这是我在阅读论文后尝试回答我的问题
我在 SAGE 中处理了一段时间的方程，并提出了这个 c 风格的伪代码：
// in case it matters: licensed under GPLv2 or later
// legend:
// sqr(x) = x*x
// sqrt(x) = square root of x
// let m1x,m1y ... m4x,m4y be the (x,y) pixel coordinates
// of the 4 corners of the detected quadrangle
// i.e. (m1x, m1y) are the cordinates of the first corner,
// (m2x, m2y) of the second corner and so on.
// let u0, v0 be the pixel coordinates of the principal point of the image
// for a normal camera this will be the center of the image,
// i.e. u0=IMAGEWIDTH/2; v0 =IMAGEHEIGHT/2
// This assumption does not hold if the image has been cropped asymmetrically
// first, transform the image so the principal point is at (0,0)
// this makes the following equations much easier
m1x = m1x - u0;
m1y = m1y - v0;
m2x = m2x - u0;
m2y = m2y - v0;
m3x = m3x - u0;
m3y = m3y - v0;
m4x = m4x - u0;
m4y = m4y - v0;
// temporary variables k2, k3
double k2 = ((m1y - m4y)*m3x - (m1x - m4x)*m3y + m1x*m4y - m1y*m4x) /
((m2y - m4y)*m3x - (m2x - m4x)*m3y + m2x*m4y - m2y*m4x) ;
double k3 = ((m1y - m4y)*m2x - (m1x - m4x)*m2y + m1x*m4y - m1y*m4x) /
((m3y - m4y)*m2x - (m3x - m4x)*m2y + m3x*m4y - m3y*m4x) ;
// f_squared is the focal length of the camera, squared
// if k2==1 OR k3==1 then this equation is not solvable
// if the focal length is known, then this equation is not needed
// in that case assign f_squared= sqr(focal_length)
double f_squared =
-((k3*m3y - m1y)*(k2*m2y - m1y) + (k3*m3x - m1x)*(k2*m2x - m1x)) /
((k3 - 1)*(k2 - 1)) ;
//The width/height ratio of the original rectangle
double whRatio = sqrt(
(sqr(k2 - 1) + sqr(k2*m2y - m1y)/f_squared + sqr(k2*m2x - m1x)/f_squared) /
(sqr(k3 - 1) + sqr(k3*m3y - m1y)/f_squared + sqr(k3*m3x - m1x)/f_squared)
) ;
// if k2==1 AND k3==1, then the focal length equation is not solvable
// but the focal length is not needed to calculate the ratio.
// I am still trying to figure out under which circumstances k2 and k3 become 1
// but it seems to be when the rectangle is not distorted by perspective,
// i.e. viewed straight on. Then the equation is obvious:
if (k2==1 && k3==1) whRatio = sqrt(
(sqr(m2y-m1y) + sqr(m2x-m1x)) /
(sqr(m3y-m1y) + sqr(m3x-m1x))
// After testing, I found that the above equations
// actually give the height/width ratio of the rectangle,
// not the width/height ratio.
// If someone can find the error that caused this,
// I would be most grateful.
// until then:
whRatio = 1/whRatio;
更新：这里是这些方程是如何确定的：
以下是SAGE中的代码。可通过http://www.sagenb.org/home/pub/704/在线访问。（Sage 在求解方程方面非常有用，并且可以在任何浏览器中使用，请查看）
# CALCULATING THE ASPECT RATIO OF A RECTANGLE DISTORTED BY PERSPECTIVE
#
# BIBLIOGRAPHY:
# [zhang-single]: "Single-View Geometry of A Rectangle
# With Application to Whiteboard Image Rectification"
# by Zhenggyou Zhang
# http://research.microsoft.com/users/zhang/Papers/WhiteboardRectification.pdf
# pixel coordinates of the 4 corners of the quadrangle (m1, m2, m3, m4)
# see [zhang-single] figure 1
m1x = var('m1x')
m1y = var('m1y')
m2x = var('m2x')
m2y = var('m2y')
m3x = var('m3x')
m3y = var('m3y')
m4x = var('m4x')
m4y = var('m4y')
# pixel coordinates of the principal point of the image
# for a normal camera this will be the center of the image,
# i.e. u0=IMAGEWIDTH/2; v0 =IMAGEHEIGHT/2
# This assumption does not hold if the image has been cropped asymmetrically
u0 = var('u0')
v0 = var('v0')
# pixel aspect ratio; for a normal camera pixels are square, so s=1
s = var('s')
# homogenous coordinates of the quadrangle
m1 = vector ([m1x,m1y,1])
m2 = vector ([m2x,m2y,1])
m3 = vector ([m3x,m3y,1])
m4 = vector ([m4x,m4y,1])
# the following equations are later used in calculating the the focal length
# and the rectangle's aspect ratio.
# temporary variables: k2, k3, n2, n3
# see [zhang-single] Equation 11, 12
k2_ = m1.cross_product(m4).dot_product(m3) / m2.cross_product(m4).dot_product(m3)
k3_ = m1.cross_product(m4).dot_product(m2) / m3.cross_product(m4).dot_product(m2)
k2 = var('k2')
k3 = var('k3')
# see [zhang-single] Equation 14,16
n2 = k2 * m2 - m1
n3 = k3 * m3 - m1
# the focal length of the camera.
f = var('f')
# see [zhang-single] Equation 21
f_ = sqrt(
-1 / (
n2[2]*n3[2]*s^2
) * (
(
n2[0]*n3[0] - (n2[0]*n3[2]+n2[2]*n3[0])*u0 + n2[2]*n3[2]*u0^2
)*s^2 + (
n2[1]*n3[1] - (n2[1]*n3[2]+n2[2]*n3[1])*v0 + n2[2]*n3[2]*v0^2
)
)
)
# standard pinhole camera matrix
# see [zhang-single] Equation 1
A = matrix([[f,0,u0],[0,s*f,v0],[0,0,1]])
#the width/height ratio of the original rectangle
# see [zhang-single] Equation 20
whRatio = sqrt (
(n2*A.transpose()^(-1) * A^(-1)*n2.transpose()) /
(n3*A.transpose()^(-1) * A^(-1)*n3.transpose())
)
c代码中的简化方程由下式确定
print "simplified equations, assuming u0=0, v0=0, s=1"
print "k2 := ", k2_
print "k3 := ", k3_
print "f := ", f_(u0=0,v0=0,s=1)
print "whRatio := ", whRatio(u0=0,v0=0,s=1)
simplified equations, assuming u0=0, v0=0, s=1
k2 := ((m1y - m4y)*m3x - (m1x - m4x)*m3y + m1x*m4y - m1y*m4x)/((m2y
- m4y)*m3x - (m2x - m4x)*m3y + m2x*m4y - m2y*m4x)
k3 := ((m1y - m4y)*m2x - (m1x - m4x)*m2y + m1x*m4y - m1y*m4x)/((m3y
- m4y)*m2x - (m3x - m4x)*m2y + m3x*m4y - m3y*m4x)
f := sqrt(-((k3*m3y - m1y)*(k2*m2y - m1y) + (k3*m3x - m1x)*(k2*m2x
- m1x))/((k3 - 1)*(k2 - 1)))
whRatio := sqrt(((k2 - 1)^2 + (k2*m2y - m1y)^2/f^2 + (k2*m2x -
m1x)^2/f^2)/((k3 - 1)^2 + (k3*m3y - m1y)^2/f^2 + (k3*m3x -
m1x)^2/f^2))
print "Everything in one equation:"
print "whRatio := ", whRatio(f=f_)(k2=k2_,k3=k3_)(u0=0,v0=0,s=1)
Everything in one equation:
whRatio := sqrt(((((m1y - m4y)*m2x - (m1x - m4x)*m2y + m1x*m4y -
m1y*m4x)/((m3y - m4y)*m2x - (m3x - m4x)*m2y + m3x*m4y - m3y*m4x) -
1)*(((m1y - m4y)*m3x - (m1x - m4x)*m3y + m1x*m4y - m1y*m4x)/((m2y -
m4y)*m3x - (m2x - m4x)*m3y + m2x*m4y - m2y*m4x) - 1)*(((m1y -
m4y)*m3x - (m1x - m4x)*m3y + m1x*m4y - m1y*m4x)*m2y/((m2y - m4y)*m3x
- (m2x - m4x)*m3y + m2x*m4y - m2y*m4x) - m1y)^2/((((m1y - m4y)*m2x -
(m1x - m4x)*m2y + m1x*m4y - m1y*m4x)*m3y/((m3y - m4y)*m2x - (m3x -
m4x)*m2y + m3x*m4y - m3y*m4x) - m1y)*(((m1y - m4y)*m3x - (m1x -
m4x)*m3y + m1x*m4y - m1y*m4x)*m2y/((m2y - m4y)*m3x - (m2x - m4x)*m3y
+ m2x*m4y - m2y*m4x) - m1y) + (((m1y - m4y)*m2x - (m1x - m4x)*m2y +
m1x*m4y - m1y*m4x)*m3x/((m3y - m4y)*m2x - (m3x - m4x)*m2y + m3x*m4y
- m3y*m4x) - m1x)*(((m1y - m4y)*m3x - (m1x - m4x)*m3y + m1x*m4y -
m1y*m4x)*m2x/((m2y - m4y)*m3x - (m2x - m4x)*m3y + m2x*m4y - m2y*m4x)
- m1x)) + (((m1y - m4y)*m2x - (m1x - m4x)*m2y + m1x*m4y -
m1y*m4x)/((m3y - m4y)*m2x - (m3x - m4x)*m2y + m3x*m4y - m3y*m4x) -
1)*(((m1y - m4y)*m3x - (m1x - m4x)*m3y + m1x*m4y - m1y*m4x)/((m2y -
m4y)*m3x - (m2x - m4x)*m3y + m2x*m4y - m2y*m4x) - 1)*(((m1y -
m4y)*m3x - (m1x - m4x)*m3y + m1x*m4y - m1y*m4x)*m2x/((m2y - m4y)*m3x
- (m2x - m4x)*m3y + m2x*m4y - m2y*m4x) - m1x)^2/((((m1y - m4y)*m2x -
(m1x - m4x)*m2y + m1x*m4y - m1y*m4x)*m3y/((m3y - m4y)*m2x - (m3x -
m4x)*m2y + m3x*m4y - m3y*m4x) - m1y)*(((m1y - m4y)*m3x - (m1x -
m4x)*m3y + m1x*m4y - m1y*m4x)*m2y/((m2y - m4y)*m3x - (m2x - m4x)*m3y
+ m2x*m4y - m2y*m4x) - m1y) + (((m1y - m4y)*m2x - (m1x - m4x)*m2y +
m1x*m4y - m1y*m4x)*m3x/((m3y - m4y)*m2x - (m3x - m4x)*m2y + m3x*m4y
- m3y*m4x) - m1x)*(((m1y - m4y)*m3x - (m1x - m4x)*m3y + m1x*m4y -
m1y*m4x)*m2x/((m2y - m4y)*m3x - (m2x - m4x)*m3y + m2x*m4y - m2y*m4x)
- m1x)) - (((m1y - m4y)*m3x - (m1x - m4x)*m3y + m1x*m4y -
m1y*m4x)/((m2y - m4y)*m3x - (m2x - m4x)*m3y + m2x*m4y - m2y*m4x) -
1)^2)/((((m1y - m4y)*m2x - (m1x - m4x)*m2y + m1x*m4y -
m1y*m4x)/((m3y - m4y)*m2x - (m3x - m4x)*m2y + m3x*m4y - m3y*m4x) -
1)*(((m1y - m4y)*m3x - (m1x - m4x)*m3y + m1x*m4y - m1y*m4x)/((m2y -
m4y)*m3x - (m2x - m4x)*m3y + m2x*m4y - m2y*m4x) - 1)*(((m1y -
m4y)*m2x - (m1x - m4x)*m2y + m1x*m4y - m1y*m4x)*m3y/((m3y - m4y)*m2x
- (m3x - m4x)*m2y + m3x*m4y - m3y*m4x) - m1y)^2/((((m1y - m4y)*m2x -
(m1x - m4x)*m2y + m1x*m4y - m1y*m4x)*m3y/((m3y - m4y)*m2x - (m3x -
m4x)*m2y + m3x*m4y - m3y*m4x) - m1y)*(((m1y - m4y)*m3x - (m1x -
m4x)*m3y + m1x*m4y - m1y*m4x)*m2y/((m2y - m4y)*m3x - (m2x - m4x)*m3y
+ m2x*m4y - m2y*m4x) - m1y) + (((m1y - m4y)*m2x - (m1x - m4x)*m2y +
m1x*m4y - m1y*m4x)*m3x/((m3y - m4y)*m2x - (m3x - m4x)*m2y + m3x*m4y
- m3y*m4x) - m1x)*(((m1y - m4y)*m3x - (m1x - m4x)*m3y + m1x*m4y -
m1y*m4x)*m2x/((m2y - m4y)*m3x - (m2x - m4x)*m3y + m2x*m4y - m2y*m4x)
- m1x)) + (((m1y - m4y)*m2x - (m1x - m4x)*m2y + m1x*m4y -
m1y*m4x)/((m3y - m4y)*m2x - (m3x - m4x)*m2y + m3x*m4y - m3y*m4x) -
1)*(((m1y - m4y)*m3x - (m1x - m4x)*m3y + m1x*m4y - m1y*m4x)/((m2y -
m4y)*m3x - (m2x - m4x)*m3y + m2x*m4y - m2y*m4x) - 1)*(((m1y -
m4y)*m2x - (m1x - m4x)*m2y + m1x*m4y - m1y*m4x)*m3x/((m3y - m4y)*m2x
- (m3x - m4x)*m2y + m3x*m4y - m3y*m4x) - m1x)^2/((((m1y - m4y)*m2x -
(m1x - m4x)*m2y + m1x*m4y - m1y*m4x)*m3y/((m3y - m4y)*m2x - (m3x -
m4x)*m2y + m3x*m4y - m3y*m4x) - m1y)*(((m1y - m4y)*m3x - (m1x -
m4x)*m3y + m1x*m4y - m1y*m4x)*m2y/((m2y - m4y)*m3x - (m2x - m4x)*m3y
+ m2x*m4y - m2y*m4x) - m1y) + (((m1y - m4y)*m2x - (m1x - m4x)*m2y +
m1x*m4y - m1y*m4x)*m3x/((m3y - m4y)*m2x - (m3x - m4x)*m2y + m3x*m4y
- m3y*m4x) - m1x)*(((m1y - m4y)*m3x - (m1x - m4x)*m3y + m1x*m4y -
m1y*m4x)*m2x/((m2y - m4y)*m3x - (m2x - m4x)*m3y + m2x*m4y - m2y*m4x)
- m1x)) - (((m1y - m4y)*m2x - (m1x - m4x)*m2y + m1x*m4y -
m1y*m4x)/((m3y - m4y)*m2x - (m3x - m4x)*m2y + m3x*m4y - m3y*m4x) -
1)^2))
# some testing:
# - choose a random rectangle,
# - project it onto a random plane,
# - insert the corners in the above equations,
# - check if the aspect ratio is correct.
from sage.plot.plot3d.transform import rotate_arbitrary
#redundandly random rotation matrix
rand_rotMatrix = \
rotate_arbitrary((uniform(-5,5),uniform(-5,5),uniform(-5,5)),uniform(-5,5)) *\
rotate_arbitrary((uniform(-5,5),uniform(-5,5),uniform(-5,5)),uniform(-5,5)) *\
rotate_arbitrary((uniform(-5,5),uniform(-5,5),uniform(-5,5)),uniform(-5,5))
#random translation vector
rand_transVector = vector((uniform(-10,10),uniform(-10,10),uniform(-10,10))).transpose()
#random rectangle parameters
rand_width =uniform(0.1,10)
rand_height=uniform(0.1,10)
rand_left =uniform(-10,10)
rand_top =uniform(-10,10)
#random focal length and principal point
rand_f = uniform(0.1,100)
rand_u0 = uniform(-100,100)
rand_v0 = uniform(-100,100)
# homogenous standard pinhole projection, see [zhang-single] Equation 1
hom_projection = A * rand_rotMatrix.augment(rand_transVector)
# construct a random rectangle in the plane z=0, then project it randomly
rand_m1hom = hom_projection*vector((rand_left ,rand_top ,0,1)).transpose()
rand_m2hom = hom_projection*vector((rand_left ,rand_top+rand_height,0,1)).transpose()
rand_m3hom = hom_projection*vector((rand_left+rand_width,rand_top ,0,1)).transpose()
rand_m4hom = hom_projection*vector((rand_left+rand_width,rand_top+rand_height,0,1)).transpose()
#change type from 1x3 matrix to vector
rand_m1hom = rand_m1hom.column(0)
rand_m2hom = rand_m2hom.column(0)
rand_m3hom = rand_m3hom.column(0)
rand_m4hom = rand_m4hom.column(0)
#normalize
rand_m1hom = rand_m1hom/rand_m1hom[2]
rand_m2hom = rand_m2hom/rand_m2hom[2]
rand_m3hom = rand_m3hom/rand_m3hom[2]
rand_m4hom = rand_m4hom/rand_m4hom[2]
#substitute random values for f, u0, v0
rand_m1hom = rand_m1hom(f=rand_f,s=1,u0=rand_u0,v0=rand_v0)
rand_m2hom = rand_m2hom(f=rand_f,s=1,u0=rand_u0,v0=rand_v0)
rand_m3hom = rand_m3hom(f=rand_f,s=1,u0=rand_u0,v0=rand_v0)
rand_m4hom = rand_m4hom(f=rand_f,s=1,u0=rand_u0,v0=rand_v0)
# printing the randomly choosen values
print "ground truth: f=", rand_f, "; ratio=", rand_width/rand_height
# substitute all the variables in the equations:
print "calculated: f= ",\
f_(k2=k2_,k3=k3_)(s=1,u0=rand_u0,v0=rand_v0)(
m1x=rand_m1hom[0],m1y=rand_m1hom[1],
m2x=rand_m2hom[0],m2y=rand_m2hom[1],
m3x=rand_m3hom[0],m3y=rand_m3hom[1],
m4x=rand_m4hom[0],m4y=rand_m4hom[1],
),"; 1/ratio=", \
1/whRatio(f=f_)(k2=k2_,k3=k3_)(s=1,u0=rand_u0,v0=rand_v0)(
m1x=rand_m1hom[0],m1y=rand_m1hom[1],
m2x=rand_m2hom[0],m2y=rand_m2hom[1],
m3x=rand_m3hom[0],m3y=rand_m3hom[1],
m4x=rand_m4hom[0],m4y=rand_m4hom[1],
)
print "k2 = ", k2_(
m1x=rand_m1hom[0],m1y=rand_m1hom[1],
m2x=rand_m2hom[0],m2y=rand_m2hom[1],
m3x=rand_m3hom[0],m3y=rand_m3hom[1],
m4x=rand_m4hom[0],m4y=rand_m4hom[1],
), "; k3 = ", k3_(
m1x=rand_m1hom[0],m1y=rand_m1hom[1],
m2x=rand_m2hom[0],m2y=rand_m2hom[1],
m3x=rand_m3hom[0],m3y=rand_m3hom[1],
m4x=rand_m4hom[0],m4y=rand_m4hom[1],
)
# ATTENTION: testing revealed, that the whRatio
# is actually the height/width ratio,
# not the width/height ratio
# This contradicts [zhang-single]
# if anyone can find the error that caused this, I'd be grateful
ground truth: f= 72.1045134124554 ; ratio= 3.46538779959142
calculated: f= 72.1045134125 ; 1/ratio= 3.46538779959
k2 = 0.99114614987 ; k3 = 1.57376280159
