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580 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 19, NO. 6, JUNE 1997 In Defense of the Eight-Point Algorithm Richard I. Hartley Abstract—The fundamental matrix is a basic tool in the analysis of scenes taken with two uncalibrated cameras, and the eight-point algorithm is a frequently cited method for computing the fundamental matrix from a set of eight or more point matches. It has the advantage of simplicity of implementation. The prevailing view is, however, that it is extremely susceptible to noise and hence virtually useless for most purposes. This paper challenges that view, by showing that by preceding the algorithm with a very simple normalization (translation and scaling) of the coordinates of the matched points, results are obtained comparable with the best iterative algorithms. This improved performance is justified by theory and verified by extensive experiments on real images. Index Terms—Fundamental matrix, eight-point algorithm, condition number, epipolar structure, stereo vision. 1 INTRODUCTION 2 OUTLINE OF THE EIGHT-POINT ALGORITHM 2.1 Notation 0162-8828/97/$10.00 © 1997 IEEE ————————————————
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Page 1: 580 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND …bebis/CS485/Handouts/hartley.pdfB C ' ' 7.1.6Minimizing Point Displacement " D E! ¨ Ë L = Ì Í L = Î Ï Ð ð ¨ Ñ C # ¯ % ,

580 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 19, NO. 6, JUNE 1997

In Defense of the Eight-Point AlgorithmRichard I. Hartley

Abstract—The fundamental matrix is a basic tool in the analysis of scenes taken with two uncalibrated cameras, and the eight-pointalgorithm is a frequently cited method for computing the fundamental matrix from a set of eight or more point matches. It has theadvantage of simplicity of implementation. The prevailing view is, however, that it is extremely susceptible to noise and hencevirtually useless for most purposes. This paper challenges that view, by showing that by preceding the algorithm with a very simplenormalization (translation and scaling) of the coordinates of the matched points, results are obtained comparable with the bestiterative algorithms. This improved performance is justified by theory and verified by extensive experiments on real images.

Index Terms—Fundamental matrix, eight-point algorithm, condition number, epipolar structure, stereo vision.

� � � � � � � � � � � � � � � � � � � � �1 INTRODUCTION

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Fig. 1. Houses Images. The epipoles are a long way from the image centers.

Fig. 2. Statue image. An outdoor scene with the epipoles well away from the center.

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Graphical Presentation of the ResultsÅ � � � � < � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � #� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � Å � � � � � ! � � � � � @ # � � � � � � � � � � � � � � � �� � � � C C � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � @ � � � � � � � � � � � � � " � � ! � � � � � � � � � D � � � � ñ � � � � � � � � � � � � � � � � E � �� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � " � � � � � �� � � � � � � ! � � � � � � � � � � ! � � � � � � C C � � � � � � � � � � ! � � � � �@ � " � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �7.3.1 Effect of Normalization on the Condition NumberÅ � � � < � � � � � � � � � � � � � � � � � C � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � # � � � ! � � � � � � � � � � � � � � � � � � � D � � � 4 � � � E � " � � � � � � � � ! � � � � � � � � � � � � � � � � ' � � � � # � � � � � � � � � � � � � � �� � � � � � ' � � � � � " � � � � � � ! � � � � � � � � � � � � � � � � � C Ü �7.3.2 Effect of Normalization on the Two Stages

of the AlgorithmŠ� � � = � � � � � � � � � � � � � � � � � � � � � � ' � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � # � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � �

Fig. 3. Grenoble museum. The epipoles are close to the image.

Fig. 4. Corridor scene. In the corridor scene the epipoles are right in the image.

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Eight-Point AlgorithmsÅ � � � > � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � ' � � � � �" � � � � � � � � � � � � � � � � � � � # � � � � � # � � � � � � # � � � � � � � � � # � � � � � � � � � � � ñ � � � � � � � � � � � � � � � � � � � ö � � � � � � � � � � � � � � � � � � � � � � � � Å � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �$ � � � � � � � � � � � � � � � � � � � � � D � � � � � � � � � � � � � # � � � � � � � �� � � � � E # � � � � � � � � � � � � � � � # � � � � � � � � � � � � � � � � � � � � � � �D � � � � � � � � � � E � � � � � � � � � � � � � � � � � ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ' � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � # � � � � ! � � � � � � � � � � � � ' � � � � � � � � � � � � �

Fig. 5. Calibration jig. In this calibration jig, the matched points were known extremely accurately.

Fig. 6. Effect of normalization on the condition number. Fig. 7. Effect of normalization on the two stages of the algorithm.

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7.3.4 Comparison of the Eight-Point Algorithm With theOptimal AlgorithmÅ � � � B � � � � � � � � � Å � � � > # � � � � � � � � � � � � � � � � � � � � � �

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7.3.6 Comparison With Other Algorithms" � � � � � � B � # � � C � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � # � � � � � � � � � � � ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � �

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Fig. 8. Comparison of normalized and unnormalized eight-point algorithms.

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Fig. 9. Comparison of the eight-point algorithm with the optimal algorithm.

Fig. 10. Isotropic vs. non-isotropic scaling.

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� � � � � � � � � � � D ; E � � � � � � � � � � � � � � � � � � � � � A � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � \ � � � � � � � � � � � # \ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � # � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Fig. 11. Comparison with other algorithms.

(a) (b)Fig. 12. Reconstruction error. (a) A comparison of the unnormalized and the normalized eight-point algorithms. (b) The normalized eight-point andoptimal algorithms.

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1 � ü � � � � � � � � ü û � � � � 5 � � � � ü � � W � ü � � i þ � � � � K � � � 1 � � � � � � � üW � ü û � 5 � � 5 X i ý � � � � � � ü � � � � � � ü � # � � û � � � � � � � ý � � ü � � � � / + V �Y E � Y � � � + + , �� � ` � a � a û � � � � � 1 � ü � � � � � � � � ü û � X ý � ý � � � � L � � � � ü � K � � 5 Y ü 4 � � � � �ü � � � ü K þ � � � � � � � # � � û � � � � � � � ý � � ü � � � * + * J � Y E � Y � � � + + I �� � � � � � Y � � � � ü � � K � � X � � � � 5 � � � � � � � � ü � � � ü � � � � � � � 4 � � � � � � � � ü � 5F � � � � # M ( 8 N O Z ) N 8 > R @ ' ( 8 : ) U [ O Z O \ 8 ( ] = ^ 8 : 8 > P > _ % ( < % > N ) D ý ý � � / J -* ` * � ! � � ü � 1 � � � � 5 � � � � � � � 0 � ü � � + + , �� � * � ! � � � 3 � � � 5 � � K � � � a � � � � � � � � � 5 1 � 2 � W � � � � K � � E � � � � ü � � �4 � � � � � � � � � � ü � � � ü � � � � � � � W � ü � � � � # 7 8 9 : ' & ) ( ; < = < 8 > U @ 7 7 ; A B c D � � � * � � E � � � � � � � � � � � / ` � � � ý � � � � � � � F � � � � � � � + + V � ý ý � / . - + I �� � , � � � j � W � K � � � H � � û � ! � � " � � ü � � L � � � � ü � K � � � � � � � � � � � � � � � � � � #M ^ < g O k ( % > = O S O Z 8 N O l 8 > R O � � � � � , , * � ý ý � � - V J � � + + ` �� � V � 2 � � � ! � � � 3 � ! � � � � � � � � K � � � � � � � H � � H K � � � 5 2 � � F � ü ü � � � � � � �$ ' 9 ) ( < N % g S ) N < : ) = < > 7 m k ^ ) f ( & 8 Q Z N < ) > & < Q < N 7 8 9 : ' & < > T � � � � � � � 5 � � �

X � � � � � 5 þ � � � � � � 5 � � 4 � � � ! � � � � + / / �� � . � n � X � � ü H � � � � � f > P > & ( 8 R ' N & < 8 > & 8 $ ' 9 ) ( < N % g f > % g o = < = D * � 5 � 5 � E � �G � � H þ j � û � 2 � � � K � � 5 � � � � � + / + �� � I � � � G � � � � � 5 � � � � � � � � � � 4 � � e � � � � � � 5 X ü � � � ü � � � � � û � � �

1 � � � � � � � � � W � ü � � � ! � � � � � ü � � � � � � � � 5 0 � " � � ü 2 � ü û � � � � 5� � � � � � � � # P @ @ @ k ( % > = O M % & & ) ( > f > % g o = < = % > R p % N ^ < > ) P > & ) g g < T ) > N ) � � � � I � ý ý � � , - * J � � + / V �� � J � � � Y � � � � ü � � K � � W � � � � � � � � � � � � � � � � � 1 � ü � � � � � # M ( 8 N O q f S M f

P 9 % T ) [ > R ) ( = & % > R < > T \ 8 ( ] = ^ 8 : � � + + J �� � / � � � � � 3 � � H ü � � � � � � � ü ü � � � � � � � � � � � ü � � � ü � � � � ü ü � � � 5 1 � ü � � # 7 8 9 U: ' & ) ( r ( % : ^ < N = % > R P 9 % T ) M ( 8 N ) = = < > T � � � � + � ý ý � . I - J � � � + J + �� � + � L � � � L � � � � � � 5 � � � � F � � � � � � � p % & ( < s 7 8 9 : ' & % & < 8 > = D * � 5 � 5 � 3 � � �ü � � � � � � W 5 � þ j � û � � � ý H � � 4 � � � ! � � � � + / + �� * ` � � � Y � � � � ü � � K � � X ü � � � ü � � � � � � � � � ü � � � � � � � � ! � � ü � � � � � � 4 � � � � � �� � � ü � 5 � � � � � � � # 7 8 9 : ' & ) ( ; < = < 8 > U @ 7 7 ; A B C D � E � � � � � � � � � � �. / / � � ý � � � � � � � F � � � � � � � + + * � ý ý � . J + - . / J �

Page 14: 580 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND …bebis/CS485/Handouts/hartley.pdfB C ' ' 7.1.6Minimizing Point Displacement " D E! ¨ Ë L = Ì Í L = Î Ï Ð ð ¨ Ñ C # ¯ % ,

HARTLEY: IN DEFENSE OF THE EIGHT-POINT ALGORITHM 593

� * � � � � Y � � � � ü � � K � � ! � � " � � ü � � � � � � � ü � � � ü � � � � � 5 Y � � � � � � ü � � � �W � � ü � ý � � Y � � � � � # P @ @ @ k ( % > = O M % & & ) ( > f > % g o = < = % > R p % N ^ < > ) P > & ) g g < UT ) > N ) � � � � � I � ý ý � � � ` , I - � � ` V � � 0 � ü � � + + V �� * * � Y � X � � � ü û � � � � � 5 � � � H � ü � û ý � 5 þ � W � � � W � � û � � � L � � ý û � � � � � � � � � �� � � � ü � � � � K ü � � � # � � û � � � � � � � ý � � ü * + I � W Y � � � � � � � � � � � � � ü � �

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7 8 > Q O 7 8 9 : ' & ) ( ; < = < 8 > � � + + . � ý ý � � � ` I V - � � ` J ` �� * I � � � Y � � � � ü � � K � � � � � � � � � W � ü û � 5 � � � � � � � � ü � � � ü � � � � � � � � � � � � � 5 ! � � � ü � # M ( 8 N O P > & t g 7 8 > Q O 7 8 9 : ' & ) ( ; < = < 8 > � � + + . � ý ý � / / * - / / J �� * J � Y � X � � � ü û � � � � � 5 � � û � � � 1 � � � � � � � � � 1 � ü � Y � ý � ü � K � � � � ü � # M ( 8 N O

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Richard I. Hartley received the BSc degree inmathematics from the Australian National Uni-versity and MSc and PhD degrees also inumathematics from the University of Toronto. Healso holds an MS degree in computer sciencefrom Stanford University. He has held variousresearch positions in mathematics at the Univer-sity of Frankfurt, Germany; Columbia University,New York; and Melbourne University, Australia,carrying out research in the area of 3D geometricand algebraic topology. Since 1985, he has been

employed at GE’s Corporate Research and Development Center,where he has carried out research in the areas of VLSI CAD, rapidvprototyping of electronic systems, DSP circuit design, and computervision. His interests include automated techniques for DSP chip andmultichip module design. He was the lead developer of the Parsifalsilicon design system, which was used extensively within GE, and hehas recently published a book on digit-serial computation, describingthat work. In recent years, Dr. Hartley has concentrated much of his

wresearch effort in the areas of computer vision and photogrammetry,particularly related to camera modeling and structure from motion fromuncalibrated images, as well as industrial and medical applications ofcomputer vision techniques. He is the author of over 70 research pa-pers in the areas of photogrammetry, geometric topology, geometricvoting theory, computational geometry, and computer-aided design. Heuholds 30 U.S. patents in the areas of CAD, circuit design, DSP design,and industrial and medical imaging.