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    <p>Here is my attempt at answering my question after reading the paper</p> <ul> <li>Zhengyou Zhang , Li-Wei He, "Whiteboard scanning and image enhancement" <a href="http://research.microsoft.com/en-us/um/people/zhang/papers/tr03-39.pdf" rel="noreferrer">http://research.microsoft.com/en-us/um/people/zhang/papers/tr03-39.pdf</a></li> </ul> <p>I manipulated the equations for some time in SAGE, and came up with this pseudo-code in c-style:</p> <pre><code> // 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; </code></pre> <h2>Update: here is how these equations where determined:</h2> <p>The following is code in <a href="http://www.sagemath.org/" rel="noreferrer">SAGE</a>. It can be accessed online at <a href="http://www.sagenb.org/home/pub/704/" rel="noreferrer">http://www.sagenb.org/home/pub/704/</a>. (Sage is really useful in solving equations, and useable in any browser, check it out)</p> <pre><code># 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()) ) </code></pre> <p>The simplified equations in the c-code where determined by</p> <pre><code>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)) </code></pre> <pre><code> # 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 </code></pre>
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    1. COThanks for the code and the nice problem!
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