Abstract:
In one embodiment of the invention, a pattern recognition apparatus comprises a unit for inputting a pattern of a to-be recognized category; and a processor with a memory for: generating input subspace; calculating and storing reference subspaces; storing constraint subspaces for extracting features; projecting the input subspace and the reference subspaces respectively onto the constraint subspaces; calculating similarities between the respective reference subspaces and the input subspace in such projected state; combining the similarities in respect of the constraint subspaces on each of the reference subspaces; and identifying the to-be recognized category with a category corresponding to one of the reference subspaces, if the combined similarity between the one of reference subspace and the input subspace is highest among them.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
   This application is based upon and claims the benefit of priority from the prior Japanese Patent Application No. 2003-376267, filed on Nov. 5, 2003; the entire contents of which are incorporated herein by reference. 
   BACKGROUND OF THE INVENTION 
   This invention relates to an art that extracts effective features for the pattern recognition, and thereby carries out stably the pattern recognition. 
   The pattern recognition art that determines a category of the unknown pattern is needed in various fields. As one of the pattern recognition art, Watanabe et al (S. Watanabe, N. Pakvasa, Subspace method of pattern recognition, Proc. 1st Int. J. Conf. on Pattern Recognition, 1973) propose the subspace method. The subspace method is advantageous in that feature extraction and classification can be executed at the same time and extension is easy from two categories to a plurality of categories. In the subspace method, a similarity is determined by angle between an input vector converted from an unknown pattern and a reference subspace. The reference subspace is generated by the principal component analysis from a previously obtained vector of one category. When the similarity is equal to or greater than a threshold, the input vector can be determined the category. 
   JP-A-11(1999)-265452 and Maeda et al (K. Maeda, T. Watanabe, A Pattern Matching Method with Local Structure, IEICE Trans. D-II Vol. J68-D, No. 3, 345-352, 1985) propose the mutual subspace method which determines similarity by angle between the input subspace and the reference subspace. The mutual subspace method is more robust against pattern variations and noise because of using an input subspace instead of an input vector. The similarity S between subspace P and subspace Q is calculated by the following equation.
 
S=cos 2  θ  (1)
 
where θ represents the angle between P and Q. This angle is called canonical angle.
 
   If two subspaces are equal, then θ=0. Described in JP-A-11(1999)-265452 cited before, cos 2  θ is obtained by determining a maximum eigenvalue of the following matrix X. 
                 Xa   =     λ   ⁢           ⁢   a             (   2   )                 X   =     (     x   ij     )       ,     (     i   ,     j   =     1   ~   N         )             (   3   )                 (     x   ij     )     =       ∑     1   ≤   k   ≤   N       ⁢       (       ψ   i     ,     ϕ   k       )     ⁢     (       ϕ   k     ,     ψ   j       )                 (   4   )               
where ψ i  represents an i-th basic vector on the subspace P. φ j  represents an j-th basic vector on the subspace Q. N represents the number of dimensions of the subspace.
 
   Furthermore, in order to enhance the recognition accuracy for the mutual subspace method, JP-A-2000-30065 and Fukui et al (K. Fukui, O. Yamaguchi, K. Suzuki, K. Maeda, Face Recognition under Variable Lighting Condition with Constrained Mutual Subspace Method—Learning of Constraint Subspace to Reduce Influence of Lighting Changes—, IEICE Trans. D-II Vol. J82-D-II, No. 4, 613-620, 1999) propose the constrained mutual subspace method. This technique is that the input subspace and the reference subspace are projected onto a constraint subspace for emphasizing extra-category variation which is considered effective for the pattern recognition. The similarity S c  under the constrained mutual subspace method, determined by an angle θ c  between subspace P c  and subspace Q c  which are projected onto a constraint subspace C (Equation (5)).
 
S C =cos 2  θ C    (5)
 
   The procedure of projection onto a constraint subspace is detailed in JP-A-2000-30065 and Maeda et al cited on p.2. The procedure of generating a constraint subspace is described in the JP-A-2000-30065. 
   When the constraint subspace is used for the pattern recognition, recognition performance becomes unstable because the similarity of a certain category becomes low. If the constraint subspace is changed, the similarity of another category becomes low. For example on the face image recognition system, the person who is occurred with such problem is prone to higher false rejection rate than other persons. 
   SUMMARY OF THE INVENTION 
   Therefore, the present invention proposes a method of using a plurality of constraint subspaces for the pattern recognition. By using a plurality of constraint subspaces, the influence of the above problem is diminished. It is expected to decrease the false rejection rate greatly. 
   Therefore, the present invention provides a pattern recognition apparatus capable of carrying out stable pattern recognition, and a method for the same. 
   According to one embodiment of the present invention, a pattern recognition apparatus comprises: a unit for inputting at least two input patterns; and a processor with a memory for; generating an input subspace from the input patterns; storing reference subspaces which are generated from reference patterns; storing a plurality of constraint subspaces for extracting an effective feature for pattern recognition; projecting the input subspace and the reference subspaces onto each one of the constraint subspaces; calculating similarities between thus projected input subspace and thus projected reference subspaces, on the each one of constraint subspaces; combining the similarities obtained by using the plurality of constraint subspaces, as to find a combined similarity between the input subspace and each of the reference subspaces; and determining a category of the input subspace by comparing the combined similarities. 
   Such construction enables correct pattern recognition in respect of the category of input subspace by using a plurality of constraint subspaces. 
   According to another embodiment of the present invention, a pattern recognition apparatus comprises: a unit for inputting an input pattern; and a processor with a memory for; generating an input vector from the input pattern; storing reference subspaces which are generated from reference patterns; storing a plurality of constraint subspaces for extracting an effective feature for pattern recognition; projecting the input vector and the reference subspaces onto each one of the constraint subspaces; calculating similarities between thus projected input vector and thus projected reference subspaces, on the each one of constraint subspaces; combining the similarities obtained by using the plurality of constraint subspaces, as to find a combined similarity between the input vector and each of the reference subspaces; and determining a category of the input subspace by comparing the combined similarities. 
   Such construction enables correct pattern recognition in respect of the category of input vector by using a plurality of constraint subspaces. 
   By the above embodiments, the pattern recognition is stably executed by utilizing a plurality of feature extractions in order to obtain effective information for the pattern recognition. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a configuration diagram of a pattern recognition apparatus  100 ; 
       FIG. 2  is a configuration diagram of a pattern recognition apparatus  200 ; 
       FIG. 3  is a configuration diagram of a pattern recognition apparatus  300 ; 
       FIG. 4  is a configuration diagram of a pattern recognition apparatus  400 ; 
       FIG. 5  is a configuration diagram of a constraint subspaces learning device  500 ; 
       FIG. 6  is a configuration diagram of constraint subspaces learning device  600 ; 
       FIG. 7  is a chart showing a flow of face image recognition; 
       FIG. 8  is a configuration diagram of a face image recognition apparatus  800 ; 
       FIG. 9  is a chart showing a learning flow of constraint subspaces; 
       FIG. 10  is a chart showing a flow of face image recognition taking account of reducing calculation time; 
       FIG. 11  is a chart showing a learning flow of constraint subspaces based on ensemble learning; and 
       FIG. 12  is a configuration diagram of a constraint subspaces learning device  1200 . 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   The present invention provides a method of using a plurality of constraint subspaces and a learning method of constraint subspaces from learning patterns. These methods are a technical idea proposed for the first time in the present invention. This is newly termed as a “multiple constrained mutual subspace method”. Concerning this, descriptions of using a plurality of constraint subspaces are made in the following first to fourth embodiments. The learning method of constraint subspaces will be described in the fifth and sixth embodiments. 
   First Embodiment 
   In method of using a plurality of constraint subspaces, a similarity combining is needed. The similarity combining methods are divided into two, depending upon whether to combining those based on a fixed weight, or to combining those based on a dynamic weight. At first, the former method of recognition is described as a first embodiment. 
   The present embodiment is shown as a pattern recognition apparatus  100  in  FIG. 1 . The pattern recognition apparatus  100  comprises a pattern input unit  101 , an input subspace generating unit  102 , a reference subspace storing unit  103 , a constraint subspace storing unit  104 , a subspace projecting unit  105 , a similarity calculating unit  106 , a similarity combining unit  107 , and a determining unit  108 . Incidentally, the subspace projecting unit  105 - 1 , . . . ,  105 -M and the similarity calculating unit  106 - 1 , . . . ,  106 -M each exist in the number of M of the constraint subspaces stored in the constraint subspace storing unit  104 . Meanwhile, the units  101 - 108  are to be realized by a program stored on a computer. 
   (1) Pattern Input Unit  101   
   The pattern input unit  101  acquires patterns of an unknown category, and stores it to a memory after conversion into a feature vector. Pattern acquisition maybe at all times. Note that the “pattern” refers to physical information capable of specifying a category, which includes face image, fingerprints, voice, characters and DNA. 
   (2) Input Subspace Generating Unit  102   
   The input subspace generating unit  102 , when feature vectors are acquired in a predefined number, generates an input subspace based on principal component analysis. 
   (3) Reference Subspace Storing Unit  103   
   The reference subspace storing unit  103  stores in the number R of reference subspaces which are generated by the principal component analysis. 
   (4) Subspace Projecting Units  105 - 1 , . . . ,  105 -M 
   Each of the subspace projecting units  105  projects the input subspace and reference subspaces stored in the reference subspace storing unit  103 , onto one of the constraint subspaces stored in the constraint subspace storing unit  104 . The procedure of projection is detailed in JP-A-2000-30065 and Maeda et al cited on p. 2. 
   (5) Similarity Calculating Units  106 - 1 , . . . ,  106 -M 
   Each of the similarity calculating unit  106  calculates similarities between the reference subspaces and the input subspace in a state having been projected onto one of the constraint subspaces, by the mutual subspace method. 
   (6) Similarity Combining Unit  107   
   The similarity combining unit  107  takes an average value, maximum value, minimum value or median value from a plurality of similarities obtained by using constraint subspaces (hereinafter, referred to as a combined similarity) 
   In this description, calculating an average, etc. of similarities for each reference subspace is referred to as “combining”. 
   When the combining is made by taking the average value, a similarity S E  is determined by Equation (6). 
                   S   E     =       1   M     ⁢       ∑     1   ≤   i   ≤   M       ⁢       cos   2     ⁢     θ     C   i                     (   6   )               
where M represents the number of constraint subspaces, and θ ci  represents an angle between P ci  and Q ci  which are the subspace P, Q projected onto a constraint subspace C i . Meanwhile, when the combining is made by taking a maximum value, a minimum value or a median value, the similarities S E  can be respectively determined by the following equation.
 S E =max{cos 2  θ C     1    . . . , cos 2  θ C     M   }  (7) S E =min{cos 2  θ C     1    . . . , cos 2  θ C     M   }  (8) S E =median{cos 2  θ C     1    . . . , cos 2  θ C     M   }  (9) 
   Besides, there is a method that, after selecting candidates by the constrained mutual subspace method using each constraint subspaces, category is finally determined by decision-by-majority or logical sum from the candidates. 
   Incidentally, combining the similarities is carried out by the number R of the reference subspaces stored in the reference subspace storing unit  103 . Each combined similarity represents a similarity between the reference subspace and the input subspace. 
   (7) Determining Unit  108   
   The determining unit  108  determines the category of the input subspace from the combined similarity which is the highest value and greater than a preset threshold. 
   Second Embodiment 
   Now a second embodiment is explained on the basis of  FIG. 2 . 
   The first embodiment was on the former of the foregoing similarity combining methods. Now explained is, as a second embodiment, the latter method that combines similarities through a dynamic weight relying upon the input/reference subspaces. 
   This embodiment is shown as a pattern recognition apparatus  200  in  FIG. 2 . 
   The pattern recognition apparatus  200  comprises various units  201 - 208  functioning similarly to the units  101 - 108  of the first embodiment, and a weight calculating unit  209 . Incidentally, the subspace projecting unit  205  and the similarity calculating unit  206  each exist in the number of M of the constraint subspaces stored in the constraint subspace storing unit  204  similarly to the first embodiment. 
   The weight calculating unit  209  selects an optimal constraint subspace for the input/reference subspace from a plurality of constraint subspaces, or makes a weighting with an adaptation of the input/reference subspace to the constraint subspace. In this case, the similarity S E  in the similarity combining unit  207  is in a weighed sum, to be determined by Equation (10). 
                   S   E     =       ∑     1   ≤   i   ≤   M       ⁢       ω   i     ⁢     cos   2     ⁢     θ     C   i                   (   10   )                   ∑     1   ≤   i   ≤   M       ⁢     ω   i       =   1           (   11   )               
where wi (1≦i≦M) represents a weight on each constraint subspace (total sum is assumably 1.0).
 
   As for how to calculate a weight wi in the weight calculating unit  209 , it can be considered to use an angle θ C     1i    between the input/reference subspace and a subspace C 1i  defined by the learning patterns for the constraint subspace. This is based on the fact that recognition accuracy increases when the variation of learning patterns and the variation of input/reference patters are similar. The subspace C 1i  is generated from learning patterns of the constraint subspace C 1 . In this case, weight wi can be expressed by Equation (12). 
   
     
       
         
           
             
               
                 
                   w 
                   i 
                 
                 = 
                 
                   
                     
                       cos 
                       2 
                     
                     ⁢ 
                     
                       θ 
                       
                         C 
                         
                           l 
                           i 
                         
                       
                     
                   
                   
                     
                       ∑ 
                       
                         1 
                         ≤ 
                         i 
                         ≤ 
                         M 
                       
                     
                     ⁢ 
                     
                       
                         cos 
                         2 
                       
                       ⁢ 
                       
                         θ 
                         
                           C 
                           
                             l 
                             i 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 12 
                 ) 
               
             
           
         
       
     
   
   So far explained is the second embodiment. 
   Third Embodiment 
   Now a third embodiment is explained on the basis of  FIG. 3 . 
   The similarity calculating unit  106  of the first embodiment, although using the mutual subspace method, can be replaced with a similarity calculating unit using the subspace method. This case is described as a third embodiment. 
   The present embodiment is shown by a pattern recognition apparatus  300  in  FIG. 3 . The pattern recognition apparatus  300  comprises a pattern input unit  301 , a reference subspace storing unit  302 , a constraint subspace storing unit  303 , projecting units  304 , similarity calculating units  305 , a similarity combining unit  306 , and a determining unit  307 . In the apparatus, number of the projecting units  304  and the number of the similarity calculating units  305  are identical with the number (“M”) of the constraint subspaces stored in the constraint subspace storing unit  303 . 
   (1) Pattern Input Unit  301   
   The pattern input unit  301  has the similar function to the pattern input unit  101  of the first embodiment. 
   (2) Reference Subspace Storing Unit  302   
   The reference Subspace Storing Unit  302  has the similar function to the reference Subspace Storing Unit  102  of the first embodiment. 
   (3) Projecting Units  304   
   Each of the projecting input vector and units  304  projects the reference subspaces in the number of R stored in the reference subspace storing unit  302 , onto one of the constraint subspaces stored in the constraint subspace storing unit  303 . 
   (4) Similarity Calculating Units  305   
   Each of the similarity calculating units  305  calculates similarities between the input vector and the reference subspace projected onto one of the constraint subspace, by the subspace method. 
   (5) Similarity Combining Unit  306   
   The similarity combining unit  306  as the similar function to the determining unit  107  of the first embodiment 
   (6) Determining Unit  307   
   The determining unit  307  has the similar function to the determining unit  108  of the first embodiment. 
   Fourth Embodiment 
   Now a fourth embodiment is explained on the basis of  FIG. 4 . 
   In the second embodiment, replacement is similarly possible with a similarity calculating unit using the subspace method based. This case is explained as a fourth embodiment. 
   The present embodiment is shown as a pattern recognition apparatus  400  in  FIG. 4 . The pattern recognition apparatus  400  comprises units  401 - 407  functioning similarly to the units  301 - 307  of the third embodiment, and a weight calculating unit  408 . In the apparatus, number of the projecting units  404  and the number of the similarity calculating units  405  are identical with the number (“M”) of the constraint subspaces stored in the constraint subspace storing unit  403 , similarly to the third embodiment. 
   The weight calculating unit  408  selects an optimal constraint subspace for the input vector or reference subspace from a plurality of constraint subspaces, or makes a weighting with an adaptation of an input vector or reference subspace and constraint subspace. 
   So far explained is the fourth embodiment. 
   Incidentally, the pattern recognition apparatus  100 ,  200 ,  300 ,  400  may store a program for realizing the functions on a storage medium of HDD, FDD, CD, DVD, memory or the like, to be invoked onto the computer when carrying out a recognition. 
   Fifth Embodiment 
   Now a fifth embodiment is explained on the basis of  FIG. 5 . 
   The fifth and sixth embodiments describe means of efficiently leaning a plurality of constraint subspaces from leaning patterns. 
   H. Aso et al (H. Aso, K. Tsuda, N. Murata, statistics for the pattern recognition and the machine learning, Iwanami Shoten, 2003) describes ensemble learning, including bugging and boosting as a representative method. In bugging, sampling is repeatedly made from learning patterns, to learn a plurality of classifiers, in boosting; the next classifier is generated by giving weights to the mistaken patterns in the previous classifier. The below describes a learning method of constraint subspaces in the framework of ensemble learning. 
   At first, the fifth embodiment describes an introduction of the bugging framework to learn constraint subspaces. 
   The present embodiment is shown as constraint subspaces learning apparatus  500  in  FIG. 5 . The constraint subspaces learning apparatus  500  comprises a learning subspaces storing unit  501 , a learning subspace selecting unit  502 , a constraint subspace learning unit  503  and a constraint subspace storing unit  504 . 
   (1) Learning Subspace Storing unit  501   
   The learning subspace storing unit  501  stores learning subspaces in the number of K prepared for ensemble learning. The learning subspaces are generated by a principal component analysis from learning patterns belonging to the category. 
   (2) Learning Subspace Selecting Unit  502   
   The learning subspace selecting unit  502  randomly selects learning subspaces in the number of T from the learning subspace storing unit  501 . Incidentally, when the learning subspaces of the same category are included among the learning subspaces in the number of T, selection is made once again. 
   (3) Constraint Subspace Learning Unit  503   
   The constraint subspace learning unit  503  generates a constraint subspace from the learning subspaces in the number of T, by the method described in JP-A-2000-30065 and Fukui et al cited on p. 2. 
   (4) Constraint Subspace Storing Unit  504   
   The learning subspace selecting unit  502  and the constraint subspace learning unit  503  are repeatedly used until constraint subspaces in the number of M are stored to the constraint subspace storing unit  504 . 
   Sixth Embodiment 
   Now a sixth embodiment is explained on the basis of  FIG. 6 . 
   The sixth embodiment concerns an introduction of a boosting framework to learn constraint subspaces. 
   The present embodiment is shown as a constraint subspace learning apparatus  600  in  FIG. 6 . The constraint subspace learning apparatus  600  comprises a learning subspace storing unit  601 , a learning weight calculating unit  602 , a constraint subspace learning unit  603  and a constraint subspace storing unit  604 . 
   (1) Learning Subspace Storing Unit  601   
   The learning subspace storing unit  601  has the same function as the learning subspace storing unit  501 . 
   (2) Learning Weight Calculating Unit  602   
   The learning weight calculating unit  602  determines a weight D t (j) of a learning subspace P j , for generating the constraint subspace in the constraint subspace learning unit  603 . The newest weight D t (j) is determined by the following equation, by using the newest constraint subspace C t-1  stored in the constraint subspace storing unit  604 . 
                     D   t     ⁡     (   j   )       =       S   j   ′         ∑     1   ≤   j   ≤   K       ⁢     S   j   ′                 (   13   )                 S   j   ′     =       ∑       1   ≤   k   ≤   K     ,     i   ≠   k         ⁢       cos   2     ⁢     θ     C     t   -     1   jk                       (   14   )               
where θc t-1jk  represents an angle between the learning subspace P j  and P k  after projection onto the constrained subspace C t-1 . Due to this, the learning subspaces which are similar on the constraint subspace C t-1  have an increasing weight. In the next constraint subspace C t , the similar learning subspaces no longer become similar. Equation (14) may be introduced with a condition that a sum is taken from the angle θc t-1jk  equal to or greater than a definite threshold. Incidentally, the initial weight is assumably given D 1 (j)=1/K.
 
   (3) Constraint Subspace Learning Unit  603   
   The constraint subspace learning unit  603  generates the constraint subspace by the method as described in JP-A-2000-30065 and Fukui et al cited on p. 2, after multiplying the weight D t (j) on basic vector of a learning subspace P j . 
   (4) Constraint Subspace Storing Unit  604   
   The learning weight calculating unit  602  and the constraint subspace learning unit  603  are repeatedly used until constraint subspaces in the number M are stored to the constraint subspace storing unit  604 . 
   One of the methods of using the constraint subspaces which are made in the sixth embodiment, similarities are combined through the use of reliability. In this case, the similarity combining unit  107  of the first embodiment uses Equation (15). 
   
     
       
         
           
             
               
                 
                   S 
                   E 
                 
                 = 
                 
                   
                     ∑ 
                     
                       1 
                       ≤ 
                       t 
                       ≤ 
                       M 
                     
                   
                   ⁢ 
                   
                     
                       α 
                       t 
                     
                     ⁢ 
                     
                       cos 
                       2 
                     
                     ⁢ 
                     
                       θ 
                       
                         C 
                         t 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 15 
                 ) 
               
             
           
         
       
     
   
   The reliability α t  may be a ratio that the angle θc t-1jk  of Equation (14) does not exceed a constant threshold. Otherwise, instead of using reliability, similarities may be combined by a method with an average value as described in the similarity combining unit  107 . 
   Detailed Example of the First Embodiment 
   Now explained is pattern recognition using face images, as a detailed example of the first embodiment. The flow of this detailed example is shown in  FIG. 7 . In  FIG. 8  is shown a configuration of the face image recognition apparatus  800 . 
   The face image recognition apparatus  800  comprises a face input unit  801 , an input subspace generating unit  802 , a reference subspace storing unit  803 , a constraint subspace storing unit  804 , a subspace projecting unit  805 , an similarity calculating unit  806 , a similarity combining unit  807 , and a face determining unit  808 . 
   (1) Face Input Unit  801   
   The face input unit  801  acquires a face image by a camera (step  701  in  FIG. 7 ), clips a facial region out of the image (step  702  in  FIG. 7 ), and raster-scans the facial region into a vector (step  703  in  FIG. 7 ). 
   The facial region can be determined by a positional relationship of facial feature points, such as the pupils and nostrils, as described in JP-A-9 (1997)-251534 and Osamu Yamaguchi et al (Osamu Yamaguchi, Kazuhiro Fukui, “Smartface”—A Robust Face Recognition System under Varying Facial Pose and Expression, IEICE Trans. D-II Vol. J84-D-II, No. 6, 1045-1052, 2001). Meanwhile, by successively getting face images, facial regions can be obtained at all times. 
   (2) Input Subspace Generating Unit  802   
   The input subspace generating unit  802 , after getting vectors in the predefined number (step  704  in  FIG. 7 ), determines an input subspace by the principal component analysis ( 705  in  FIG. 7 ). 
   (3) Reference Subspace Storing Unit  803   
   The reference subspace storing unit  803  is stored with reference subspaces in the number of R. 
   (4) Constraint Subspace Storing Unit  804   
   The constraint subspace storing unit  804  is stored with constraint subspaces in the number of M. In order to improve recognition performance, constraint subspaces are generated taking into consideration the followings. 
   The cause of the performance decline includes the variation in appearance due to lighting, ornaments and the like. In order to provide a resistance to lighting variation, learning patterns require face images taken by changing the lighting conditions. 
   Meanwhile, in order to provide a resistance to the variation due to the ornaments (glasses) worn on the face, learning patterns require face images taken by changing the ornament. 
     FIG. 9  shows a flow of learning constraint subspaces. 
   At first, prepared are learning subspaces generated by learning patterns which are acquired in various lighting conditions and ornaments conditions. From those, selected are learning subspaces of the persons satisfying the defined criterion (e.g. wearing glasses) (step  901  in  FIG. 9 ). 
   After selecting learning subspaces of a constant number of persons (step  902  in  FIG. 9 ), the basic vectors of the learning subspaces are taken as an input to principal component analysis (step  903  in  FIG. 9 ). 
   The eigenvectors, obtained as a result of the principal component analysis, are selected in ascending order of the eigenvalue to get the basic vectors of constraint subspace (step  904  in  FIG. 9 ). 
   (5) Subspace Projecting Unit  805   
   The subspace projecting unit  805  projects the input subspace and the reference subspaces of the R persons stored in the reference subspace storing unit  803 , onto one of the constraint subspaces stored in the constraint subspace storing unit  804  (step  706  in  FIG. 7 ). 
   The procedure of projection may use a method described in JP-A-2000-30065 and Fukui et al cited on p.2. 
   (6) Similarity Calculating Unit  806   
   The similarity calculating unit  806  calculates the similarities between the reference subspace and input subspace projected onto one constraint subspace by the mutual subspace method (step  707  in  FIG. 7 ). 
   The subspace projecting unit  805  and the similarity calculating unit  806  exist in the number of M to carry out parallel. 
   Otherwise, a subspace projecting unit  805  and a similarity calculating unit  806  are repetitively used sequentially by the number M of constraint subspaces (step  708  in  FIG. 7 ). 
   (7) Similarity Combining Unit  807   
   The similarity combining unit  807  combines the similarities by the method described in the similarity combining unit  107  (step  709  in  FIG. 7 ). Incidentally, combining the similarities is carried out by the number R of the reference subspaces stored in the reference subspace storing unit  803 . 
   (8) Face Determining Unit  808   
   The face determining unit  808  determines the person of the input subspace from the combined similarity which is the highest value and greater than a preset threshold. In other cases, output is as a person not registered in the reference subspace storing unit. Output is by notification on screen or by sound with using a monitor or speaker. 
   (9) Outline 
   The above (1) to (8) is outlined. 
   At first, reference subspaces in the number of R are previously prepared. These are reference subspaces G 1 , G 2 , . . . , GR made from the facial regions in the number of R persons. 
   Meanwhile, the constraint subspace includes constraint subspaces S 1 , S 2 , . . . , SM in the number of M, e.g. constraint subspaces that glasses are worn and constraint subspaces that lighting is applied. 
   Here, inputted is a facial region of the person X to make an input subspace on X. 
   Then, the input subspace on X and reference subspace G 1  are projected on to a constraint subspace S 1 , to thereby determine a similarity B 1 - 1 . 
   Next, the input subspace on X and the reference subspace G 1  are projected onto a constraint subspace S 2 , to thereby determine a similarity B 2 - 1 . 
   Subsequently, similarities B 1 - 1 , B 2 - 1 , . . . , BM- 1 , BM- 2 , . . . , BR-M are determined in the similar manner. Namely, determined are similarities in the number of M×R. 
   Then, a combined similarity is determined on each reference subspace. Namely, the similarities B 1 - 1 , B 2 - 1 , . . . , BM- 1  are combined together, to determine a combined similarity J 1  between the input subspace on X and the reference subspace G 1 . Also, determined is a combined similarity J 2  between the input subspace on X and the reference subspace G 2 . Subsequently, combined similarities J 3 , . . . , JR are determined in the similar manner. 
   Then, the reference subspace, having a combined similarity highest in the combined similarities J 1 , . . . , JR in the number of R and its value is greater than a preset threshold, provides the corresponding person. 
   (9) Modification Taking Account of Reducing the Amount of Calculation 
   In the case of sequentially calculating similarities by using a plurality of constraint subspaces, there is increase in calculation time. In order to reduce the amount of calculation, it is possible to narrow down the persons who are calculated similarities. There is shown in  FIG. 10  a flow of face image recognition taking account of reducing the amount of calculation. 
   At first, the input subspace and reference subspaces are projected onto one of the constraint subspaces in the number of M (step  1007  in  FIG. 10 ). Second, similarities are calculated by the mutual subspace method (step  1008  in  FIG. 10 ). Selected are the persons in the number of X (X&lt;R) in descending order of the similarity (step  1009  in  FIG. 10 ). R represents the number of the registered persons. Only on those persons, similarities are calculated by using the remaining constraint subspaces in the number of M- 1  (steps  1010 - 1011  in  FIG. 10 ). 
   The face image recognition is true for the second to fourth embodiments, similarly. 
   Detailed Example of the Fifth Embodiment 
   Now, one of the learning methods of constraint subspaces is explained to be used in face image recognition, as a detailed example of the fifth embodiment. There is shown in  FIG. 11  a flow of this detailed example while in  FIG. 12  a configuration of a constraint subspace learning apparatus  1200 . 
   The constraint subspace learning apparatus  1200  comprises a learning subspace storing unit  1201 , a learning subspace selecting unit  1202 , a constraint subspace learning unit  1203  and a constraint subspace storing unit  1204 . 
   (1) Learning Subspace Storing Unit  1201   
   The learning subspace storing unit  1201  stores learning subspaces in the number of K. The learning subspaces generated from face images taken by varying lighting conditions or wearing ornaments, such as glasses. 
   (2) Learning Subspace Selecting Unit  1202   
   The learning subspace selecting unit  1202  randomly selects a learning subspace from the learning subspace storing unit  1201  (step  1101  in  FIG. 11 ). This is repeated until persons in the number of T (T&lt;K) are selected (step  1102  in  FIG. 11 ). If thus selected ones of the learning subspaces in number of T include two or more of those for same person, selection should be made anew. 
   (3) Constraint Subspace Learning Unit  1203   
   The constraint subspace learning unit  1203  generates a constraint subspace by the principal component analysis from the basic vectors of selected learning subspaces (step  1103  in  FIG. 11 ). The eigenvectors obtained as a result of the principal component analysis are selected in ascending order of the eigenvalue and taken as the basis vectors of constraint subspace (step  1104  in  FIG. 11 ). 
   (4) Constraint Subspace Storing Unit  1204   
   Until constraint subspaces in the number of M are stored to the constraint subspace storing unit  1204 , the learning subspace selecting unit  1202  and constraint subspace learning unit  1203  are used repeatedly (step  1105  in  FIG. 11 ). 
   Learning constraint subspaces using a face image is true for the sixth embodiment. 
   Modification 
   Whereas face images of persons are used as input patterns in the above embodiments, fingerprints, voice, letters, DNA or the like may be used if identification of is feasible.