1.
Introduction
This paper deals with the comparison of the constrained mutual subspace method (CMSM) [5] and the orthogonal mutual subspace method (OMSM) [6], including the nonlin- ear extensions of these methods with kernel trick. These methods used here are the extensions of the mu- tual subspace method (MSM), which is a generalization of the subspace method (SM) [1]   [3]. SM is used to calculate the similarity as the minimum canonical angle θ 1 between an input vector and a reference subspace which represents the variation of the learning set where an n x n image pattern is represented as a vector in n×n-dimensional space I . In the case of MSM, the canonical angle is calculated be- tween an input subspace and reference subspace as the sim- ilarity. MSM can deal with various recognition problems of the sets of images, such as a video image, multi-view images obtained from a multiple camera system. MSM has a high ability for absorbing the changes of pat- tern to achieve higher performance compared to that of SM using a single image. Since the generated subspaces are inde- pendent to each other in the case of MSM, the classification performance is s till insufficient for many applications as is the case with other methods based on PCA [3].  Although each subspace represents the distribution of the training patterns well in terms of a least mean square approximation, there is no reason to assume a priori that it is the optimal subspace. in terms of classification performance. To improve the ability of classification of MSM, the con- strained mutual subspace method (CMSM) [5] and the or- thogonal mutual subspace method (OMSM) [6] have been proposed. The validity of these methods is due to the or- thogonalization of class subspaces. In the case od CMSM, the projection of each class sub- space is onto the generalized differenece subspace, it extracts a common subspace of all the class subspaces from each sub- space, so that the canonical angles between subspaces are en- larged to approach to the orthogonal relation. On the other hand, OMSM orthogonalizes directly all the class subspaces based on the framework of the orthogonalization method pro- posed by Fukunaga & Koontz [19], [20]. Since the distribution of each class can not be represented by a linear subspace without overlaping with another sub- space, the performance of CMSM and OMSM are poor as the methods based linear subspace space, when the distribu- tion of patterns has high nonlinear structures. In order to overcome this problem, it is valid to represent such the distribution by a nonlinear subspace generated by kernel nonlinear PCA [8]. It is vividly stated that SM and MSM have extended to the kernel subspace method (KSM) [11], [12] and the kernel mutual subspace method (KMSM) [14], respectively, based on the frame work of the kernel PCA. Moreover CMSM has been also extended to the kernel CMSM (KCMSM) [13] to achieve higher performance compared to KSM and KMSM. The basis algorithm of OMSM is almost the same as that of CMSM except a part of the process of calculating similarity. Therefore in this paper, we will try to extend OMSM to the kernel nonlinear method, and name the extended method the kernel nonlinear orthogonal mutual subspace method (KOMSM). KOMSM has an interesting characteristic that it can al- ways orthogonalize all subspaces perfectly regardless of the number of classes and the dimension of each class subspace when the Gaussian kernel function is used in kernel trick. This is because the Gaussian kernel fucntion maps a pattern onto the infinite dimensional feature space. In contrast, the oroginal linear OMSM cannot realize the perfect orthogo- nalization of subspaces unless the product of the number of classes and the dimension of each subspace is less than the dimension of the linear feature space. The nonlinear kernel methods have a lack of requiring a lot of computing cost depending the number of learning pat- terns and classes. Especially, the computing cost of KOMSM is extremely high, because it has to calculate all the eigen- values and the eigenvectors of a matrix with the size equal to to the number of classes and the dimension of each class. Thus, the learning for KOMSM is more difficult as the size of the matrix becomes larger with increasing the number and the dimesion of classes. By contrast, since KCMSM requires only a part of the eigenvectors corresponding to the larger eigenvalues of all eigenvectors of the kernel matrix, it does not have such the serious problem. The paper is organized as follows. Firstly we outline the algorithms of MSM, CMSM and OMSM in sections 2, 3 and 4, respectively. In section 5, we describe the kernel trick for nonlinear extension of CMSM and OMSM. In section 6 we describe the algorithm of KCMSM. Then, in section 7, we extend OMSM to the nonlinear kernel OMSM (KOMSM). In section 8 we compare the performance of all the meth- ods. The section 9 is the discussion. Finally we conclude the paper.
Mutual subspace method
2i
In this section, we firstly describe the concept of canonical angle between subspaces. Next we outline the algorithm of MSM based on the canonocal angles. 2.1   Canonical angles between subspaces Assume that P and Q are an N-dimensional input sub- space and an M-dimensional reference subspaces, in the f- dimensional space I , which are generated from the pattern set of input and reference, respectively.  We can obtain N canonical angles (for convenience N ≤ M ) between P and Q [15]. Let Φ i and Ψ i denote the i-th f-dimensional orthonor- mal basis vectors of the subspaces P and Q, respectively. The value cos 2 θ i of the i-th smallest canonical angle θ i (i = 1, . . , N) is obtained as the i-th largest eigenvalue λ i of the following N × N  matrix X [5], [15]:
X = (x j ), x j = ∑ k =1 (Ψ j · Φ k · Ψ j ) The maximum eigenvalue of these matrices represents cos 2 θ  of the minimum canonical angle θ 1 , whereas the sec- ond maximum eigenvalue represented cos 2 θ2 of the minimum angle θ 2 in the direction perpendicular to that of θ 1  cos 2 θ i for i = 3, . . , N , are calculated similarly.
where u i ∈ P,v i ∈ Q,||u i |キ0,||v i |キ0,  ( · )  and || · || represent an inner product and a norm respectively. 2.2 Definition of similarity between two sub- spaces We consider the value of the mean of the canonical angles, S [t] =  ± ∑ i=1 cos θ i, as the similarity between two sub- spaces. The similarity S [ t ] has the following characteristics. In the case that two subspaces coincide completely with each other, S [ t ] is 1.0, since all canonical angles are zero.  The sim- ilarity S[t] becomes smaller as the two subspaces separate. Finally, the similarity S[t] is zero when the two subspaces are orthogonal to each other. Note that the value S[t] reflects the structual similarity between two subspaces.
Constrained mutual subspace method
3.
To improve the performance of the MSM, we project an input subspace P and reference subspaces Q onto the constraint subspace C that satisfies a constraint condition: "it includes only the essential component for recognition between class subspaces", and then the canonical angles between the projected input subspace and reference sub- spaces as shown in Fig.1 are calculated.  This MSM with the projection onto the constraint subspace C is called the constrained mutual subspace method  (CMSM) [5].
3. 1
Generation of constraint subspace
The constraint subspace is a generic name of the subspace which satisfies the above condition. We can use several dif- ferent subspaces as constraint subspace.  In this paper, we actually employ the generalized difference subspace. The generalized different subspace is generated as fol- Given k (≥2) N -dimensional subspaces, the general- lows. ized difference subspace D is defined as the subspace which results by removing the principal component subspace M of all subspaces from the sum subspace S of these sub- spaces as shown in Fig.2.  According to this definition, D is spanned by N a eigenvectors d i ( i = N × k − N a , . . , N × k ) corresponding to the Nd smallest eigenvalue, of the matrix G = E k 1 P i of projection matrices P i , where the projec- j =1 Φj Φj T, Φj is the j-th orthonormal tion matrix P t = ∑ - basis vector of the i-th class subspace. The eigenvectors, d i correspond to the i-th eigenvalue λ i in descending order. The effectiveness of the generalized difference subspace can be explained in several ways.  First speaking qualitatively, the generalized difference subspace is the subspace which represents the difference among multiple k(≥2) subspaces as an extension of the difference subspace defined as the dif- ference between two subspaces [5]. Therefore, the projection onto the generalized difference subspace is corresponding to extracting selectively the difference between subspaces. Fi- nally, the projection onto a generalized difference subspace enables CMSM to have a higher classification ability besides the ability to tolerate variations in the face patterns. From the other viewpoint, the projection onto the gen- eralized difference subspace D corresponds to removing the principal (common) component subspace M from the sum subspace S. Therefore, this projection has the effect of ex- panding the canonical angles between subspaces and forms a relation between subspaces which is close to the orthogo- nal relation, thus improving the performance of classification based on canonical angles [5].  This orthogonal process is use- ful for the methods based on the canonical angles. Since the orthogonal degree between the projected subspaces changes with the dimension nc , a proper dimension has to be set by the pre-experiments.
3.2   Process flow of CMSM
The flow of CMSM is as follows. In the learning step, all the reference subspaces are generated from learning patterns using K L expansion or Gram-Schmidt method, and they are projected onto the generalized difference subspace. The subspace P is projected onto the generalized difference subspace D by the following steps: 1. N orthogonal basis vectors of the subspace P are pro- jected onto the generalized difference D . 2. The length of each projected vector is normalized. Gram-Schmidt orthogonalization is applied to the nor- 3. malized vectors to obtain N orthogonal basis vectors of the subspace PD . In the recognition step, the input subspace P in is generated from input image patterns. Then, the input subspace P in is projected onto the generalized difference subspace D . Then, we compute the canonical angles between the projected input subspace P in and the projected reference subspace P in a database using MSM, and the registered reference subspace. that has the highest similarity is determined to be that of the identified class given the similarity is above a threshold. Instead of using the above method of calculating canoni- cal angles, we can use the procedure as described in [5].  In this method, firstly the set of input patterns and the refer- ence patterns are projected, and then these subspaces are generated from the projected patterns.
Orthogonal mutual subspace method
4.
The essence of OMSM is to carry out MSM using the class subspaces orthogonalized with each other in advance. We can orthogonalize class subspaces using the framework of the orthogonalization method proposed by Fukunaga & Koontz [19], [20]. This was used for the orthogonal subspace method (OSM). In this method the orthogonalization is re- alized by applying a transformation matrix for whitening. OMSM also utilizes this framework to orthogonalize class subspaces. However, strictly speaking, the whitening matrix used in the OMSM differs from that used in the classical OSM. The whitening matrix used in OSM is calculated from the autocorrelation matrix of the mixture of all the classes. On the other hand, in the case of OMSM, the whitening ma- trix is calculated from the sum matrix of all the projection matrixes of the class subspaces. 4.1   Calculation of the whitening matrix In this section, we will describe how to calculate the whitening matrix O. In the following, we will consider the case that r m -dimensional class subspaces are orthogonalized in  f -dimensional space I . At first, we define the f×f sum matrix G of the r projection matrixes by the following equation.
where the f×f matrix P i represents the projection matrix which is corresponding to the projection onto the subspace Pi of the class i. Using the eigenvectors and the eigenvalues of the sum matrix G, the v×f whitening matrix O is defined by the following equation.
where v = rxm (v = f, if v > f), and A is the vxv di- agonal matrix with the i-th highest eigenvalue of the matrix G as the i-th diagonal component, and H is the f x v matrix whose i-th column vector is the eigenvector of the matrix G corresponding to the i-th highest eigenvalue. The suffix "To" represents transposition of a matrix. We can confirm that the matrix O can whiten the matrix G by the following transformations of equation.
where the matrix Ivxv is a vxv unit matrix. In the above transformation, we used the relation that G = BAB™. The process flow of OMSM based on the whitening ma- trix is as follows.  First, all the basis vectors of an input subspace are orthogonalized by applying the whitening ma- trix O. Then, the transformed basis vectors are orthogo- nalized again by Gram-Schmidt orthogonalization to form the orthogonalized input subspace. One should note that if the multiple of the number r of classes and the dimension m of each class is smaller than the dimension f of feature space, the Gram-Schmidt orthogonalization method is not required, as transformed basis vectors are orthogonal to each other. Similarly, all the basis vectors of reference subspaces are orthogonalized and orthogonalized reference subspaces are generated from the transformed basis vectors.  Finally, MSM is applied to the orthogonalized input subspace and the orthogonalized reference subspaces.
5.  Kernel PCA
In order to understand the nonlinear extension of the linear method mentioned before, we review kernel Principal Com- ponent Analysis (KPCA) in this section. The nonlinear function φ maps the patterns x = ( x 1 , . . , x f ) T  of an f -dimensional input space I onto an f ð - dimensional feature space F: φ : R f → R f +, x → φ(x). To perform PCA on the nonlinear mapped patterns, we need to calculate the inner product (φ(x) · φ(y)) between the func- tion values. However, this calculation is difficult, because the dimension of the feature space F can be very large, possibly infinite. However, if the nonlinear map φ is defined through a kernel function k(x,y) which satisfies Mercer's conditions, the inner products (φ(x) φ(y)) can be calculated from the in- ner products ( x · y ). This technique is known as the "kernel trick".  A common choice is to use Gaussina kernel func- tion [8]:
The function φ with the above exponential kernel function maps an input pattern onto an infinite feature space F . The PCA of the mapped patterns is called kernel PCA [8].  We should note that a linear subspace generated by the kernel PCA can be regarded as a nonlinear subspace in the input space I. Given the N -dimensional nonlinear subspace V k of class k generated from m training patterns x i , ( i = 1, . . ,m ), the N orthonormal basis vectors e , , ( i = 1, . . , N ), which span the nonlinear subspace V k , can be represented by the linear combination of the m φ(x ),(i = 1,...,m) as follows
where the coefficient a k , is the j-th component of the eigen- vector a i corresponding to the i-th largest eigenvalue λ i of the m×m kernel matrix K defined by the following equa- tions:
where a i is normalized to satisfy λ i (a i · a i )=1. We can com- pute the projection of the mapped φ(x) onto the i-th or- thonormal basis vector e k of the nonlinear subspace V k of
44 class k by the following equation:

Kernel constrained mutual subspace
6.
method
6.1   Generation of the kernel difference subspace It is possible to compute the projection of a mapped pat- tern φ(x) onto the kernel generalized difference subspace D φ using the kernel trick, since it consists of the inner prod- ucts in the feature space F.  Let the Ng-dimensional D* be generated from the r N-dimensional nonlinear subspace Firstly we calculate the orthonormal V k , (k  =  1, . . ,r). bases of kernel generalized difference subspace from all the orthonormal basis vectors of r nonlinear subspaces, namely, r × N basis vectors. This calculation corresponds to the PCA of all basis vectors. Define E to be a matrix, which contains all basis vectors as columns such as:
Secondly, we solve the eigenvalue problem of the matrix D defined by the following equations.
Where E i represents the i-th column of the matrix E. The inner product between the i-th orthonormal basis vec- tor ek of the class k subspace and the j-th orthonormal basis vector e j " of the class k* subspace can be obtained as the linear combination of kernel functions k(x k , x k * ) as follows:
The i-th orthonormal basis vector d 2 of the kernel general- ized difference subspace D ® can be represented by a linear combination of the vectors E j (j = 1, . . ,r×N).
Where the coefficient b 5 is j -th component of the eigenvector b i corresponding to the i-th smallest eigenvalue β i of matrix D under the condition that the vector b i is normalized to satisfy that βέ(bέ · bέ)=1. Let E j denote the η( j )-th basic vector of class ζ( j ). The above equation is converted to the following equation:

6.2
Projection onto the kernel difference sub- space Although it is impossible to calculate the orthonormal ba- sis vector d 2 of the kernel generalized difference subspace D p , the projection of the mapped pattern φ(x) onto this vector d c can be calculated from an input pattern x and all mxr training patterns x k (s = 1, . . ,m, k = 1, . . ,r).
Note that we can compute k(x ξ ( j ) ,x) through Eq.8 easily. Finally, each component of the projection τ(φ(x)) of the mapped φ(x) onto the N 2 (< r×N )-dimensional kernel gen- eralized difference subspace is represented as the following.

6.3
Algorithm of KCMSM
We construct KCMSM by applying linear MSM to the pro- jection τ(φ(x)).  Fig.3 shows a schematic of the KCMSM algorithm. In the training stage, the mapped patterns φ ( x k i ) of the patterns x k , (i = 1,...,m) belonging to class k , are pro- jected onto the kernel difference subspace D .  Then, the N φ -dimensional linear reference subspace P ρ * of each class k is generated from the mapped patterns τ(φ(x,f)) using PCA. In the recognition stage, we generate the linear input subspace P in  on the D f from the input patterns x i,(i = 1, . . , m ).  Then we compute the similarity S , defined in Sec.2.3, between the input subspace PD* and each refer- ence subspace P k ° . Finally the reference subspace with the highest similarity S is determined to be that of the identified object, given the similarity S' is above a threshold.
7.
Kernel orthogonal mutual subspace
method
7.1 Generation of kernel whitening matrix The procedure of generating a nonlinear kernel whitening matrix Oるis almost the same as that of the linear whitening matrix O. Actually O 4 can be generated by only replacing the linear subspaces with nonlinear subspaces in the calcu- lation. In the following, we will explain how to generate the n k × n k  whitening matrix O  from all the basis vectors of r d -dimensional nonlinear class subspaces V k ( k = 1 ～ r ), that is, r×d basis vectors. This calculation coincides to the PCA for the basis vectors of all the classes. Assume that a class k nonlinear subspace V k is generated from m learning patterns x k ( i = 1 ～ m ). The d basis vectors e ξ ( i = 1 ～d ), which expand the subspace V k , are defined by the following equations.
where a ξ j is the coefficient shown in the equation (11). Sim- ilarly the other nonlinear class subspaces are generated. Next, assume that E is the matrix where all basis vectors are arranged as the column component.
Then, we solve the eigenvalue problem of the matrix Q defined by the following equations.
where Ei means the i-th column component of the matrix E. In the above equation, the inner product between i-th basis vector e k of the class k and j -th basis vector e k * of the class k* can be actually calculated as the linear combination of a kernel function value k(x k , x k *) of x k and x k *.
The i-th row vector Ooj of the kernel whitening matrix O s can be represented as the linear combination of the basis vectors E j ( j = 1 ～ r×d ) using the eigenvector b , corre- sponding to the eigenvalue β i as the combination coefficient.
where the vector b i is canonized to satisfy that β i (b i · b i ) is equal to 1.0. Moreover, assume that E j is the η(j)-th ba- sis vector of the class ζ(j). Then the above equation can be changed as follows.
Although this vector O si cannot be calculated practically, the inner product (φ(x) · Oφ i ) with the mapped vector φ(x) can be calculated. 7.2  Whitening the mapped patterns We transform the mapped vector φ(x) by the kernel whitening matrix Oるas the follows. This transformation can be calculated using an input vector x and all r×m learning vectors x k (s = 1~m, k = 1~r) using the following equations.
Finally, each component of the whitening transformed vec- tor χ(φ(x)) of the mapped vector φ(x) is represented as fol- lows.

7.3  Algorithm of KOMSM
We construct the nonlinear kernel orthogonal mutual sub- space method (KOMSM) by applying the linear MSM to the linear subspaces generated from the whitening transformed patterns {χ(φ(x))}. The process flow of KOMSM is as fol- lows.
In the learning stage:
1 The nonlinear mapped φ(x ) of all the patterns x f (i = 1 ～ m) belonging to class k are transformed by the kernel whitening matrix O ♂ . 2 The basis vectors of the n-dimensional nonlin- ear orthogonal k reference subspace P k + are obtained as the eigenvectors of the autocorrelation matrix generated from the set of the whitening transformed pattern set {x(¢(x;)),...,x(¢(x;n))}, corresponding to the n highest values. Similarly the other nonlinear orthogonal reference 3 subspaces are generated on the nonlinear feature space F . In the recognition stage: The nonlinear orthogonal input subspace P in is 1 also generated from the set of the whitening transformed pattern set {χ(φ(x i n )), . . . ,χ(φ(x m ))}. The canonical angles between the input nonlinear 2 orthogonal subspace P in * and the nonlinear reference orthog- onal subspaces P k * are calculated as the similarity Finally the object class is determined as the nonlin- 3 ear reference orthogonal subspace with the highest similarity S , given that S is above a threshold value. In the above process, it is possible to replace the processes of generating the nonlinear orthogonal subspaces with the following processes. Firstly the nonlinear input subspace and the nonlinear reference subspaces are generated from the set of the nonlinear mapped patterns. Next the basis vectors of the generated subspaces are transformed by the kernel whitening matrix.  Finally the whitening transformed basis vectors are orthogonalized by the Gram-Schmidt method so that they span a nonlinear subspace.
Evaluation experiments
8.
We compared the performances of MSM, CMSM, OMSM, KMSM, KCMSM, and KOMSM using the open data base of the multi-view image (Cropped-close128 of ETH-80) [18] and the data set of front face images collected by our self. 8.1   3D object recognition (Experiment-I) We selected 30 similar models from the ETH-80 database as shown in Fig.4 and used them for the evaluation. The images of each model were captured from 41 views as shown in Fig.5. The view directions are the same for all models. All images are cropped, so that they contain only the object without any border area. The odd numbered images (21 frames) and the even num- bered images (20 frames) were used for training and evalua- tion, respectively.  We prepared 10 datasets for each model by making the start frame number i change from 1 to 10, where one set contains 10 frames from i-th frame to i + 9-th. The total number of the evaluation trials is 9000(=10×30×30). The evaluation was performed in terms of the recognition rate and separability, which is a normalized index of clas- sification ability.  A higher value allows choosing a larger rejection threshold. The equal error rate (EER) represents the error rate at the point where the false accept rate (FAR) is equal to the false reject rate (FRR). The experimental condition is as follows.  We converted the 180×180 pixels color images to 15×15 pixels monochrome images and used them as the evaluation data. Thus, the di- mension f of a pattern is 225(=15×15). The dimensions of the input subspace and the reference subspaces were set to 7 for all the methods.  Gaussian kernel function with σ 2 = 0.05 defined by Eq.(8) was used for all the kernel methods. P in and P k  were generated from the patterns projected on the generalized difference subspace. The difference subspace D and the whitening matrix O were generated from thirty 20- dimensional linear class subspaces. We varied the dimension Na of D between 180 and 205 to compare the performance. The kernel difference subspace D s and the kernel whitening matrix O 6 were generated from thirty 20-dimensional non- linear class subspaces. We varied the dimension N 2 of D 2 between 100 and 550.
8.2  Experimental-I results
(1)Recognition rate
Table 1 and Table 2 show the recognition rate and the sep- arability of each method respectively. The notation used in the tables is "method type – dimension of the difference sub- space". For example, KCMSM-550 means the KCMSM with a 550-dimensional kernel difference subspace.  S [t] denotes the similarity defined using the multiple eigenvalues. The recognition of multiple view images is typically a non-
linear problem.  This is clearly shown by the experimen- tal results that the performance of the nonlinear methods (KMSM, KCMSM and KOMSM) is superior to that of the linear methods (MSM, CMSM and OMSM). In particular, the high separability of KCMSM and KOMSM are remark- able. This implies that they can maintain high performance even in the case of large number of classes. The performance of MSM was improved by the nonlinear extension to KMSM where the recognition rate increased from 76.3% to 87.0% and the separability increased from 0.082 to 0.429. KCMSM improved the recognition rate fur- ther to 89.7% and increased the separability by a value of about 0.2 in comparison to KMSM. This confirms the effec-
tiveness of the projection onto the kernel difference subspace, which serves as a feature extraction step in the feature space F. KOMSM also improved the performance of OMSM largely where the recognition rate increased from 54.0% to 88.0% and the separability increases from 0.116 to 0.621. The re- sults shows clearly the effectiveness of the orthogonalization of the class subspaces. The effectivenesses of KCMSM and KOMSM also appear in the result of ERR. We can observe that the performance of OMSM was ex- tremely low.  This may be due to the existence of the common subspace of class subspaces. Although the matrix G has to have 225 positive eigenvalues if there is no common of class subspaces, it actually has only 211 positive eigenvalues. It is very interesting that the performances of KCMSM and KOMSM are in the same level regardless of their dif- ferent principles of orthogonalization. We will consider this result from the view of the degree of orthogonalization later. The classification ability of the KCMSM and KOMSM were improved while increasing t of the similarity S [ t ].  These results show that the similarity S [1] using a single image is not sufficient for classification of the models with the simi- lar 3D shapes, as S [1] utilizes the information obtained from only a single view. (2)Mean orthogonal degree between subspaces In the previous section, we have described that the perfor- mances of subspace-based methods depend on the geomet- rical relation between class subspaces. To confirm this, we defined the mean orthogonal degree, ω , which represents the relation between class subspaces. The value of ω is actually defined by the following equation: ω = 1.0 – cos 2 θ , where Ô is the mean value of the canonical angles θ i between sub- spaces projected onto the general difference subspace or the nonlinear generalized difference subspace. The ω is closer to 1.0, when the relation between subspaces becomes closer to orthogonal relation. Fig.6 shows the changes of the orthogonal degree ω s of CMSM and KCMSM in the case that thirty 7-dimensional class subspaces are projected onto the generalized difference subspace or the nonlinear generalized difference subspace. The different subspace and the kernel different subspace were generated from thirty 20-dimensional linear subspaces and nonlinear subspaces, respectively. In the figure, the horizon- tal axis represents the dimensions of the general difference subspace D and the kernel difference subspace D δ . The ver- tical axis represents the mean orthogonal degree ω .  The lines labeled as CMSM and KCMSM show the changes of ω of the both methods respectively. The ω of MSM is as low as 0.160. The ω of CMSM in- creased to 0.725 by projecting class subspaces onto the 170- dimensional general difference subspace. On the other hand, the ω of OMSM was still as low as 0.611 despite the orthog- onalization with the whitening matrix. KOMSM obtained the maximum ω , that is, 1.0. This means that KOMSM achieved the perfect orthogonalization as expected theoretically. On the other hand, ω of KCMSM with the 450-dimensional kernel different subspace also ob- tained 0.991 (84.5 degree in angle). The fact that ω of KCMSM is also almost 1.0 is corre- sponding well to the experimental result that KCMSM and KOMSM have the same level performance.  Moreover, we can see that the dimension with the largest orthogonal degree ω is well corresponding to that with the best classification per¬ formance. 8.3 Recognition of face image (Experiment-II) Next we conducted the evaluation experiment of all the methods using the face images of 50 persons captured under 10 kinds of lighting. We cropped the 15 ×15 pixels face im- ages from the 320 × 240 pixels input images based on the positions of pupils and nostrils [?] The normalized face patterns of subjects 1-25 in lighting conditions L1-L10 were used for generating the difference subspace D , the kernel difference subspace D s , the whiten- ing matrix O and the kernel whitening matrix Oみ. The face patterns extracted from the images of the other subjects, 26- 50, in lighting conditions L1-L10 were used for evaluation.
The number of the data of each person is 150～180 frames for each lighting condition. The data was divided into 15〜18 sub datasets by every 10 frames.  The input sub- space and reference subspaces were generated from these sub datasets. Finally, the total number of the evaluation trials is 18468(=10C2×25×15〜18). The dimension of the input subspace and reference sub- spaces were set to 7 for all the methods. The difference subspace D and the whitening matrix O were generated from twenty five 60-dimensional linear sub- spaces of 1～25 persons. Similarly the kernel difference sub- space D s and the kernel whitening matrix O s were generated from twenty five 60-dimensional nonlinear class subspaces The dimensions of the generalized difference subspace and the kernel generalized difference subspace were set to 450 and 1050, respectively. We used a Gaussian kernel with σ 2 = 1.0 for all the nonlinear methods. The table 4 shows the recognition rate and the EER of each method  The high performances of KCMSM and KOMSM also appear in this experiment.  The performance of the linear MSM is high. This means that the task of face recognition can be regarded as a linear separable problem.
Discussion
9.
9.1   Computing time
KOMSM needs to calculate all the eigenvalues and all the eigen vectors of the kernel matrix K. It becomes harder to calculate these, while the size of the kernel matrix K be- comes larger depending on the number of classes. On the other hand, the calculation is easy in KCMSM since the pro- jection of a pattern onto the general difference subspace can be indirectly calculated using only the eigenvectors corre- sponding to the highest eigenvalues.
9.2 Selection of the optimal dimension of the gen-
eral difference subspace
The performances of CMSM and KCMSM depend on the dimensions of the general difference subspace and the kernel difference subspace. Although the dimensions of them were selected by the pre-experiments, it is desirable that the opti- mal dimensions of these difference subspaces can be selected automatically in advance for real applications where the di- mensions N å and N a of the above difference subspaces are large. From the prior discussions, it seems to be natural to con- sider that the optimal dimensions of the difference subspaces should be determined based on the degree of the orthogonal- ization between subspaces. The orthogonalization of OMSM was realized by whitening the distribution of all the basis vec- tors of subspaces. This leads to an idea that the distribution of the projections of all the basis vectors of subspaces on the general different subspace should be as uniform as possible in order to achieve higher orthogonalization of subspaces. Since the distribution of the projection of the basis vec¬ tors in each direction is represented by the eigenvalue of the matrix G   We believe that the measuring the uniformity of all the eigenvalues is valid to select the optimal dimension in advance.
Conclusion
10.
In this paper, we have compared the performances of CMSM, OMSM, and their nonlinear extensions: KCMSM and KOMSM. We confirmed that the effectivenesses of these methods are commonly derived from the orthogonalization of subspaces by the experimental result that the dimension with the largest orthogonal degree was well corresponding to that with the best classification performance. The evalua- tion experiments showed that KCMSM and KOMSM have the high recognition performance compared with CMSM and OMSM, and the performances of KCMSM and KOMSM are almost in the same level in spite of having different principle of orthogonailization. Since, however, the computing time of KCMSM is less than that of KOMSM, KCMSM has the ad- vantage over KOMSM for practical applications with many objects to be classified.