proportions of a perspective-deformed rectangle

Update

After reading your update, and looking at the first reference (Whiteboard scanning and image enhancement), I see where the missing point is.

The input data of the problem is a quadruple (A,B,C,D), AND the center O of the projected image. In the article, it corresponds to the assumption u0=v0=0. Adding this point, the problem becomes constrained enough to get the aspect ratio of the rectangle.

The problem is then restated as follows: Given a quadruple (A,B,C,D) in the Z=0 plane, find the eye position E(0,0,h), h>0 and a 3D plane P such that the projection of (A,B,C,D) on P is a rectangle.

Note that P is determined by E: to get a parallelogram, P must contain parallels to (EU) and (EV), where U=(AB)x(CD) and V=(AD)x(BC).

Experimentally, it seems that this problem has in general one unique solution, corresponding to a unique value of the w/h ratio of the rectangle.

Previous Post

No, you can't determine the rectangle ratio from the projection.

In the general case, a quadruple (A,B,C,D) of four non collinear points of the Z=0 plane is the projection of infinitely many rectangles, with infinitely many width/height ratios.

Consider the two vanishing points U, intersection of (AB) and (CD) and V, intersection of (AD) and (BC), and the point I, intersection of the two diagonals (AC) and (BD). To project as ABCD, a parallelogram of center I must lie on a plane containing the line parallel to (UV) through point I. On one such plane, you can find many rectangles projecting to ABCD, all with a different w/h ratio.

See these two images done with Cabri 3D. In the two cases ABCD is unchanged (on the gray Z=0 plane), and the blue plane containing the rectangle is not changed either. The green line partially hidden is the (UV) line and the visible green line is parallel to it and contains I.

Here is my attempt at answering my question after reading the paper

• Zhengyou Zhang , Li-Wei He, "Whiteboard scanning and image enhancement" http://research.microsoft.com/en-us/um/people/zhang/papers/tr03-39.pdf

I manipulated the equations for some time in SAGE, and came up with this pseudo-code in c-style:

// 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;

Update: here is how these equations were determined:

The following is code in SAGE. It can be accessed online at http://www.sagenb.org/home/pub/704/. (Sage is really useful in solving equations, and useable in any browser, check it out)

# 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())
              ) 

The simplified equations in the c-code where determined by

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