ieee transactions on pattern analysis and …bebis/mathmethods/lda/belhumeur.pdf · eigenfaces vs....

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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 19, NO. 7, JULY 1997 711 Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection Peter N. Belhumeur, Joao P. Hespanha, and David J. Kriegman Abstract—We develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression. Taking a pattern classification approach, we consider each pixel in an image as a coordinate in a high-dimensional space. We take advantage of the observation that the images of a particular face, under varying illumination but fixed pose, lie in a 3D linear subspace of the high dimensional image space—if the face is a Lambertian surface without shadowing. However, since faces are not truly Lambertian surfaces and do indeed produce self-shadowing, images will deviate from this linear subspace. Rather than explicitly modeling this deviation, we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation. Our projection method is based on Fisher’s Linear Discriminant and produces well separated classes in a low-dimensional subspace, even under severe variation in lighting and facial expressions. The Eigenface technique, another method based on linearly projecting the image space to a low dimensional subspace, has similar computational requirements. Yet, extensive experimental results demonstrate that the proposed “Fisherface” method has error rates that are lower than those of the Eigenface technique for tests on the Harvard and Yale Face Databases. Index Terms—Appearance-based vision, face recognition, illumination invariance, Fisher’s linear discriminant. 1 INTRODUCTION 0162-8828/97/$10.00 © 1997 IEEE ———————————————— Fig. 1. The same person seen under different lighting conditions can appear dramatically different: In the left image, the dominant light source is nearly head-on; in the right image, the dominant light source is from above and to the right.

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Page 1: IEEE TRANSACTIONS ON PATTERN ANALYSIS AND …bebis/MathMethods/LDA/belhumeur.pdf · Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection Peter N. Belhumeur,

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 19, NO. 7, JULY 1997 711

Eigenfaces vs. Fisherfaces: RecognitionUsing Class Specific Linear Projection

Peter N. Belhumeur, Joao�

P. Hespanha, and David J. Kriegman

Abstract—We develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression.Taking a pattern classification approach, we consider each pixel in an image as a coordinate in a high-dimensional space. We takeadvantage of the observation that the images of a particular face, under varying illumination but fixed pose, lie in a 3D linearsubspace of the high dimensional image space—if the face is a Lambertian surface without shadowing. However, since faces arenot truly Lambertian surfaces and do indeed produce self-shadowing, images will deviate from this linear subspace. Rather thanexplicitly modeling this deviation, we linearly project the image into a subspace in a manner which discounts those regions of the

�face with large deviation. Our projection method is based on Fisher’s Linear Discriminant and produces well separated classes in alow-dimensional subspace, even under severe variation in lighting and facial expressions. The Eigenface technique, another methodbased on linearly projecting the image space to a low dimensional subspace, has similar computational requirements. Yet, extensiveexperimental results demonstrate that the proposed “Fisherface” method has error rates that are lower than those of the Eigenfacetechnique for tests on the Harvard and Yale Face Databases.

�Index Terms—Appearance-based vision, face recognition, illumination invariance, Fisher’s linear discriminant.

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

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0162-8828/97/$10.00 © 1997 IEEED

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• E F G H I J F K L M H L G N O J F J F G P G Q J G L R K L P K S T I J H J O K Q H U V O M O K Q H Q W P K Q J L K U X Y G T J ZK R [ U G \ J L O \ H U [ Q ] O Q G G L O Q ] X ^ H U G _ Q O ` G L M O J a X b G N c H ` G Q X P E d e f g d h i g e j Z[ h S H O U k l m G U F I S G I L X n L O G ] S H Q o p a H U G Z G W I X F G M T H Q F H p a H U G Z G W I Zq r s t u v w x y z w { v { x | { } ~ � � { � � ~ � � � w { | x u { } � � q r w � ~ � � � � � { v � � � { s } { } � � w r v v { y �z r s v { � � � � � r t � � r s �� � w x s � � w � r z x � s � s � � z r x s x s � w { y w x s z u � � z � x u r w z x v � { � y � { r u { u { s } { � � r x � z � �z w r s u y r � x � v � � y t z { w � � w � � r s } w { � { w { s v { � � � � � � � � � � t � � { w ~ � � � � � �Fig. 1. The same person seen under different lighting conditions canappear dramatically different: In the left image, the dominant lightsource is nearly head-on; in the right image, the dominant light sourceis from above and to the right.

Page 2: IEEE TRANSACTIONS ON PATTERN ANALYSIS AND …bebis/MathMethods/LDA/belhumeur.pdf · Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection Peter N. Belhumeur,

712 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 19, NO. 7, JULY 1997

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Fig. 2. A comparison of principal component analysis (PCA) andFisher’s linear discriminant (FLD) for a two class problem where datafor each class lies near a linear subspace.

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BELHUMEUR ET AL.: EIGENFACES VS. FISHERFACES: RECOGNITION USING CLASS SPECIFIC LINEAR PROJECTION 715

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Fig. 3. The highlighted lines of longitude and latitude indicate the lightsource directions for Subsets 1 through 5. Each intersection of a lon-gitudinal and latitudinal line on the right side of the illustration has acorresponding image in the database.

Fig. 4. Example images from each subset of the Harvard Database used to test the four algorithms.

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716 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 19, NO. 7, JULY 1997

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Extrapolating from Subset 1Method Reduced Error Rate (%)

Space�

Subset 1 Subset 2 Subset 3Eigenface 4 0.0 31.1 47.7

10 0.0 4.4 41.5Eigenface 4 0.0 13.3 41.5w/o 1st 3� 10 0.0 4.4 27.7

Correlation 29 0.0 0.0 33.9Linear Subspace 15 0.0 4.4 9.2

Fisherface 4 0.0 0.0 4.6

Fig. 5. Extrapolation: When each of the methods is trained on images with near frontal illumination (Subset 1), the graph and corresponding table showthe relative performance under extreme light source condit

�ions.

Page 7: IEEE TRANSACTIONS ON PATTERN ANALYSIS AND …bebis/MathMethods/LDA/belhumeur.pdf · Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection Peter N. Belhumeur,

BELHUMEUR ET AL.: EIGENFACES VS. FISHERFACES: RECOGNITION USING CLASS SPECIFIC LINEAR PROJECTION 717

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& � � � � � � � � � � � � � � � � � � � 9 � � � � � � � � � �� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � 8 � � � � "� � � � � � � � � � � � � / � � � * � � � � � � � ) � � � � � � � � � � � ! � � � � � 9 � � � � � � � � � � � ! � � � � � � � � � � � � $ � � � "

Interpolating between Subsets 1 and 5Method Reduced Error Rate (%)

Space Subset 2 Subset 3 Subset 4Eigenface 4 53.3 75.4 52.9

10 11.11 33.9 20.0Eigenface 4 31.11 60.0 29.4w/o 1st 3 10 6.7 20.0 12.9

Correlation 129 0.0 21.54 7.1Linear Subspace 15 0.0 1.5 0.0

Fisherface 4 0.0 0.0 1.2

Fig. 6. Interpolation: When each of the methods is trained on images from both near frontal and extreme lighting (Subsets 1 and 5), the graph andcorresponding table show the relative performance under intermediate lighting conditions.

Fig. 7. The Yale database contains 160 frontal face images covering 16 individuals taken under 10 different conditions: A normal image underambient lighting, one with or without glasses, three images taken with different point light sources, and five different facial expressions.

Page 8: IEEE TRANSACTIONS ON PATTERN ANALYSIS AND …bebis/MathMethods/LDA/belhumeur.pdf · Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection Peter N. Belhumeur,

718 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 19, NO. 7, JULY 1997

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Fig. 8. As demonstrated on the Yale Database, the variation in performance of the Eigenface method depends on the number of principal compo-nents retained. Dropping the first three appears to improve performance.

”Leaving-One-Out” of Yale DatabaseMethod Reduced Error Rate (%)

Space�

Close Crop Full FaceEigenface 30 24.4 19.4Eigenfacew/o 1st 3 30 15.3 10.8

Correlation 160 23.9 20.0Linear

Subspace48 21.6 15.6

Fisherface 15 7.3 0.6

Fig. 9. The graph and corresponding table show the relative performance of the algorithms when applied to the Yale Database which containsvariation in facial expression and lighting.�

Page 9: IEEE TRANSACTIONS ON PATTERN ANALYSIS AND …bebis/MathMethods/LDA/belhumeur.pdf · Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection Peter N. Belhumeur,

BELHUMEUR ET AL.: EIGENFACES VS. FISHERFACES: RECOGNITION USING CLASS SPECIFIC LINEAR PROJECTION 719

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MGlasses Recognition� � � � � � � � � � � � � � � � � 4 � � � � � � � � � � � � � � � � �

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TABLE 1C�

OMPARATIVE RECOGNITION ERROR RATES FOR G�

LASSES/�

NO G�

LASSES RECOGNITION USING THE YALE DATABASE

Glasses RecognitionMethod Reduced Space Error Rate

(%)�

PCA 10 52.6Fisherface 1 5.3

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7 . 8 � ) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � "� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � $ � � " � � ( $ � � � � � � � / � � � * � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � $ � � � �

9 � ! � � � � � � $ � � � $ � � � � � � � � � � � � � � � �% � � � � � � � � - = � ! ! � � � � � � ) � � � � � � � � � � � $ � � � � � � � � � � � � � � ? � � � � � � � � � � � � � � � � � � � � � "� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � �( � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 0 � � � �� � � 0 � � ! � � � $ � � � � � � � � � � � � � � � � � � � � � * � � " � + � � 7 � ) � � + � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ) � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � 0 � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � 4 � � � � � ¦ ¥ ¥ ­ - « « © � ¥ ¤ � � � � � � " � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ¨ " � � $ � � � "� � � � � � � 4 � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � "� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � "� � � � � � � � � � � � $ � � � � Ú � 1 � � ) � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � $ � � � � � � � � � � � � � � � � � � � � � � ! " � � � � � � � � � � � � � � � � � � + �

ACKNOWLEDGMENTS> ' 3 � � � � � � ! � � � � � � � � � � ( � , � � � � 2 ( ( = � + " C 7 "� " � + C + � > = � � � � ! � � � � � � � � � � � � ° * ' � � � � � �

Fig. 10. The left image is an image from the Yale Database of a personwearing glasses. The right image is the Fisherface used for determin-ing if a person is wearing glasses.

Page 10: IEEE TRANSACTIONS ON PATTERN ANALYSIS AND …bebis/MathMethods/LDA/belhumeur.pdf · Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection Peter N. Belhumeur,

720 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 19, NO. 7, JULY 1997

* � � � ) � � � � � � � ¡ � � � 9 ? * " C � � : � � � � ( ) , * � ¡ � � �) + C : � � " C + " � " � � < � � � � ( � , ¡ � � � 2 ( ( = � + " C 7 " � " � � � + 2 � ² � � � � � ! � � � � � � � � � � � � ° * ' � � � � � � * � � � ) � � � � � � � � � � � ' � & � & � & " C � 7 ; C C � � � � � , ' � ' � � � � + "C 1 " � " � 1 � 7 8 � � � � � � � ! � � � � � � 0 � � � � � 0 > � � = � � � � � � � �� � � � � � � � � � = � � � � � � 2 � � � � � � � � ( � � � � � � � � � 2 � � � �@ � � � � � � � � � � � � � � � � � � � � � �

REFERENCES�

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� Å Ë � � � ß « ¯ ¨ �     ¥ ¡ ¨ © Ï � Æ § ³ ³ ¥ § £ ­ ¶ ¡ ² � � � ² § ³ ¨ ¥ ´ ¥ § ¨ · ¶ � ¡ ´ ¯ « � ¦ ¹ ¦ � Ï � ° ¢   ¡ ´ � ¦ £ º ¼ [ [ [ E L H Q M Z » H J J G L Q Ñ Q H U a M O M H Q W å H \ F O Q G ¼ Q J G U U O ] G Q \ G £ ¹ §   � � ¿ £¨ § � � Á £ ¢ ¢ � � £ Á à Š � £ Á ¿ ¾ £ â ² ´ � � Ä Ä ¾ �� Å À � Ý � ± � Ì ¥   × � « ´ ¡ ¨ © ¤ � Ô ¡ ¨ ³ £ ­ ¸ � � ¡   Â Ï ¥ ° � ¶ ¡ ² � � � ² § ³ ¨ ¥ ´ ¥ § ¨ ª ¬ ¦ ´ � °Ö ¦ ¥ ¨ ³ � ¯ ¦ ´ § ° Î Ø ª Ç ® ¡ « © Ó ¡ « � £ º » L K \ Z ¼ [ [ [ æ K L n M F K T K Q P K S T I J G LÑ L \ F O J G \ J I L G M R K L å H \ F O Q G » G L \ G T J O K Q £ � Ä Ä ¾ £ ¢ ¢ � ¿ ½ Â Ë Ë �� Å ½ � ß � Ù � Æ � ® § « ¨ £ P K S T I J G L V O M O K Q � � ¡ ° × « ¥ © ³ � £ ± ¡ ¦ ¦ � · ± Ç Ï Æ « � ¦ ¦ £ � Ä ½ Ë �� Å Ä � ¤ � ± � ª ¥   ¹ � « £ Y G J G L S O Q O Q ] Ð F H T G H Q W Ê G R U G \ J H Q \ G _ M O Q ] å I U J O T U G ¼ S hH ] G M £ Æ � Í ´ � � ¦ ¥ ¦ £ ± ¡ ¦ ¦ ¡ ² � ¯ ¦ � ´ ´ ¦ Ç ¨ ¦ ´ ¥ ´ ¯ ´ � § µ Ï � ² � ¨ §   § ³ ¬ £ � Ä ½ Á �� ¾ Á � � � Ý � ¤ § § © � ¡ ° £ ­ ¸ ¨ ¡   ¬ ¦ ¥ ¨ ³ Ç ° ¡ ³ � ¦ § µ � ¯ « ¹ � © ª ¯ « µ ¡ ² � ¦ £ º Ñ L J O R O \ O H U¼ Q J G U U O ] G Q \ G £ ¹ §   � � À £ ¢ ¢ � � � À  � à Á £ � Ä ½ � �� ¾ � � ª � Ò ¡ ¬ ¡ « ¡ ¨ © ® � ± ¯ « ¡ ¦ � £ ­ Í ¥ ° � ¨ ¦ ¥ § ¨ ¡   ¥ ´ ¬ § µ Ç     ¯ ° ¥ ¨ ¡ ´ ¥ § ¨ ¥ ¨ ¸ ¢ ¢ � ¡ « ¡ ¨ ² � ± ¡ ´ ² � ¥ ¨ ³ £ º ¼ [ [ [ P K Q R Z Ê K m K J O \ M H Q W Ñ I J K S H J O K Q £ � Ä Ä Ë �� ¾ Å � Æ � Ò � ß �   � ¯ ° � ¯ « ¡ ¨ © Í � Ý � Ù « ¥ � ³ ° ¡ ¨ £ ­ ¤ � ¡ ´ ¥ ¦ ´ � � ª � ´ § µ Ç ° ¡ ³ � ¦ § µ¡ ¨ â × á � ² ´ ¯ ¨ © � « ¡     Æ § ¦ ¦ ¥ ×   � Ø ¥ ³ � ´ ¥ ¨ ³ � § ¨ © ¥ ´ ¥ § ¨ ¦ ç £ º ¼ [ [ [ » L K \ ZP K Q R Z P K S T I J G L V O M O K Q H Q W » H J J G L Q Ê G \ K ] Q O J O K Q £ � Ä Ä Ë �Peter N. Belhumeur received his ScB degree(with highest honors) in computer and informationengineering from Brown University in 1985. Hereceived an SM and PhD from Harvard Universityin 1991 and 1993, respectively, where he studiedunder a Harvard Fellowship. In 1994, he spend ahalf-year as a Postdoctoral Fellow at the Univer-sity of Cambridge’s Sir Isaac Newton Institute forMathematical Sciences.

Currently, he is an assistant professor of elec-trical engineering at Yale University. He teaches

�courses on signals and systems, pattern and object recognition, andcomputer vision. He is a member of the Yale Center for ComputationalVision and Control and a member of the Army research Center for Imag-ing Science. He has published over twenty papers on image processingand computational vision. He is a recipient of a U.S. National ScienceFoundation Career Award; he has been awarded a Yale University JuniorFaculty Fellowship for the Natural Sciences; and he won the IEEE BestPaper Award for his work on characterizing the set of images of an objectunder variable illumination.

Joaoè P. Hespanha received the Licenciatura andMS degrees in electrical and computer engineer-ing from Instituto Superior Técnico, Lisbon, Portu-gal, in 1991 and 1993, respectively, and a secondMS degree in electrical engineering from YaleUniversity, New Haven, Connecticut, in 1994,where he is currently pursuing the PhD degree inengineering and applied science.

From 1987 to 1989, he was a research assis-tant at Instituto de Engenharia de Sistemas e

�Computadores (INESC) in Lisbon, Portugal, and,

from 1989 to 1990, was an instructor at Fundo para o DesenvolvimentoTecnológico (FUNDTEC) in the areas of electronics, computer science,and robotics. From 1992 to 1993, he was a working partner in Sociedadede Projectos em Sistemas e Computadores, Lda., also in Lisbon. Hisresearch interests include nonlinear control, both robust and adaptive,hybrid systems, switching control, and the application of vision to robotics.

David J. Kriegman received the BSE degree(summa cum laude) in electrical engineering andcomputer science in 1983 from Princeton Uni-versity, where he was awarded the Charles Ira�Young Award for electrical engineering research.He received the MS degree in 1984 and PhD in1989 in electrical engineering from StanfordUniversity, where he studied under a HertzFoundation Fellowship.

Currently, he is an associate professor at theCenter for Computational Vision and Control in the

Departments of Electrical Engineering and Computer Science at YaleUniversity and was awarded a U.S. National Science Foundation YoungInvestigator Award in 1992. His paper on characterizing the set of imagesof an object under variable illumination received the best paper at the1996 IEEE Conference on Computer Vision and Pattern Recognition. Hehas published over sixty papers on object representation and recognition,illumination modeling, geometry of curves and surfaces, structure frommotion, robot planning, and mobile robot navigation.