape[0]+5, V.shape[1]-4):
# NOTE: hstack **absolutately** necessary here as double indexing after
# array indexing is **not** possible with numpy (it returns a copy)
Cd[:,:,2][i==(j-index)] = numpy.hstack([border,
_connect_1d(V.diagonal(index)), border])
# Direction: -45 degrees (/)
Vud = numpy.flipud(V)
Cdud = numpy.flipud(Cd[:,:,3])
for index in reversed(range(V.shape[1]-5, -V.shape[0]+4, -1)):
# NOTE: hstack **absolutately** necessary here as double indexing after
# array indexing is **not** possible with numpy (it returns a copy)
Cdud[:,:][i==(j-index)] = numpy.hstack([border,
_connect_1d(Vud.diagonal(index)), border])
return Cd
def binarise(self, G):
"""Binarise vein images using a threshold assuming distribution is diphasic
This function implements Step 3 of the paper. It binarises the 2-D array
``G`` assuming its histogram is mostly diphasic and using a median value.
Parameters:
G (numpy.ndarray): A 2-dimensional 64-bit array ``G`` containing the
result of the filtering operation. ``G`` has the dimensions of the
original image.
Returns:
numpy.ndarray: A 2-dimensional 64-bit float array with the same
dimensions of the input image, but containing its vein-binarised version.
The output of this function corresponds to the output of the method.
"""
median = numpy.median(G[G>0])
Gbool = G > median
return Gbool.astype(numpy.float64)
def _view_four(self, k, suptitle):
'''Display four plots using matplotlib'''
import matplotlib.pyplot as plt
k[k<=0] = 0
k /= k.max()
plt.subplot(2,2,1)
plt.imshow(k[...,0], cmap='gray')
plt.title('Horizontal')
plt.subplot(2,2,2)
plt.imshow(k[...,1], cmap='gray')
plt.title('Vertical')
plt.subplot(2,2,3)
plt.imshow(k[...,2], cmap='gray')
plt.title('+45 degrees')
plt.subplot(2,2,4)
plt.imshow(k[...,3], cmap='gray')
plt.title('-45 degrees')
plt.suptitle(suptitle)
plt.tight_layout()
plt.show()
def _view_single(self, k, title):
'''Displays a single plot using matplotlib'''
import matplotlib.pyplot as plt
plt.imshow(k, cmap='gray')
plt.title(title)
plt.tight_layout()
plt.show()
def __call__(self, image):
"""Reads the input image, extract the features based on Maximum Curvature of the fingervein image, and writes the resulting template"""
finger_image = image[0] #Normalized image with or without histogram equalization
finger_image = image[0]
finger_mask = image[1]
return self.maximum_curvature(finger_image, finger_mask)
import time
start = time.time()
kappa = self.detect_valleys(finger_image, finger_mask)
#self._view_four(kappa, "Valley Detectors - $\kappa$")