robust direct motion estimation considering discontinuity

Robust direct motion estimation considering discontinuity

Jong-Eun Ha a,*, In-So Kweon b,1

a MECA Group, Technology/R&D Center, Samsung Corning Co. Ltd., 472 Shin-Dong, Paldal-Gu, Suwon-Shi,

Kyunggi-Do 442-390, South Koreab Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Kusong-dong,

Yusong-gu, Taejon, South Korea

Received 15 September 1999; received in revised form 18 April 2000

Abstract

In this paper, we propose a robust motion estimation algorithm using uncalibrated 3D motion model considering

depth discontinuity. Most of the previous direct motion estimation algorithms with 3D motion model compute the

depth value through the local smoothing, which result in erroneous results at depth discontinuity. In this paper, we

overcome this problem at depth discontinuity by adding discontinuity preserving regularization term to the original

equation. Robust estimation enables motion segmentation through the dominant camera motion compensation.

Experimental results show the improved result at the depth discontinuity. Ó 2000 Elsevier Science B.V. All rights

reserved.

Keywords: Optic ¯ow; Discontinuity; Direct method; Uncalibrated

1. Introduction

Analysis of image motion plays an importantrole in many areas of computer vision: scene mo-tion detection, object segmentation, tracking, andthe recovery of scene structure. Typical gradientbased optic-¯ow algorithms are based on thebrightness constancy assumption: invariance ofrecorded image brightness along motion trajecto-ries. However, this assumption provides a singleconstraint for two unknowns at each pixel.

Horn and Schunck (1981) introduce a spatialconstraint by regularization on the optical ¯ow®elds, often called the smoothness constraint, tocompute a dense, smoothly varying velocity ®eld.This method gives a globally smooth motion ®eld,but it also blurs the motion ®eld at the discontinu-ity. In general, optic ¯ow ®eld is piecewise contin-uous rather than locally constant or globallysmooth. Nagal and Enkelmann (1986) proposed anoriented smoothness constraint to attenuatesmoothing across strong intensity edges. Schunck(1989) identi®es motion boundary by clusteringlocal gradient-based constraints. Bartolini and Piva(1997) proposed median based relaxation optic ¯owalgorithm to reduce the strength of the smoothingstep, especially across motion boundaries.

Motion discontinuities can be treated explic-itly by introducing line ®eld to be computed

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Pattern Recognition Letters 21 (2000) 999±1011

* Corresponding author. Tel.: +82-331-219-7874; fax: +82-

331-219-7085.

E-mail addresses: [email protected] (J.-E. Ha),

[email protected] (I.-S. Kweon).1 Tel.: +82-2-958-3465; fax: +82-2-960-0510.

0167-8655/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 7 - 8 6 5 5 ( 0 0 ) 0 0 0 5 9 - 3

simultaneously to image motion. This approach hasoften been embedded in a Markov random ®elds(MRF) modeling framework. MRF modeling givesa means to organize velocities and motion discon-tinuities by allowing the introduction of genericknowledge of a local and contextual nature. MRFframework for image motion estimation has beenexploited in the work of Konrad and Dubois (1992),Heitz and Bouthermy (1990). Robust estimationtechniques by Black and Anandan (1993), Boberand Kittler (1994), Odobez and Bouthemy (1995),and Ayer et al. (1994) can also be exploited to tacklemotion discontinuity. Local approach can deal withmotion discontinuity explicitly but their perfor-mance degrades in areas with a low intensity gra-dient and homogenous regions.

Bergen et al. (1992) pointed out that an explicitrepresentation of the motion model might lead tomore accurate computation of motion ®elds. Di-rect method imposes an explicit motion model inaddition to the optic ¯ow constraint, and it gives abetter performance in the homogenous region.Previous direct methods using 3D motion modelcompute the depth ®eld through the localsmoothing. Horn and Weldon (1988) and Hanna(1991) proposed the direct method using the cali-brated 3D motion model, and they obtain thedepth ®eld using the local smoothing. Szeliski andCoughlan (1997) proposed a spline based imageregistration method based on uncalibrated 3Dmotion model, but his framework uses the splineso that a smooth depth map is obtained from theinterpolation of the spline's control points. Allthese methods employ the least-squares estima-tion. Ayer (1995) proposed robust direct estima-tion using uncalibrated 3D motion model, buttheir motion model is based on the instantaneousmotion and depth map is obtained explicitly fromthe motion parameter at each step. Also he as-sumes locally constant depth map.

In this paper, we propose a robust direct ap-proach using uncalibrated 3D motion model con-sidering depth discontinuity. We consider thedepth discontinuity through the regularizationwith discontinuity in the motion ®eld. In addition,the dominant motion of camera is directly givendue to the robust estimation framework with un-calibrated 3D motion model. On the other hand,

direct estimation using a simple linear model re-quires additional processing to extract the domi-nant motion of the camera. The proposedalgorithm can be easily extended to the motionsegmentation problem.

2. Related direct method

In this section, we review the direct methodusing calibrated or uncalibrated 3D motion modeland their shortcomings at depth discontinuity. Thebasic assumption behind any optic ¯ow algorithmis the brightness constancy

I�x; t� � I�xÿ u�x�; t ÿ 1�:Direct method usually obtains the motion ®eld

through the minimization of sum of squared dif-ference (SSD) error over a local image area or theentire image using the explicit motion mode.

E�fug� �X

x

�I�x; t� ÿ I�xÿ u�x�; t ÿ 1��2: �1�

The motion model u(x) is chosen according toapplication and its explicit form is

u�x� � u�x : fhig�; �2�where fhig is a vector representing the modelparameters. Thus, motion estimation problemreduces to the estimation of model parameters.

In the case of the general perspective projectionmodel, the image motion induced by a rigidlymoving object can be written as:

u�x� � 1

Z�x�At � Bx;

A � ÿf 0 x

0 ÿf y

� �;

B � �xy�=f ÿ�f 2 � x2�=f y

�f 2 � y2�=f �xy�=f ÿx

� �;

�3�

where f is the focal length, t the translation vector,x the angular velocity vector, and Z is the depth.The estimation for this motion model requires twoparts: estimation of the global parameters and theestimation of the local parameters. Bergen et al.(1992) obtained the local depth parameters

1000 J.-E. Ha, I.-S. Kweon / Pattern Recognition Letters 21 (2000) 999±1011

explicitly through the minimization of the follow-ing local component of the error measure

Elocal �X5�5

E�t;x; 1=Z�: �4�

Di�erentiating Eq. (4) with respect to 1=Z�x�and setting the result to 0 gives

1=Z

�ÿP5�5�rI�TAt DI ÿ �rI�TAti=Zi � �rI�TBxÿ �rI�TBxi

� �P

5�5 �rI�TAt� �2

:

�5�

To re®ne the global motion parameters, theminimization is performed over the entire imageand 1=Z�x� of Eq. (5) is used, thus through theGauss±Newton minimization only t and x is up-dated.

Eglobal �Ximage

E�t;x; 1=Z�: �6�

After updating the global motion parameters,local depth is obtained explicitly using Eq. (5).Local depth of Eq. (5) is based on the locallyconstant depth, and this assumption is violated atthe depth of discontinuity.

Ayer (1995) proposed a robust direct estimationwith uncalibrated 3D motion model, he updatesonly the motion parameters using the iterativereweighted least squares (IRLS) and depth is ob-tained using an explicit equation from the least-squares. Szeliski and Coughlan (1997) presented aregistration algorithm using the spline withuncalibrated 3D motion model. Projective depth isestimated at each control point and the pixel-wisedepth map is obtained using the interpolationthrough the control points. All these methodsproduce a smooth depth map ignoring the depthdiscontinuity.

All previous direct approaches using calibrat-ed or uncalibrated 3D motion model give anerroneous result in depth discontinuity due to thelocally constant depth assumption. In this paper,we present a robust direct method using uncali-brated 3D motion model considering depthdiscontinuity.

3. Robust direct estimation using uncalibrated 3D

motion model considering depth discontinuity

Direct model-based motion estimation obtainsthe motion ®eld through the minimization of theregistration error using the motion model explicitly.Robust estimation of motion ®eld is obtainedthrough the minimization of the following equation

E�fug� �Xi;j2R

q�I2�xi � ui; yi � vi� ÿ I1�xi; yi�; r�:

�7�The function q is the robust M-estimator and r

is the scale factor that adjusts the shape of robustM-estimator. We can use various motion modelsaccording to a speci®c application. We use theuncalibrated 3D motion model that is proposed byHartley et al. (1992) and Faugeras (1992) to copewith unknown parameters of cameras and to dealwith the perspective e�ect.

u�x1; y1� � m0x1 � m1y1 � m2 � z�x1; y1�m8

m6x1 � m7y1 � 1� z�x1; y1�m10

ÿ x1;

v�x1; y1� � m3x1 � m4y1 � m5 � z�x1; y1�m9

m6x1 � m7y1 � 1� z�x1; y1�m10

ÿ y1;

�8�

where m � fm1; . . . ;m10g are the motion parame-ters of an uncalibrated camera and z�x; y� is theprojective depth. Above motion model is valid forany pinhole camera model and even can cope withtime varying internal camera parameters. Theprojective coordinates are related to the true Eu-clidean coordinates through the 3D projectivecollineation, which can be recovered by a self-calibration algorithm using only projective infor-mation. Previous direct approaches obtained themotion and structure parameters through theminimization of Eq. (7). In this paper, we considerthe following two formulae that take into accountthe depth discontinuity:

E �X

i

qD�I2�xi � ui; yi � vi� ÿ I1�xi; yi�; rD�

�X

i

Xt2Ni

qs�zt ÿ zi; rS�; �9�

E �X

i

qD�I2�xi � ui; yi � vi� ÿ I1�xi; yi�; rD�

�X

i

Xt2Ni

qS�ut ÿ ui; rS�; �10�

J.-E. Ha, I.-S. Kweon / Pattern Recognition Letters 21 (2000) 999±1011 1001

where Ni represents the neighborhood of the cur-rent pixel, and we consider four neighborhoods ofeast, west, south and north. qD and qS representrobust M-estimator for the data conservation termand spatial coherence term, and we use equal ro-bust estimator for both of them. Eq. (9) is usefulwhen we use the calibrated 3D motion model. Butin case of the uncalibrated 3D motion model thedi�erence in projective depth has no physicalmeaning. Therefore, we use the formulation ofEq. (10) to consider the depth discontinuity in directmotion estimation with uncalibrated 3D motionmodel.

Black and Rangarajan (1996) shows that robustq-functions are closely related to the traditionalline-process approaches for coping with disconti-nuities. For many q-functions it is possible to re-cover an equivalent formulation in terms of analogline processes. Based on this observation, throughthe second term in Eq. (10), we can take into ac-count the discontinuity in the robust direct esti-mation with uncalibrated 3D motion model.Therefore, we can impose a global constraint forthe image motion through the uncalibrated 3Dmotion model and can recover optic ¯ow pre-serving discontinuity.

The objective function of Eq. (10) has a non-convex form and it has many local minima. We usethe graduated non-convexity (GNC) algorithm byBlake and Zisserman (1987) to minimize this non-convex object function. GNC algorithm ®nds thesolution by varying the functional form. In therobust estimation, this adjustment of the func-tional form is possible through the adjustment ofthe scale parameters. In each ®xed scale, a gradientbased method can ®nd the local minimum. We usethe simultaneous over relaxation (SOR) as thelocal minimizer.

The update formula of each parameter is

pn�1 � pn ÿ c1

T �p�oEop; �11�

where 0 < c < 2 is an overrelaxation parameterthat is used to overcorrect the estimate of pn�1 atstage n� 1. When 0 < c < 2, the method is provento converge. The term T �p� is an upper bound onthe second partial derivative of E.

The gradient of the function with respect to themotion and depth is

oEomk�X

i

q0D�ei; rD� I2xoui

omk

�� I2y

ovi

omk

��X

i

Xt2Ni

q0S�ui

�ÿ ut;rS� oui

omk

�ÿ out

omk

��;

�12�

oEozk�X

i

q0D�ei; rD� I2xoui

ozk

�� I2y

ovi

ozk

��X

i

Xt2Ni

q0S�ui

�ÿ ut; rS� oui

ozk

�ÿ out

ozk

��;

�13�where ei � I2�xi � ui; yi � vi� ÿ I1�xi; yi�.

We use the Lorentzian q-function:

q�x; r� � log 1

�� 1

2

xr

� �2�; �14�

w�x; r� � oqox� 2x

2r2 � x2: �15�

3.1. Initialization

There are many unknowns, 11 motion param-eters and projective depth at each pixel. Initial-ization of the these unknowns is a di�cultproblem. We initialize the motion parameters un-der the assumption that there is no motion be-tween two images, i.e., m0 � m4 � 1 and all othervalues are 0. The initial projective depth is set to aconstant.

Other initialization is also possible. For exam-ple, in the work of Szeliski and Coughlan (1997),the uncalibrated motion parameters are initializedaccording to a given image sequence using a prioriinformation. While we do not use such informa-tion, thus our algorithm is more ¯exible.

3.2. Propagation of motion parameters in pyramidstructure

A coarse-to-®ne strategy is usually employedto handle large displacements by constructing a


pyramid of spatially ®ltered and sub-sampled im-ages. In an optic ¯ow algorithm that computes�u; v� at each pixel, the result of coarse pyramidlevel is propagated with constant multiplier, usu-ally 2, to the next ®ner level. In a direct methodwith uncalibrated 3D motion model, we shouldtransfer the motion parameter of uncalibratedcamera and depth at each pixel to the next ®nerlevel.

We use linear interpolation when we trans-fer depth to the next ®ner pyramid level. Themotion transfer equation to the ®ner pyramid levelis obtained using the following facts. Whenwe have obtained the corresponding points �x1; y1�;�x2; y2� at the pyramid level N ÿ 1, their positions atthe next ®ner pyramid level N are �x01; y01� ��2x1; 2y1�, �x02; y 02� � �2x2; 2y2�. These correspond-ing points should satisfy the motion model ofEq. (8). By substituting these into Eq. (8) we obtain

x02 �m0x01 � m1y01 � 2m2 � 2z�x1; y1�m8

�m6=2�x01 � �m7=2�y01 � 1� z�x1; y1�m10

;

y02 �m3x01 � m4y01 � 2m5 � 2z�x1; y1�m9

�m6=2�x01 � �m7=2�y01 � 1� z�x1; y1�m10

:

�16�

From Eq. (16), the transfer formula of motionparameters to the next ®ner pyramid level are

m2 2m2; m5 2m5; m8 2m8;

m9 2m9; m6 m6

2; m7 m7

2

�17�

Here, other parameters have the same values.

3.3. Update of the motion and depth parameters ateach pyramid level

We ®rst update the motion parameters in N -iteration, then we update the depth in N -iterationat each pyramid level. The scales rD; rS are de-creased using rn�1 � cr00 and c is set 0.95 at eachpyramid level. The motion parameters are updatedwith following sequences: fm2;m5;m0;m1;m3;m4;m6;m7;m8;m9;m10g. The projective depth at eachpixel is updated by two passings even and oddgroups, as shown in Fig. 1.

4. Experimental results

In this section, we ®rst compare our algorithmwith other two algorithms: one by Black andAnandan (1996) and another by Szeliski andCoughlan (1997) using the yosemite image

Fig. 1. Two updating groups in projective depth updating.

Fig. 2. The yosemite image sequence: (a) yosemite 11 and (b) yosemite 12.


sequence. Black and Anandan (1993)'s algorithmis a local ¯ow algorithm, and it obtains the pixel-wise optic ¯ow using the robust formulation ofdata conservation an spatial coherence terms.Szeliski and Coughlan (1997)'s algorithm com-putes the pixel-wise optic ¯ow using the uncali-

brated 3D motion in a direct framework, and it isbased on the least-squares formulation.

Fig. 2 represents yosemite images 11 and 12and we excluded the upper cloudy part because ithas no true optic ¯ow ®eld. With the true optic¯ow ®eld, three methods are compared in terms

Table 1

Flow error

(mean/std)

Flow error (percentage)

<1� <2� <3� <5� <10�

Black and Anandan (1996) 4:46�/4:21� 6% 22% 40% 75% 93%

Szeliski and Coughlan (1997) 4:11�/12:5� 11.2% 36.6% 57.1% 81.4% 97.2%

Proposed approach 4:02�/4:75� 6.8% 21.6% 38.5% 73.6% 96.8%

Fig. 3. SRI tree images.

Fig. 4. The result by Black and Anandan (1996)'s robust ¯ow algorithm: (a) disparity map of u, (b) disparity map of v and (c) disparity

map by magnitude of the disparity.


of two error measures: the average angle errorand the standard deviation of angle error of theoptic ¯ow used in the work of Barron et al.(1994). The angle error of the optic ¯ow is ob-tained using cosÿ1�ht he�, where ht is the true 3Dunit velocity and he is the estimated 3D unit ve-locity. The 3D unit velocity is de®ned ash � �u; v; 1�T= ��

u2 � v2 � 1p

, where u and v repre-sent the component of the motion vector. Weused frD; rSg � f25=

��2p

; 0:4=��2p g as the initial

scale parameters and four pyramid levels for theyosemite image.

Flow error of Black and Anandan (1996)'salgorithm is from the original result of their paperand the result of Szeliski and Coughlan (1997)'salgorithm is computed from our own implemen-

Fig. 5. The result by Szeliski and Coughlan (1997)'s mixed global and local algorithm: (a) disparity map of u, (b) disparity map of v

and (c) disparity map by magnitude of the disparity.

Fig. 6. The result by proposed algorithm: (a) disparity map of u, (b) disparity map of v and (c) disparity map by magnitude of the

disparity.

Fig. 7. Local areas for the ¯ow comparision (window size

40� 40).


tation of their algorithm. Our algorithm gives acomparable result as shown in Table 1. Szeliskiand Coughlan (1997)'s algorithm is based on the

least-squares formulation so that it cannot dealwith the outliers, thus occasionally gives lagererror. Black and Anandan (1996)'s algorithm

Fig. 8. Local ¯ow characteristic of each algorithm for the region A in Fig. 7: the ®rst row is by Black and Anandan (1996), the second

row is by Szeliski and Coughlan (1997), and the third row is by proposed algorithm and the ®rst column shows the computed disparity

of u and the second column is disparity of v.


Fig. 9. Local ¯ow characteristic of each algorithm for the region B in Fig. 7: the ®rst row is by Black and Anandan (1996), the second

row is by Szeliski and Coughlan (1997), and the third row is by proposed algorithm and the ®rst column shows the computed disparity

of u and the second column is disparity of v.


employs the robust estimation, but their algo-rithm is based on the local interaction and it hasno global constraint like that of the proposedalgorithm imposed by the uncalibrated 3D mo-tion model. Proposed algorithm does not producelocally large error and gives a globally stable ¯ow,which is veri®ed in the following experiments.

Fig. 3 shows two sample images of the SRI treesequence. This sequence is obtained through thetranslation of camera in horizontal direction, thuswe can directly relate the disparity of u with thedepth of the given scene. Figs. 4±6 represent thedisparity map each by Black and Anandan

(1996)'s robust ¯ow, Szeliski and Coughlan(1997)'s mixed global and local approach, and theproposed algorithm. We used frD; rSg � f10=

��2p

;1:0=

��2p g as the initial scale parameters and three

pyramid levels for the SRI tree image.The proposed algorithm gives the minimum

vertical disparity and globally stable results.Black's algorithm is based on the local charac-teristics and it gives locally sensitive results.Szeliski's algorithm is based on the least-squaresand it produces large erroneous results whenoutlier exists locally. The proposed algorithm usesthe uncalibrated 3D motion model and robust

Fig. 10. (a) Original di�erence, (b) compensated di�erence image by proposed algorithm and (c) compensated di�erence image by

Black and Anandan (1996), and mean absolute brightness error of intensity di�erence is: (a) 14.9, (b) 4.33 and (c) 5.96.

Fig. 11. Epipolar lines from the uncalibrated motion parameters.


Fig. 12. The disparity map of SRI tree image sequence produced by the proposed algorithm using the same parameter.

Fig. 13. (a) and (b) are two frames from the original image sequences that contain camera motion and independently moving objects,

(c) original di�erence image and (d) compensated di�erence image by proposed algorithm.


estimation to handle more e�ectively the locallynoise data.

Fig. 7 shows the two selected local areas for the¯ow comparison appear in Figs. 8 and 9.

Black and Anandan (1996)'s algorithm and ourproposed algorithm give a discontinuity preservingoptic ¯ow and Szeliski's algorithm gives a smoothoptic ¯ow at the depth discontinuity as shown inFigs. 8 and 9. Szeliski and Coughlan (1997)'smixed global and local approach gives smoothdepth ®eld by interpolating the depth of controlpoints. Black and Anandan (1996)'s and proposedalgorithm gives a discontinuity preserving optic¯ow ®eld and proposed algorithm gives moreglobally globally stable results, this is veri®ed inFigs. 8 and 9.

Fig. 10 represents the original di�erence imageand the motion compensated di�erence image bythe proposed algorithm and Black and Anandan(1996)'s algorithm, respectively. Our algorithmgives improved results, particularly at the depthdiscontinuity. Fig. 11 represents the epipolar lineobtained directly from the uncalibrated motionparameters. Marked points on the epipolar linesrepresent the corresponding points. We obtainedthe Fundamental matrix from the two perspectivecamera projection matrices. The true epipolar linesof the SRI tree sequence are parallel. The com-puted epipolar lines in Fig. 11 are approximatelyparallel, which demonstrate that the obtained un-calibrated motion parameters are reasonably ac-curate.

Fig. 12 represents the performance of the pro-posed algorithm in successive image sequence us-ing the same parameter settings. The disparitymaps show consistent result at the successive im-ages.

Fig. 13 shows an image sequence obtained in anindoor scene where two objects swing indepen-dently while the camera moves. Using the pro-posed algorithm, we can segment out theindependently moving objects directly. Traditionalapproaches ®rst obtain the optic ¯ow and thenproceed the bottom-up approaches employing themotion model. Our algorithm directly compen-sates the dominant camera motion while detectingindependently moving objects. Since we use theuncalibrated 3D motion model, if there exists a

dominant 3D motion of camera we can extractmoving objects directly.

5. Conclusion

In this paper, we proposed a robust direct mo-tion estimation using uncalibrated 3D motionmodel considering depth discontinuity. Previousdirect method using 3D motion model obtainedlocally smooth depth ®eld, thus giving erroneousmotion ®eld at depth discontinuity. By introducingdiscontinuity preserving regularization termthrough the robust estimation, we can obtain im-proved ¯ow at the depth discontinuity. Also,proposed algorithm is based on the robust esti-mation of uncalibrated 3D motion model, thus wecan easily extract independently moving objectswith respect to egomotion using the compensatedresidual image produced by the proposed algo-rithm.

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