eigenfaces vs fisherfaces recognition using class specific linear projection 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

su 3 richardson gaudin models three level systems we present the exact solution of the richardson gaudin models associated with the su 3 lie algebra the derivation is based on a gaudin algebra valid for any simple lie algebra in the rational trigonometric and hyperbolic cases for the rational case additional cubic integrals of motion are obtained whose number is added to that of the quadratic ones to match as required from the integrability condition the number of quantum degrees of freedom of the model we discuss different su 3 physical representations and elucidate the meaning of the parameters entering in the formalism by considering a bosonic mapping limit of one of the su 3 copies we derive new integrable models for three level systems interacting with two bosons these models include a generalized tavis cummings model for three level atoms interacting with two modes of the quantized electric field

comments on globally maximizing locally minimizing unsupervised discriminant projection with application to face and palm biometrics this paper develops an unsupervised discriminant projection udp technique for dimensionality reduction of high dimensional data in small sample size cases udp can be seen as a linear approximation of a multimanifolds based learning framework which takes into account both the local and nonlocal quantities udp characterizes the local scatter as well as the nonlocal scatter seeking to find a projection that simultaneously maximizes the nonlocal scatter and minimizes the local scatter this characteristic makes udp more intuitive and more powerful than the most up to date method locality preserving projection lpp which considers only the local scatter for clustering or classification tasks the proposed method is applied to face and palm biometrics and is examined using the yale feret and ar face image databases and the polyu palmprint database the experimental results show that udp consistently outperforms lpp and pca and outperforms lda when the training sample size per class is small this demonstrates that udp is a good choice for real world biometrics applications

general tensor discriminant analysis and gabor features for gait recognition traditional image representations are not suited to conventional classification methods such as the linear discriminant analysis lda because of the undersample problem usp the dimensionality of the feature space is much higher than the number of training samples motivated by the successes of the two dimensional lda 2dlda for face recognition we develop a general tensor discriminant analysis gtda as a preprocessing step for lda the benefits of gtda compared with existing preprocessing methods such as the principal components analysis pca and 2dlda include the following 1 the usp is reduced in subsequent classification by for example lda 2 the discriminative information in the training tensors is preserved and 3 gtda provides stable recognition rates because the alternating projection optimization algorithm to obtain a solution of gtda converges whereas that of 2dlda does not we use human gait recognition to validate the proposed gtda the averaged gait images are utilized for gait representation given the popularity of gabor function based image decompositions for image understanding and object recognition we develop three different gabor function based image representations 1 gabord is the sum of gabor filter responses over directions 2 gabors is the sum of gabor filter responses over scales and 3 gaborsd is the sum of gabor filter responses over scales and directions the gabord gabors and gaborsd representations are applied to the problem of recognizing people from their averaged gait images a large number of experiments were carried out to evaluate the effectiveness recognition rate of gait recognition based on first obtaining a gabor gabord gabors or gaborsd image representation then using gdta to extract features and finally using lda for classification the proposed methods achieved good performance for gait recognition based on image sequences from the university of south florida usf humanid database experimental comparisons are made with nine state of the art classification methods in gait recognition

subalgebras of c algebras iii multivariable operator theory a d contraction is a d tuple t1 td of mutually commuting operators acting on a common hilbert space h such that t1 1 t2 2 td d 1 2 d for all 1 2 d h these are the higher dimensional counterparts of contractions we show that many of the operator theoretic aspects of function theory in the unit disk generalize to the unit ball bd in complex d space including von neumann s inequality and the model theory of contractions these results depend on properties of the d shift a distinguished d contraction which acts on a new h2 space associated with bd and which is the higher dimensional counterpart of the unilateral shift h 2 and the d shift are highly unique indeed by exploiting the noncommutative choquet boundary of the d shift relative to its generated c algebra we find that there is more uniqueness in dimension d 2 than there is in dimension one

random sampling for subspace face recognition subspace face recognition often suffers from two problems 1 the training sample set is small compared with the high dimensional feature vector 2 the performance is sensitive to the subspace dimension instead of pursuing a single optimal subspace we develop an ensemble learning framework based on random sampling on all three key components of a classification system the feature space training samples and subspace parameters fisherface and null space lda n lda are two conventional approaches to address the small sample size problem but in many cases these lda classifiers are overfitted to the training set and discard some useful discriminative information by analyzing different overfitting problems for the two kinds of lda classifiers we use random subspace and bagging to improve them respectively by random sampling on feature vectors and training samples multiple stabilized fisherface and n lda classifiers are constructed and the two groups of complementary classifiers are integrated using a fusion rule so nearly all the discriminative information is preserved in addition we further apply random sampling on parameter selection in order to overcome the difficulty of selecting optimal parameters in our algorithms then we use the developed random sampling framework for the integration of multiple features a robust random sampling face recognition system integrating shape texture and gabor responses is finally constructed

a multi manifold discriminant analysis method for image feature extraction in this paper we propose a multi manifold discriminant analysis mmda method for an image feature extraction and pattern recognition based on graph embedded learning and under the fisher discriminant analysis framework in an mmda the within class graph and between class graph are respectively designed to characterize the within class compactness and the between class separability seeking for the discriminant matrix to simultaneously maximize the between class scatter and minimize the within class scatter in addition in an mmda the within class graph can represent the sub manifold information while the between class graph can represent the multi manifold information the proposed mmda is extensively examined by using the feret ar and orl face databases and the polyu finger knuckle print databases the experimental results demonstrate that an mmda is effective in feature extraction leading to promising image recognition performance

inductive limits of semiprojective c algebras abstract we prove closure properties for the class of c algebras that are inductive limits of semiprojective c algebras most importantly we show that this class is closed under shape domination and so in particular under shape and homotopy equivalence it follows that the considered class is quite large it contains for instance the stable suspension of any nuclear c algebra satisfying the uct and with torsion free k 0 group in particular the stabilized c algebra of continuous functions on the pointed sphere is isomorphic to an inductive limit of semiprojectives

the dixmier property and tracial states for c algebras a t was partially supported by an nserc postdoctoral fellowship and through the epsrc grant ep n00874x 1 acknowledgements we are grateful to luis santiago for helpful discussions at an early stage of this investigation we would also like to thank the referee for providing helpful comments which have led to a number of improvements

nonparametric discriminant analysis for face recognition in this paper we develop a new framework for face recognition based on nonparametric discriminant analysis nda and multi classifier integration traditional lda based methods suffer a fundamental limitation originating from the parametric nature of scatter matrices which are based on the gaussian distribution assumption the performance of these methods notably degrades when the actual distribution is non gaussian to address this problem we propose a new formulation of scatter matrices to extend the two class nonparametric discriminant analysis to multi class cases then we develop two more improved multi class nda based algorithms nsa and nfa with each one having two complementary methods based on the principal space and the null space of the intra class scatter matrix respectively comparing to the nsa the nfa is more effective in the utilization of the classification boundary information in order to exploit the complementary nature of the two kinds of nfa pnfa and nnfa we finally develop a dual nfa based multi classifier fusion framework by employing the over complete gabor representation to boost the recognition performance we show the improvements of the developed new algorithms over the traditional subspace methods through comparative experiments on two challenging face databases purdue ar database and xm2vts database

large margin linear projection and face recognition unlike linear discriminant analysis the large margin linear projection lmlp classifier presented in this paper which also roots in linear fisher discriminant takes full advantage of the singularity of within class scatter matrix and classifies projected points in one dimensional space by itself theoretical analysis and experimental results both reveal that lmlp is well suited for high dimensional small sample pattern recognition problems such as face recognition

discriminative common vectors for face recognition in face recognition tasks the dimension of the sample space is typically larger than the number of the samples in the training set as a consequence the within class scatter matrix is singular and the linear discriminant analysis lda method cannot be applied directly this problem is known as the small sample size problem in this paper we propose a new face recognition method called the discriminative common vector method based on a variation of fisher s linear discriminant analysis for the small sample size case two different algorithms are given to extract the discriminative common vectors representing each person in the training set of the face database one algorithm uses the within class scatter matrix of the samples in the training set while the other uses the subspace methods and the gram schmidt orthogonalization procedure to obtain the discriminative common vectors then the discriminative common vectors are used for classification of new faces the proposed method yields an optimal solution for maximizing the modified fisher s linear discriminant criterion given in the paper our test results show that the discriminative common vector method is superior to other methods in terms of recognition accuracy efficiency and numerical stability

rapid and brief communication face recognition based on 2d fisherface approach two dimensional 2d discrimination analysis using methods such as 2d pca and image lda is of interest in face recognition because it extracts discriminative features faster than one dimensional 1d discrimination analysis however existing 2d methods generally use more discriminative features and take longer to test than 1d methods 2d pca in particular cannot make full use of the fisher discriminant criterion image lda also has drawbacks in that it cannot perform 2d principal component analysis and discards components with poor discriminative capabilities in addition existing 2d methods cannot provide an automatic strategy to choose 2d principal components or discriminant vectors in this paper we propose 2d fisherface a novel discrimination approach that combines the two stage pca lda strategy and 2d discrimination techniques it can extract face discriminative features by automatically selecting two dimensional principal components and discriminant vectors using the ar database as the test data it is shown that the proposed approach is faster and more effective than several representative 1d and 2d discrimination methods

a new and fast implementation for null space based linear discriminant analysis in this paper we present a new implementation for the null space based linear discriminant analysis the main features of our implementation include i the optimal transformation matrix is obtained easily by only orthogonal transformations without computing any eigendecomposition and singular value decomposition svd consequently our new implementation is eigendecomposition free and svd free ii its main computational complexity is from a economic qr factorization of the data matrix and a economic qr factorization of a nxn matrix with column pivoting here n is the sample size thus our new implementation is a fast one the effectiveness of our new implementation is demonstrated by some real world data sets

geometric linear discriminant analysis for pattern recognition we propose an innovative technique geometric linear discriminant analysis geometric lda to reduce the complexity of pattern recognition systems by using a linear transformation to lower the dimension of the observation space we experimentally compare geometric lda to other dimensionality reduction methods found in the literature we show that geometric lda produces the same and in many cases a significantly better linear transformation than other methods found in the literature

extracting the optimal dimensionality for local tensor discriminant analysis supervised dimensionality reduction with tensor representation has attracted great interest in recent years it has been successfully applied to problems with tensor data such as image and video recognition tasks however in the tensor based methods how to select the suitable dimensions is a very important problem since the number of possible dimension combinations exponentially increases with respect to the order of tensor manually selecting the suitable dimensions becomes an impossible task in the case of high order tensor in this paper we aim at solving this important problem and propose an algorithm to extract the optimal dimensionality for local tensor discriminant analysis experimental results on a toy example and real world data validate the effectiveness of the proposed method

theory analysis on fslda and ulda abstract it is first revealed that the fisher criterion ratio of each fslda discriminant vector must not be less than that of corresponding ulda discriminant vector so the phenomenon in yang et al pattern recognition 35 2002 2665 is not strange but certain and must be available in all experiments in addition it is also first illustrated that in fact ulda discriminant vectors are the s t orthogonal eigenvectors of a generalized eigenequation as a result the algorithms to obtain s t orthogonal eigenvectors of the generalized eigenequation are equivalent to the ulda algorithm consequently it is possible to work out ulda discriminant vectors more efficiently

1d lda vs 2d lda when is vector based linear discriminant analysis better than matrix based recent advances have shown that algorithms with 2d matrix based representation perform better than the traditional 1d vector based ones in particular 2d lda has been widely reported to outperform 1d lda however would the matrix based linear discriminant analysis be always superior and when would 1d lda be better in this paper we investigate into these questions and have a comprehensive comparison between 1d lda and 2d lda in theory and in experiments we analyze the heteroscedastic problem in 2d lda and formulate mathematical equalities to explore the relationship between 1d lda and 2d lda then we point out potential problems in 2d lda it is shown that 2d lda has eliminated the information contained in the covariance information between different local geometric structures such as the rows or the columns which is useful for discriminant feature extraction whereas 1d lda could preserve such information interestingly this new finding indicates that 1d lda is able to gain higher fisher score than 2d lda in some extreme case furthermore sufficient conditions on which 2d lda would be bayes optimal for two class classification problem are derived and comparison with 1d lda in this aspect is also analyzed this could help understand how 2d lda is expected to achieve at its best further discover its relationship with 1d lda and well support other findings after the theoretical analysis comprehensive experimental results are reported by fairly and extensively comparing 1d lda with 2d lda in contrast to the existing view that some 2d lda based algorithms would perform better than 1d lda when the number of training samples for each class is small or when the number of discriminant features used is small we show that it is not always true and show that some standard 1d lda based algorithms could perform better in those cases on some challenging data sets

regularized discriminant analysis and its application to face recognition the linear fisher discriminant analysis lda is a well known and popular statistical method in pattern recognition and classi5cation the basic idea is to optimize the discriminant criteria in which the ratio between the interand the intra class distance is maximized this approach is theoretically sound and a number of papers have shown its superior performance when applying in pattern recognition applications 1 5 however this approach su ers from the small size problem this problem occurs when the sample size is small compared with the size of feature vector which always appears in the face recognition applications this is because the typical image dimension for face image is greater than 32 32 and usually only 2 6 images are used for training thus the within class covariance estimation cw will be singular in this case the eigenvalue problem is ill posed the fisher index in t will be reached at the null space of cw and will always result in in5nity index in t a number of methods have been proposed in the last decade to overcome the limitation of lda on small sample size these methods in applying to face recognition can be roughly grouped into three categories the 5rst approach applied a dimension reduction method such as principal component analysis pca to extract important components for lda this approach is straightforward but the dimension reduction method may remove some useful information for recognition the second approach modi5es the fisher s optimization criteria theoretically this approach can solve the problem but the constraints 1 3 will limit the

rapid and brief communication two dimensional fld for face recognition this paper presents a new scheme of face image feature extraction namely the two dimensional fisher linear discriminant experiments on the orl and the umist face databases show that the new scheme outperforms the pca and the conventional pca fld schemes not only in its computational efficiency but also in its performance for the task of face recognition

face recognition using laplacianfaces we propose an appearance based face recognition method called the laplacianface approach by using locality preserving projections lpp the face images are mapped into a face subspace for analysis different from principal component analysis pca and linear discriminant analysis lda which effectively see only the euclidean structure of face space lpp finds an embedding that preserves local information and obtains a face subspace that best detects the essential face manifold structure the laplacianfaces are the optimal linear approximations to the eigenfunctions of the laplace beltrami operator on the face manifold in this way the unwanted variations resulting from changes in lighting facial expression and pose may be eliminated or reduced theoretical analysis shows that pca lda and lpp can be obtained from different graph models we compare the proposed laplacianface approach with eigenface and fisherface methods on three different face data sets experimental results suggest that the proposed laplacianface approach provides a better representation and achieves lower error rates in face recognition

a study on three linear discriminant analysis based methods in small sample size problem in this paper we make a study on three linear discriminant analysis lda based methods regularized discriminant analysis rda discriminant common vectors dcv and maximal margin criterion mmc in the small sample size sss problem our contributions are that 1 we reveal that dcv obtains the same projection subspace as both rda and wmmc weighted mmc a general form of mmc when rda s regularization parameter tends to zero and wmmc s weight parameter approaches to which builds on close relationships among these three lda based methods 2 we offer efficient algorithms to perform rda and wmmc in the principal component analysis transformed space which makes them feasible and efficient to applications such as face recognition 3 we formulate the eigenvalue distribution of wmmc on one hand the formulated eigenvalue distribution can guide practitioners in choosing wmmc s projection vectors and on the other hand the underlying methodology can be employed in analyzing the eigenvalue distribution of matrices such as aat bbt where the rows of a and b are far larger than their columns and 4 we compare their classification performance on several benchmarks to get that when the mean standard variance msv criterion is small dcv can obtain competitive classification performance to both rda and wmmc under optimal parameters but when msv is large dcv generally yields lower classification accuracy than rda and wmmc under optimal parameters

an optimization criterion for generalized discriminant analysis on undersampled problems an optimization criterion is presented for discriminant analysis the criterion extends the optimization criteria of the classical linear discriminant analysis lda through the use of the pseudoinverse when the scatter matrices are singular it is applicable regardless of the relative sizes of the data dimension and sample size overcoming a limitation of classical lda the optimization problem can be solved analytically by applying the generalized singular value decomposition gsvd technique the pseudoinverse has been suggested and used for undersampled problems in the past where the data dimension exceeds the number of data points the criterion proposed in this paper provides a theoretical justification for this procedure an approximation algorithm for the gsvd based approach is also presented it reduces the computational complexity by finding subclusters of each cluster and uses their centroids to capture the structure of each cluster this reduced problem yields much smaller matrices to which the gsvd can be applied efficiently experiments on text data with up to 7 000 dimensions show that the approximation algorithm produces results that are close to those produced by the exact algorithm

a new lda based face recognition system which can solve the small sample size problem a new lda based face recognition system is presented in this paper linear discriminant analysis lda is one of the most popular linear projection techniques for feature extraction the major drawback of applying lda is that it may encounter the small sample size problem in this paper we propose a new lda based technique which can solve the small sample size problem we also prove that the most expressive vectors derived in the null space of the within class scatter matrix using principal component analysis pca are equal to the optimal discriminant vectors derived in the original space using lda the experimental results show that the new lda process improves the performance of a face recognition system signi cantly 2000 pattern recognition society published by elsevier science ltd all rights reserved

efficient and robust feature extraction by maximum margin criterion in pattern recognition feature extraction techniques are widely employed to reduce the dimensionality of data and to enhance the discriminatory information principal component analysis pca and linear discriminant analysis lda are the two most popular linear dimensionality reduction methods however pca is not very effective for the extraction of the most discriminant features and lda is not stable due to the small sample size problem in this paper we propose some new linear and nonlinear feature extractors based on maximum margin criterion mmc geometrically feature extractors based on mmc maximize the average margin between classes after dimensionality reduction it is shown that mmc can represent class separability better than pca as a connection to lda we may also derive lda from mmc by incorporating some constraints by using some other constraints we establish a new linear feature extractor that does not suffer from the small sample size problem which is known to cause serious stability problems for lda the kernelized nonlinear counterpart of this linear feature extractor is also established in the paper our extensive experiments demonstrate that the new feature extractors are effective stable and efficient

face recognition using lda based algorithms low dimensional feature representation with enhanced discriminatory power is of paramount importance to face recognition fr systems most of traditional linear discriminant analysis lda based methods suffer from the disadvantage that their optimality criteria are not directly related to the classification ability of the obtained feature representation moreover their classification accuracy is affected by the small sample size sss problem which is often encountered in fr tasks in this paper we propose a new algorithm that deals with both of the shortcomings in an efficient and cost effective manner the proposed method is compared in terms of classification accuracy to other commonly used fr methods on two face databases results indicate that the performance of the proposed method is overall superior to those of traditional fr approaches such as the eigenfaces fisherfaces and d lda methods

local linear discriminant analysis framework using sample neighbors the linear discriminant analysis lda is a very popular linear feature extraction approach the algorithms of lda usually perform well under the following two assumptions the first assumption is that the global data structure is consistent with the local data structure the second assumption is that the input data classes are gaussian distributions however in real world applications these assumptions are not always satisfied in this paper we propose an improved lda framework the local lda llda which can perform well without needing to satisfy the above two assumptions our llda framework can effectively capture the local structure of samples according to different types of local data structure our llda framework incorporates several different forms of linear feature extraction approaches such as the classical lda and principal component analysis the proposed framework includes two llda algorithms a vector based llda algorithm and a matrix based llda mllda algorithm mllda is directly applicable to image recognition such as face recognition our algorithms need to train only a small portion of the whole training set before testing a sample they are suitable for learning large scale databases especially when the input data dimensions are very high and can achieve high classification accuracy extensive experiments show that the proposed algorithms can obtain good classification results

multilinear discriminant analysis for face recognition there is a growing interest in subspace learning techniques for face recognition however the excessive dimension of the data space often brings the algorithms into the curse of dimensionality dilemma in this paper we present a novel approach to solve the supervised dimensionality reduction problem by encoding an image object as a general tensor of second or even higher order first we propose a discriminant tensor criterion whereby multiple interrelated lower dimensional discriminative subspaces are derived for feature extraction then a novel approach called k mode optimization is presented to iteratively learn these subspaces by unfolding the tensor along different tensor directions we call this algorithm multilinear discriminant analysis mda which has the following characteristics 1 multiple interrelated subspaces can collaborate to discriminate different classes 2 for classification problems involving higher order tensors the mda algorithm can avoid the curse of dimensionality dilemma and alleviate the small sample size problem and 3 the computational cost in the learning stage is reduced to a large extent owing to the reduced data dimensions in k mode optimization we provide extensive experiments on orl cmu pie and feret databases by encoding face images as second or third order tensors to demonstrate that the proposed mda algorithm based on higher order tensors has the potential to outperform the traditional vector based subspace learning algorithms especially in the cases with small sample sizes

locality preserved maximum information projection dimensionality reduction is usually involved in the domains of artificial intelligence and machine learning linear projection of features is of particular interest for dimensionality reduction since it is simple to calculate and analytically analyze in this paper we propose an essentially linear projection technique called locality preserved maximum information projection lpmip to identify the underlying manifold structure of a data set lpmip considers both the within locality and the between locality in the processing of manifold learning equivalently the goal of lpmip is to preserve the local structure while maximize the out of locality global information of the samples simultaneously different from principal component analysis pca that aims to preserve the global information and locality preserving projections lpps that is in favor of preserving the local structure of the data set lpmip seeks a tradeoff between the global and local structures which is adjusted by a parameter alpha so as to find a sub space that detects the intrinsic manifold structure for classification tasks computationally by constructing the adjacency matrix lpmip is formulated as an eigenvalue problem lpmip yields orthogonal basis functions and completely avoids the singularity problem as it exists in lpp further we develop an efficient and stable lpmip qr algorithm for implementing lpmip especially on high dimensional data set theoretical analysis shows that conventional linear projection methods such as weighted pca maximum margin criterion mmc linear discriminant analysis lda and lpp could be derived from the lpmip framework by setting different graph models and constraints extensive experiments on face digit and facial expression recognition show the effectiveness of the proposed lpmip method

residually finite dimensional operator algebras abstract we study non selfadjoint operator algebras that can be entirely understood via their finite dimensional representations in contrast with the elementary matricial description of finite dimensional c algebras in the non selfadjoint setting we show that an additional level of flexibility must be allowed motivated by this peculiarity we consider a natural non selfadjoint notion of residual finite dimensionality we identify sufficient conditions for the tensor algebra of a c correspondence to enjoy this property to clarify the connection with the usual self adjoint notion we investigate the residual finite dimensionality of the minimal and maximal c covers associated to an operator algebra

linear dimensionality reduction via a heteroscedastic extension of lda the chernoff criterion we propose an eigenvector based heteroscedastic linear dimension reduction ldr technique for multiclass data the technique is based on a heteroscedastic two class technique which utilizes the so called chernoff criterion and successfully extends the well known linear discriminant analysis lda the latter which is based on the fisher criterion is incapable of dealing with heteroscedastic data in a proper way for the two class case the between class scatter is generalized so to capture differences in co variances it is shown that the classical notion of between class scatter can be associated with euclidean distances between class means from this viewpoint the between class scatter is generalized by employing the chernoff distance measure leading to our proposed heteroscedastic measure finally using the results from the two class case a multiclass extension of the chernoff criterion is proposed this criterion combines separation information present in the class mean as well as the class covariance matrices extensive experiments and a comparison with similar dimension reduction techniques are presented

a projective c algebra related to k theory the c algebra qc is the smallest of the c algebras qa introduced by cuntz j cuntz a new look at kk theory k theory 1 1 1987 31 51 in the context of kk theory an important property of qc is the natural isomorphism k0 a lim qc mn a our main result concerns the exponential boundary map from k0 of a quotient b to k1 of an ideal i we show if a k0 element is realized in hom qc b then its boundary is realized as a unitary in i the picture we obtain of the exponential map is based on a projective c algebra p that is universal for a set relations slightly weaker than the relations that define qc a new shorter proof of the semiprojectivity of qc is described smoothing questions related the relations for qc are addressed