Closed-form Stereo Image Rectification

Changming SunCSIRO Mathematics, Informatics and Statistics

Locked Bag 17, North Ryde, NSW 1670, Australiachangming.sun@csiro.au

ABSTRACTStereo image rectification transforms a pair of images into anew pair such that the epipolar lines in the transformed im-ages have the same direction as the image rows and matchingepipolar lines in different images have the same row indexto enable an efficient and reliable dense stereo matching. Inthis paper we propose a first closed-form algorithm to au-tomatically rectify stereo images just using the fundamen-tal matrix. The algorithm involves only direct and purelygeometric transformation processes. There is no iterativeor optimization processes for the rectification parameter es-timation in our method. Real images have been used fortesting purposes, and convincing results have been obtainedfrom our algorithm. Comparison with the state-of-the-artmethod shows that our method produces better rectificationresults.

Categories and Subject DescriptorsI.2.10 [Vision and Scene Understanding]: 3D/stereoscene analysis; I.4.8 [Scene Analysis]: Stereo

General TermsAlgorithms

KeywordsStereo image rectification, Fundamental matrix, Epipolarlines, Epipoles, Keystone effect.

1. INTRODUCTIONStereo image rectification is important in the analysis of

three dimensional scenes. The rectification process trans-forms the input images into new ones where epipolar linesare parallel to the x- axis and matching epipolar lines fromdifferent images have the same row index. This process re-moves the vertical disparities between matching points in theoriginal left and right images. The rectified images should

(c) 2012 Association for Computing Machinery. ACM acknowledges thatthis contribution was authored or co-authored by an employee, contractoror affiliate of the national government of Australia. As such, the govern-ment of Australia retains a nonexclusive, royalty-free right to publish orreproduce this article, or to allow others to do so, for Government purposesonly.IVCNZ ’2012, November 26-28 2012, Dunedin, New ZealandCopyright 2012 ACM 978-1-4503-1473-2/12/11 ...$15.00.

be positioned on the same plane and parallel to the lineconnecting the camera centers. The rectification processshould introduce minimal distortions for the rectified im-ages. For stereo matching, image rectification can increaseboth the reliability and the speed of disparity estimation.This is because in the rectified images, the relative rotationbetween the original images has been removed and the dis-parity search happens only along the image horizontal scan-lines. This rectification is also useful for human viewing ofstereoscopic images to reduce vertical parallax which oftencauses eyestrain.

The rectification process usually requires certain cameracalibration parameters or weakly calibrated (uncalibrated)epipolar geometries of the image pair and most algorithmsrequire the initial feature matching points as a part of theinputs. Some algorithms obtain the rectification matrix justfrom the image matching points.

Ayache and Hansen presented a technique for calibratingand rectifying a pair or triplet of images [1]. In their case, acamera matrix needs to be estimated. Jawed et al. presenteda hardware implementation for real time rectification usingFPGA [8]. Fusiello et al. presented a compact algorithm forrectifying calibrated stereo images [4]. All these rectificationalgorithms only work for calibrated cameras where cameraparameters are known.

There is also a large number of algorithms for uncalibratedcamera cases. Pollefeys et al. proposed a simple and efficientrectification method for general two view stereo images byreparameterizing the image with polar coordinates aroundthe epipoles [14]. A similar approach is described in [2].These two methods used a direct sampling approach basedon the epipolar geometry. A method on the rectification ofstereo images onto cylindrical surfaces was given in [16].

Hartley gave a mathematical basis and a practical algo-rithm for the rectification of stereo images from widely dif-ferent viewpoints [6]. Oram presented a similar method forrectification for any epipolar geometry [13]. All of the abovealgorithms require the fundamental matrix and the accom-panying matching points information.

Isgro and Trucco proposed a projective rectification methodwhich uses the matching points to estimate the rectificationmatrices directly without the use of epipolar geometry [7].Fusiello and Irsara proposed a quasi-Euclidean rectificationmethod which minimizes a rectification error [3]. Kumaret al. used a similar approach to obtain the rectification ma-trices directly from image matching points while consideringthe different zoom factors of the two images [9].

Loop and Zhang proposed a technique for computing recti-fication homographies for stereo vision using the fundamen-

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tal matrix between images together with all the points in theimages to minimize image distortion [10]. Robert et al. de-scribed a method for rectification considering the minimiza-tion of image distortion by preserving orthogonality and thenewly filled areas [15]. A similar approach is used in [5] forobtaining rectifying transformations that minimize resam-pling effects. Monasse et al. proposed a three-step imagerectification method which involves optimizing one parame-ter [12]. Mallon and Whelan used fundamental matrix andregularly sampled points in the images for rectification [11].All of these methods use the fundamental matrix and alsoinvolve iterative or optimization process for rectification pa-rameters estimation.

In this paper, we propose a first closed-form algorithmfor automatically rectifying two uncalibrated images onlyusing the fundamental matrix and no iterative parameteroptimization step is involved. All the steps that we usedonly involve direct geometric transformation.

2. RECTIFYING UNCALIBRATED IMAGESOur rectification process only requires the fundamental

matrix that can be obtained from matching points betweentwo images or from any other methods.

2.1 Move Epipoles to InfinityGiven the epipolar geometry defined by the fundamen-

tal matrix F12 between two images, a pair of epipoles e12

and e21 in these two images can be obtained by solvingF12e12 = 0 and FT

12e21 = 0 using a singular value decompo-sition method, and then scaling the epipoles so that the thirdelements are equal to 1. The image rectification process is toinitially transform the input images such that the epipoles inthe transformed images are at infinity on the x- axis. In theusual case of left and right stereo image arrangement, theepipoles of the transformed images are at the infinity point(1, 0, 0)T .

One needs to find the mapping functions which will trans-form the two epipoles e12 and e21 in the two original imagesinto the infinity point (1, 0, 0)T in the transformed images.In order to reduce image distortions, image transformationsneed to be as rigid as possible. In the following paragraph,we will describe the algorithm steps for transforming theepipoles into infinity. This transformation steps have beenused in several articles [6, 17].

Assuming that the image center is at (u, v, 1)T , one can usethe following transformation to shift the image coordinatesystem to the image center:

T =

0@ 1 0 −u0 1 −v0 0 1

1A . (1)

Then the image can be rotated such that the epipole aftertranslation e′12 = T e12 = (e′12[0], e′12[1], 1)T is further movedonto the x- axis. This rotation transformation takes the formof

R1 =

0@ cos θ1 sin θ1 0− sin θ1 cos θ1 0

0 0 1

1A (2)

where θ1 = arctan(e′12[1]/e′12[0]). The next step will be toproject the epipole position to infinity. This transformation

can be achieved using the following matrix:

K1 =

0@ 1 0 00 1 0

−1/k1 0 1

1A (3)

where k1 = e′′12[0] with e′′12 = R1 e′12. The combined trans-formation matrix is:

P1 = K1R1T. (4)This transformation will have the effect of projecting theepipole e12 to infinity on the x- axis. Note that the values ofθ1 and k1 are obtained from the immediate previous trans-formation operations. A similar transformation for the rightimage which maps the epipole e21 to infinity on the x- axiscan be obtained with P2 = K2R2T. The corresponding val-ues θ2 and k2 for R2 and K2 for the right image are obtainedin a similar way as θ1 and k1 for the left image.

2.2 Aligning Matching Epipolar LinesAfter moving the epipoles of the two images to infinity

by applying P1 and P2, the epipolar lines in the two trans-formed images become parallel to the horizontal scanline.However, the matching epipolar lines from the left and rightimages usually do not lie on the same scanline. In this sectionwe present our new method for aligning matching horizontalepipolar lines from the two images so that they will have thesame y value.

Three pairs of matching epipolar lines from the left andright images are needed for our alignment purpose. Thesethree pairs of matching epipolar lines can be obtained byfirstly selecting three points on a vertical line passing throughthe image center in the left image with the same spacing.The middle point can be the center of the image, and theother two points are one above and one below the centerpoint. The spacing for these points can be arbitrary. Thenepipolar lines passing through these three points in the leftimage can be calculated using the fundamental matrix. Thethree matching epipolar lines in the right image can also beobtained. These three pairs of matching epipolar lines willall be horizontal after the P1 and P2 transformations. Themiddle epipolar line in the left image goes through the imagecenter. In order to make the matching middle epipolar lineshave the same y value, we first align the middle horizontalepipolar lines between the left and right images by shiftingthe right image vertically with

Tv =

0@ 1 0 00 1 −vm0 0 1

1A (5)

where vm is the y difference between the two horizontal mid-dle matching epipolar lines.

Next we need to find the transformation for aligning theremaining two matching epipolar lines. Figure 1 illustratesthe cross sections of the two transformed images, perpen-dicular to the epipolar lines and going through the imagecenter. The x- axis of the epipolar lines is perpendicular tothe page. The locations of the three epipolar lines on im-age plane AMD are shown by the three small black squares,while the locations of the matching epipolar lines on imageplane BFE are given by the three small dots which are tobe transformed to the locations at A, M, and D. H1, H2, G1,and G2 are the y coordinates of points A, D, B, and E, re-spectively. The y coordinates of F and M are now zero afterthe Tv shift for the right image.

The transformation matrix for alignment will take a formof a projection matrix. It just transforms one point loca-

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H

C

1G1

G2 H2

E

B

F

A

M

D

Figure 1: Illustration for aligning matching epipolarlines. Points A, M, D and B, F, E are the y loca-tions of the horizontal epipolar lines. Point C is thecamera center. The x- axis of the epipolar lines isperpendicular to the page.

tion to another for the matching epipolar lines. Scaling isalso necessary in the matrix. Here is the form for aligningepipolar lines from B to A and from E to D:

A2 =

0@ 1 0 00 wy 00 k′y 1

1A (6)

where wy and k′y will be obtained using the location infor-mation of the matching epipolar lines. To transform pointB with A2, we have0@ 1 0 0

0 wy 00 k′y 1

1A0@ 0G1

1

1A=

0@ 0wyG1

k′yG1 + 1

1A ∼0@ 0

wyG1

k′yG1+1

1

1A(7)

This point matches (0, H1, 1)T . Therefore, we can havewyG1

k′yG1 + 1= H1 (8)

Similarly, for points (0, G2, 1)T and (0, H2, 1)T , we can havewyG2

k′yG2 + 1= H2 (9)

From Eqs. (8) and (9), we can solve wy and k′y as8>><>>:wy =

(G2 −G1)H1H2

(H2 −H1)G1G2

k′y =H1G2 −H2G1

(H2 −H1)G1G2

(10)

The matrix A2Tv aligns all the horizontal epipolar linesin the right image to their matching horizontal epipolar linesin the left image.

2.3 Keystone Effect CorrectionAfter the alignment step as described early, matching epipo-

lar lines are horizontal and are on the same scanline. How-ever, the spacings between the three epipolar lines used ear-lier may not be the same due to the projective transfor-mation. This is similar to the keystone effect in projectorprojections. To remove the keystone effect, i.e., to make thespacings between the horizontal epipolar lines equal, we de-velop the following new method for our rectification purposefor reducing transformation distortions. In Figure 2, dif-ferent spacings |H1| and |H2| between neighboring epipolarlines are to be transformed to the same spacing S.

With the law of sine, the relationships between some quan-

2β

H1

2H

θ

θ

S

A

α1

β1

OC f

2α

D

S

D’

A’

Figure 2: Illustration for keystone effect correction.

tities of the triangles 4AA′O and that of 4DD′O as shownin Figure 2 can be written as:

sinα1

S=

sinβ1H1

andsin(π − α2)

S=

sinβ2H2

(11)

which leads tosinα1

sin(π − α2)=H2 sinβ1H1 sinβ2

(12)

From this equation, we are to find the angle θ as shown inFigure 2 and then the length S. With β1 = π − α1 − θ andβ2 = α2 − θ, we have

sinα1

sin(π − α2)=H2 sin(π − α1 − θ)H1 sin(α2 − θ)

(13)

and with the angle sum and angle difference formulae, onehas

H1 sinα1

H2 sinα2=

sinα1 cos θ + cosα1 sin θ

sinα2 cos θ − cosα2 sin θ(14)

The solution for θ can be obtained from Eq. (14) and equa-tion sin2θ + cos2θ = 1 as the following:

θ = ± arcsinp

1/(1 + g2) (15)with

g =H1 sinα1 cosα2 +H2 cosα1 sinα2

H1 sinα1 sinα2 −H2 sinα1 sinα2. (16)

There are two solutions for θ. We choose the one whichsatisfies Eq. (14). The values of α1 and α2 can be obtainedfrom f,H1 and H2. With uncalibrated images, f is set to1. From the obtained θ, we can calculate β1 and then thelength S can be calculated as S = H1 sinα1/ sinβ1.

The keystone correction matrix can be obtained usingthe same procedure as used when calculating A2 but withG1 = S and G2 = −S. Given that there is only a projectivetransformation without scaling for keystone effect correction,one can have a simpler matrix0@ 1 0 0

0 1 00 k′ 1

1A (17)

with k′ as the unknown. The transformed point of (0, H1, 1)T

becomes0@ 1 0 00 1 00 k′ 1

1A0@ 0H1

1

1A=

0@ 0H1

k′H1 + 1

1A ∼0@ 0

H1k′H1+1

1

1A (18)

This point matches (0, S, 1)T . Hence we haveH1

k′H1 + 1= S i.e. k′ =

H1 − SH1S

(19)

Here we denote the newly obtained matrix as H for keystoneeffect correction which will be applied to both left and right

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images. That is

H =

0@ 1 0 00 1 00 H1−S

H1S1

1A (20)

2.4 Deskewing ImagesThe rectified images after keystone effect correction can be

further processed so that two orthogonal vectors in an origi-nal image are close to orthogonal in the rectified image. Thetwo orthogonal vectors in the original image can go throughthe image center, one horizontal and one vertical. Assumingthe two transformed vectors after keystone effect correction,i.e., after applying HP1 to the original orthogonal vectors,are m = (u1, v1, 0)T and n = (u2, v2, 0)T , with a skew matrix

S1 =

0@ 1 s1 00 1 00 0 1

1A , (21)

we wish to make the two transformed vectors m and n afterapplying S1 orthogonal to each other, i.e., their dot prod-uct is zero: 〈S1m,S1n〉 = 0. This gives an equation withvariable s1:

v1v2s21 + (u1v2 + u2v1)s1 + u1u2 + v1v2 = 0 (22)

and it can be solved with two solutions. The one with thesmallest absolute value is taken as s1. The matrix for theright image S2 can be obtained similarly, using HA2TvP2

instead of HP1.

2.5 Reducing Image TranslationThe rectified images obtained by applying the transforma-

tion matrix S1HP1 for the left image and S2HA2TvP2 forthe right image are almost always different from the originalimages. Our intention is to minimize the image differencebefore and after the rectification process. An image trans-lation can be carried out by using the sum of the positiondifferences for the four corners of the rectified and the orig-inal images.

The vertical shift should have the same value for boththe left and right rectified images, i.e., v′1 = v′2. They areobtained by calculating the average vertical differences ofboth the four corners in the left image and the four cornersin the right image, i.e., totaling of eight corners, before andafter applying transformations S1HP1 and S2HA2TvP2.

The horizontal shifts u′1 and u′2 for the left and right im-ages can be different and will be obtained independentlyfor each image. u′1 is obtained by calculating the averagehorizontal differences of the four corners in the left imagebefore and after applying transformations S1HP1. Simi-larly u′2 is obtained for the right image using transformationS2HA2TvP2. The transformation matrix for the shifts are

T1 =

0@ 1 0 −u′10 1 −v′10 0 1

1A and T2 =

0@ 1 0 −u′20 1 −v′20 0 1

1A(23)

2.6 Algorithm StepsThe combined transformations to achieve the final image

rectifications are through the use of T1S1HP1 andT2S2HA2TvP2 for the left and right images respectively. Ifwe put the transformations in groups, they become:

for left image: T1S1 H K1R1Tfor right image: T2S2| {z } H|{z} A2Tv| {z } K2R2T| {z }

↑ ↑ ↑ ↑Ã Â Á À

The transformations in group À project the epipoles in thetwo images to infinity and therefore all the epipolar linesbecome horizontal. Transformations in group Á for the rightimage align the matching epipolar lines between two images.The H matrix in group  corrects the keystone effect sothat the spacings between epipolar lines become regular orthe same. The transformations in group à deskew and shifttransformed images to minimize distortion. The steps of ouralgorithm for stereo image rectification are the following:

1. Obtain or read in the fundamental matrix between twoviews.

2. Uncalibrated stereo image rectification:

(a) Calculate the epipoles on the two images from theF12 matrix.

(b) Obtain the transformation matrices P1 and P2

for the left and right images using the methoddescribed in Section 2.1.

(c) Obtain alignment transformation A2Tv for theright image.

(d) Obtain keystone correction transformation H forboth images.

(e) Obtain the deskewing matrices S1 and S2, andobtain the translation matrices T1 and T2.

(f) Apply the two transformation matrices T1S1HP1

and T2S2HA2TvP2 to the left and right imagesrespectively and apply resamplings to obtain therectified images.

3. EXPERIMENTAL RESULTSThis section shows some of the stereo image rectification

results obtained using our new method described in previoussections and also gives some comparison results. A varietyof real images have been tested. The fundamental matrixfor each pair of images can be obtained using the methodproposed in [18]. After we have obtained the fundamentalmatrix for a pair of images, we can then carry out the rec-tification process using our rectification algorithm. Whenresampling the input images for rectification, bilinear inter-polation can be used.

Figure 3 shows the results for the main steps of our rec-tification process. Figure 3(a,b) show the epipolar linesoverlaid on the original stereo images. Figure 3(c,d) showthe epipolar lines transformed into horizontal lines. Notethat the matching epipolar lines are not yet aligned to eachother. Figure 3(e,f) show that the matching epipolar linesare aligned between the left and right images. Figure 3(g,h)are the final rectification results after correcting keystone ef-fect, deskewing, and shifts. Figure 4 shows another exampleof our rectification results. Figure 4(a,b) show the epipolarlines overlaid on the original stereo images, and Figure 4(c,d)are the final rectification results.

For majority of the cases, the keystone effects are mini-mum especially when the epipoles are further away from theimage centers. Figure 5 shows an example of keystone effectcorrection for the right image of a stereo pair. Figure 5(a)shows the epipolar lines with different spacings, with largerspacings on the upper part and smaller spacings on the lowerpart of the image, while Figure 5(b) shows epipolar lines withthe same spacing after keystone effect correction for the sameimage.

In [11], Mallon and Whelan compared their method withtwo other popular methods from the literature, Hartley’s [6]

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(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 3: Rectification results for the main stepsof our algorithm. (a,b) Epipolar lines overlaid onthe original stereo images; (c,d) Epipolar lines trans-formed into horizontal lines; (e,f) Matching epipolarlines aligned between the left and right images; (g,h)Final rectification results after correcting keystoneeffect, deskewing, and shifts.

and Loop and Zhang’s [10]. Mallon and Whelan’s resultsshowed that their method gives a much improved perfor-mance than the other two methods. Our experimental re-sults also confirmed that the results of [11] are better thanthat of [6] and [10]. Because of this and the limitation ofspace, in this paper we just show the results of our methodand that of Mallon and Whelan’s method which is currentlythe state-of-the-art. Both our method and Mallon and Whe-lan’s method use the same input for rectification, i.e., onlythe fundamental matrix information. Figure 6 shows thecomparison results between ours and that of Mallon andWhelan’s using the same set of fundamental matrices on the‘Slate’ dataset. The ‘Slate’ dataset is an indoor office scene.It can be seen that our method produces smaller areas ofblack pixels, i.e., more usable pixel areas in the rectified im-ages. The method in [11] does not have all the distortion

(a) (b)

(c) (d)

Figure 4: Another example of our rectification re-sults. (a,b) Epipolar lines overlaid on the originalstereo images. (c,d) Final rectification results.

reduction steps as in our method.In terms of running times, our rectification process is very

fast, taking about 81 ms for a 640×480 pixel image on aLinux PC with a 3.00GHz Intel Core2Duo CPU using the Clanguage. The majority of the CPU time was on the actualimage resampling process which takes about 76 ms. Theprocess for obtaining the rectification matrices only takesabout 4.7 ms.

4. CONCLUSIONSIn this paper, a new closed-form method for automat-

ically rectifying uncalibrated stereo images has been pre-sented. The transformation matrices applied to the originalimages are constructed just based on the epipolar geometriesbetween an image pair, i.e. only using the fundamental ma-trix. There is no iterative parameter optimization process inour methods. That is our method is in closed-form and withjust the fundamental matrix. Real images have been testedand the results validate our new method. Comparison withthe state-of-the-art method shows that our method producesbetter rectification results. Future work should include morequantitative evaluations of different rectification methods.

(a) (b)

Figure 5: An example showing the keystone effectcorrection. (a) Epipolar lines with different spac-ings. (b) Epipolar lines with the same spacing.

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(a) (b)

(c) (d)

(e) (f)

Figure 6: Rectification results of ours and Mallonand Whelan’s on the ‘Slate’ dataset. (a,b) Epipolarlines overlaid on the original stereo images. (c,d)Results using Mallon and Whelan’s method. (e,f)Results using our method.

5. ONLINE DEMOAn online web demo for stereo image rectification using

our method is available at:

vision-cdc.csiro.au/rectify2v

6. ACKNOWLEDGEMENTSThe author thanks Xiao Tan of CSIRO and the anony-

mous reviewers for their comments and suggestions. Theauthor is also grateful to Dr John Mallon from Dublin CityUniversity for sample MATLAB codes described in [11]. Theimages and related data in Figure 6 are from Dr Mallon. Thisresearch was partially supported by the CSIRO’s Transfor-mational Biology Capability Platform.

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