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rectification.py
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rectification.py
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from pylab import *
from misc import *
import scipy.optimize as opt
def planar_homography(F_ft, xy_from, xy_to):
# Find the orienting homography
epipole = homogeneous(kernel(F_ft, both = True)[1])
amin, ier = opt.leastsq(lambda a: planar_homography_error(a, F_ft, epipole, xy_from, xy_to), array([0, 0, 0]),
xtol = 1e-12, ftol = 1e-12, maxfev = 15000)
H = dot(skew_symmetric(epipole), F_ft) - dot(epipole.reshape(3, 1), amin.reshape(1, 3))
return H, epipole
def planar_homography_error(a, F, e2, xy1, xy2):
H = dot(skew_symmetric(e2), F) - dot(e2.reshape(3, 1), a.reshape(1, 3))
fit = homogeneous(dot(xy1, H.T))
return sqrt(sum((fit - xy2)**2, axis = 1))
def setup_rectification(images, F, xy1, xy2):
w, h = images.shape[2], images.shape[1]
e1, e2 = kernel(F, both = True)
# Choose second image as reference
F_ft, xy_from, xy_to = F, xy1, xy2
reference_image = 1
rectified = False
if at_infinity(e2):
if at_infinity(e1):
# If both images are rectified, do nothing
rectified = True
else:
# If only the second epipole is infinite, swap the images
F_ft, xy_from, xy_to = F_ft.T, xy_to, xy_from
reference_image = 0
if not at_infinity(e1) and not at_infinity(e2):
# Transform to the image with the epipole closer to the image center
e1, e2 = homogeneous(e1), homogeneous(e2)
dist1 = (e1[0] - w / 2)**2 + (e1[1] - h / 2)**2
dist2 = (e2[0] - w / 2)**2 + (e2[1] - h / 2)**2
if dist2 > dist1:
F_ft, xy_from, xy_to = F_ft.T, xy_to, xy_from
reference_image = 0
return reference_image, F_ft, xy_from, xy_to, rectified
def homography_transform(image, H, emptyflag = 1e3, samplingfunc = bilinear_color):
w, h = image.shape[1], image.shape[0]
bounds = array([[0, 0, 1], [w - 1, 0, 1], [w - 1, h - 1, 1], [0, h - 1, 1]])
bounds_H = homogeneous(dot(bounds, H.T))
minx, miny = int(floor(min(bounds_H[:, 0]))), int(floor(min(bounds_H[:, 1])))
maxx, maxy = int(ceil(max(bounds_H[:, 0]))), int(ceil(max(bounds_H[:, 1])))
Hinv = inv(H)
i, j = mgrid[minx:maxx, miny:maxy]
ij = vstack((i.flatten(), j.flatten(), ones(i.shape).flatten())).T
ij_H = homogeneous(dot(ij, Hinv.T))
# Transform the second image to be oriented with the first
image_H = tile(emptyflag, (maxy - miny, maxx - minx))
for i in range(ij.shape[0]):
if point_in_quad(bounds, ij_H[i]):
image_H[ij[i, 1] - miny, ij[i, 0] - minx] = samplingfunc(image, ij_H[i, 0], ij_H[i, 1])
del i, j, ij, ij_H
return image_H, bounds, bounds_H
def common_region(epipole, bounds_from, bounds_to):
# Find oriented epipolar lines to the corners of both images and their angles
lines1 = cross(epipole, bounds_from)
lines2 = cross(epipole, bounds_to)
angles1 = array([arctan2(-lines1[i, 0], lines1[i, 1]) for i in range(4)])
angles2 = array([arctan2(-lines2[i, 0], lines2[i, 1]) for i in range(4)])
minangle1, maxangle1 = min(angles1), max(angles1)
minangle2, maxangle2 = min(angles2), max(angles2)
# Find the common region
if point_in_quad(bounds_from, epipole):
if point_in_quad(bounds_to, epipole):
minangle, maxangle = -pi, pi
else:
minangle, maxangle = minangle2, maxangle2
elif point_in_quad(bounds_to, epipole):
minangle, maxangle = minangle1, maxangle1
else:
if abs(minangle1 - minangle2) <= pi:
minangle = max(minangle1, minangle2)
else:
minangle = min(minangle1, minangle2)
if abs(maxangle1 - maxangle2) <= pi:
maxangle = min(maxangle1, maxangle2)
else:
maxangle = max(maxangle1, maxangle2)
flip = False
if argmin(angles1) >= 2 and argmin(angles2) >= 2:
flip = True
if maxangle - minangle > pi and maxangle - minangle < 2 * pi:
minangle, maxangle = maxangle, minangle + 2 * pi
flip = True
return minangle, maxangle, flip
def rectification_table(H, epipole, w, h):
# Prepare the rectification table
bounds_to = array([[0, 0, 1], [w - 1, 0, 1], [w - 1, h - 1, 1], [0, h - 1, 1]])
bounds_from = homogeneous(dot(bounds_to, H.T))
box1 = cross(bounds_from, bounds_from[r_[1:4, 0]])
box2 = cross(bounds_to, bounds_to[r_[1:4, 0]])
# Find the common region in both images
minangle, maxangle, flip = common_region(epipole, bounds_from, bounds_to)
# Set the rectification table up
rectify = array([])
mindist, maxdist = 1e50, -1e50
angle = minangle
step = 1e50
while angle <= maxangle:
rectify = append(rectify, angle)
# Find the epipolar line
m = tan(angle)
l = array([m, -1, epipole[1] - m * epipole[0]]) # mx - y - b = 0
# Intersection with first image and their distances
points = homogeneous(cross(box1, l))
intersection = array([point_in_quad(bounds_from, xy) for xy in points])
dist = sqrt(sum((points[intersection] - epipole)**2, axis = 1))
if dist.shape[0] > 0:
mindist = min(mindist, min(dist))
maxdist = max(maxdist, max(dist))
step = arctan2(1.0, max(dist))
# Intersection with second image and their distances
points = homogeneous(cross(box2, l))
intersection = array([point_in_quad(bounds_to, xy) for xy in points])
dist = sqrt(sum((points[intersection] - epipole)**2, axis = 1))
if dist.shape[0] > 0:
mindist = min(mindist, min(dist))
maxdist = max(maxdist, max(dist))
step = min(step, arctan2(1.0, max(dist)))
angle = angle + step
return rectify, flip, mindist, maxdist
def rectify_image(image, Hinv, epipole, rect_table, emptyflag = 1e3, samplingfunc = bilinear_color):
rectify, flip, mindist, maxdist = rect_table
xsize = int(ceil(maxdist - mindist))
ysize = rectify.shape[0]
rect = tile(emptyflag, (ysize, xsize))
for y in range(ysize):
# Find the epipolar line angle
angle = rectify[y]
dx = cos(angle)
dy = sin(angle)
# Fill the current scanline
pixel0 = array([epipole[0] + dx * mindist, epipole[1] + dy * mindist, 1])
for x in range(xsize):
pixel = homogeneous(dot(Hinv, pixel0))
if pixel[0] >= 0 and pixel[0] < image.shape[1] and pixel[1] >= 0 and pixel[1] < image.shape[0]:
rect[y, x] = samplingfunc(image, pixel[0], pixel[1])
pixel0[0] += dx
pixel0[1] += dy
if flip:
rect = flipud(fliplr(rect))
return rect
def unrectify_image(image, H, epipole, w, h, rect_table, emptyflag = 1e3, samplingfunc = bilinear_color):
rectify, flip, mindist, maxdist = rect_table
if flip:
image = flipud(fliplr(image))
# Fill in the unrectified image
unrect = tile(emptyflag, (h, w))
for y in range(h):
for x in range(w):
p = homogeneous(dot(H, array([x, y, 1])))
if p[0] >= 0 and p[0] < w and p[1] >= 0 and p[1] < h:
l = cross(epipole, p)
angle = arctan2(-l[0], l[1])
if flip and angle < 0:
angle += 2 * pi
# Find the distance to the boundary in the direction of the epipole
dist = sqrt(sum((p - epipole)**2)) - mindist
# Find the linearly interpolated y from the angle
y_1 = argmin(abs(rectify - angle))
angle_1 = rectify[y_1]
if angle > angle_1: y_2 = min(y_1 + 1, rectify.shape[0] - 1)
else: y_2 = max(0, y_1 - 1)
angle_2 = rectify[y_2]
if y_1 == y_2: yrect = y_1
else: yrect = (abs(angle - angle_1) * y_2 + abs(angle_2 - angle) * y_1) / abs(angle_2 - angle_1)
unrect[y, x] = samplingfunc(image, dist, yrect)
return unrect
def mapping_homography(self, xy1, xy2):
A = vstack([array(
[[xy1[i, 0], xy1[i, 1], 1, 0, 0, 0, -xy1[i, 0] * xy2[i, 0], -xy1[i, 1] * xy2[i, 0], -xy2[i, 0]],
[0, 0, 0, xy1[i, 0], xy1[i, 1], 1, -xy1[i, 0] * xy2[i, 1], -xy1[i, 1] * xy2[i, 1], -xy2[i, 1]]])
for i in range(xy1.shape[0])])
return kernel(A).reshape(3, 3)