CN107679515A - A kind of three-dimensional face identification method based on curved surface mediation shape image depth representing - Google Patents
A kind of three-dimensional face identification method based on curved surface mediation shape image depth representing Download PDFInfo
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Abstract
A kind of three-dimensional face identification method based on curved surface mediation shape image depth representing, is mapped on two dimensional surface disk after being pre-processed to given three-dimensional face curved surface；For each summit in three-dimensional face, the curvature and normal vector of each apex are calculated, has respectively obtained six kinds of mediation form parameters；By six kinds of mediation form parameters on each summit, it is respectively embedded in the face after ajusting, has obtained the two-dimension human face image of six kinds of mediation shapes, be input in depth convolutional neural networks, using the depth convolutional neural networks trained, its depth characteristic is extracted；By the method for rarefaction representation, the comparison of three-dimensional face is realized.The three-dimensional face identification technology of the present invention have the advantages that simply, be easily achieved, accuracy of identification it is high, and robustness is preferable during to facial expression shape change.
Description
Technical field
The present invention relates to a kind of three-dimensional face identification method, relates generally to a kind of based on curved surface mediation shape image depthmeter
The three-dimensional face identification method shown.
Background technology
As a kind of new bio feature identification technique, three-dimensional face identification technology has in fields such as finance, security protection, anti-terrorisms
There is huge potential using value.Prior art mainly include the point, line, surface based on three-dimensional face curved surface, normal vector, curvature,
The geometric senses such as Shape Indexes are portrayed.In combination with the feature (such as Gabor wavelet conversion and local binary patterns) of engineer
Realize the final expression of three-dimensional face curved surface.Especially, existing correlation technique using surface parameterization technology (such as conformal projection and
Isometric Maps etc.) realize conversion of the three-dimensional face curved surface to information between two dimensional surface.However, the three-dimensional based on surface parameterization
Face recognition technology is difficult to obtain higher accuracy of identification at present.With the continuous development of deep learning, based on big-sample data
Depth convolutional neural networks obtained by training are proved to have good popularization performance, and its effect is appointed in a variety of computer visions
Substantially exceed Traditional Man feature in business (such as target detection and object identification).
The content of the invention
It is an object of the invention to propose a kind of three-dimensional face identification based on curved surface mediation shape image depth representing
Curved surface mediation shape image is combined by method, this method with deep learning, on the one hand, curved surface, which reconciles, to be shone upon with deep number
Theory basis and distinct physical interpretation, moreover, curved surface, which reconciles, shines upon algorithm to the topological sum border of curved surface with stronger
Robustness.On the other hand, deep learning can the stronger characteristics of image of Extraction and discrimination power, the combination of the two will be expected to greatly improve
Performance based on surface parameterization three-dimensional face recognition algorithm.
To achieve the above object, the present invention adopts the following technical scheme that：
A kind of three-dimensional face identification method based on curved surface mediation shape image depth representing, comprises the following steps：
1) data prediction：A three-dimensional face curved surface is given, it is pre-processed；
2) reconcile and shine upon：The three-dimensional face curved surface that will have been pre-processed, is shone upon using mediation, and three-dimensional face curved surface is mapped to
On two dimensional surface disk；
3) for each summit in three-dimensional face, the curvature and normal vector of each apex is calculated, is respectively obtained
Six kinds of mediation form parameters, it is shape index, curvature, normal vector on the x, tri- directions of y, z that six kinds, which reconcile form parameters,
Projection and depth；
4) face that will be mapped on two dimensional surface disk is ajusted, the face after being ajusted；Step 3) is required
Six kinds of mediation form parameters on each summit obtained, are respectively embedded in the face after ajusting, and have obtained six kinds of mediation shapes
Two-dimension human face image；
5) two-dimension human face image for six kinds of mediation shapes for obtaining step 4) is input in depth convolutional neural networks, profit
With the depth convolutional neural networks trained, its depth characteristic is extracted；
6) face alignment：The depth characteristic of image is obtained according to step 5), by the method for rarefaction representation, realizes three-dimensional people
The comparison of face.
Further improve of the invention is that pretreatment detailed process is：Noise remove, prenasale detection, face cutting
And left and right inner eye corner point automatic detection.
Further improve of the invention is, three-dimensional face curved surface is mapped into the detailed process on two dimensional surface disk D
It is as follows：Surface is triangular mesh dough sheet M, now reconciles and shines upon for discrete situation, for the function being defined within summitFunction f mediation energy is drawn, is written as：
Wherein, wijFor side [vi,vj] cotangent weight；Given [vi,vj] it is face [vi,vj,vk] and face [vi,vj,vl] adjacent
Side, θkIt is with vkIt is summit in [vi,vj,vk] drift angle in triangle；θlIt is with vlIt is summit in [vi,vj,vl] triangle
Drift angle in shape, now the weight on side be then referred to as：If [vi,vj] it is to be in borderline one
Side, then now unique adjacent triangle only has [vi,vj,vk], then now：Pushed up for all inside
Point, discrete Laplace's equation are changed into：
Solve to obtain the function f on summit to above-mentioned Laplace's equation, according to the function f on summit, by three-dimensional
Face curved surface is mapped on two circle of position disks.
Further improve of the invention is that shape index and curvature obtain especially by procedure below：Calculate each
The curvature and normal vector of apex；It is a discrete curved surface containing ginseng approx to think the region around the summitAccording to the coordinate of the three-dimensional face curved surface pre-processed
Fit parameter A, B, C, D, E, F and G in above formula；Further according to matrixCharacteristic root point is carried out to it
Solution, obtain the i.e. maximum principal curvatures K of characteristic root of maximum1, and minimum characteristic root is minimum principal curvatures K2；Introduce two kinds of parameters：
Shape index Shapeindex and curvature Curvedness, its calculation formula are respectively：
The maximum principal curvatures tried to achieve, minimum principal curvatures are brought into above-mentioned formula, just obtained on face curved surface one
The shape index and curvature of apex；
Be projected through procedure below of the normal vector on x, tri- directions of y, z obtains：The normal direction value on each summit
Its calculation formula is：
Wherein, viIt is summit i, vi,jIt is by vertex viAnd vjSide determined by connection, each top is calculated according to the formula
Point place normal vector, the normal vector for three-dimensional vector, i.e. projection of the normal vector on x, tri- directions of y, z.
Further improve of the invention is, the third dimension on each summit in the three-dimensional face curved surface pre-processed is sat
It is denoted as a new parameter, referred to as depth.
The present invention, which further improves, to be, the detailed process that the face that will be mapped on two dimensional surface disk is ajusted
For：Two inner eye corner points and prenasale are chosen, according to the coordinate information of these three points, determines a spin matrixThen the spin matrix is multiplied by respectively to all summits, the face after being ajusted.
Further improve of the invention is that, by the method for rarefaction representation, the detailed process for obtaining the comparison of face is：
Given one belongs to inhomogeneous data acquisition system comprising N number of, and the dictionary for defining sparse representation model is D=[d1,d2,...,dN],
Then for any one image y, it is the dictionary of sparse representation model to have y=Dx+ ε, D, and x is sparse coefficient, and ε is reconstruction error；
Wherein, sparse coefficient x is solved by following formula：
In formula, L represents the openness of coefficient；Assuming thatThe solution of above formula, then minimum reconstruction error vector ri(y) it is
Minimum reconstruction error vector representation image y classification, wherein, δiRepresent the indicator function for belonging to i classes；
Minimum reconstruction error vector gives test face y identity, so as to complete the comparison of three-dimensional face.
Compared with prior art, the device have the advantages that：Curved surface is reconciled shape image and depth by the present invention
Habit is combined, on the one hand, curved surface, which reconciles, to be shone upon with deep mathematical theory basis and distinct physical interpretation, moreover, curved surface
Algorithm is shone upon in mediation has stronger robustness to the topological sum border of curved surface.On the other hand, deep learning being capable of Extraction and discrimination
The stronger characteristics of image of power, the combination of the two will be expected to greatly improve the property based on surface parameterization three-dimensional face recognition algorithm
Energy.By the experiment on disclosed three-dimensional face identification database in the world, demonstrate based on six kinds of mediation shape images
The validity of the validity that three-dimensional face represents and the depth representing based on mediation shape image.Identified with existing three-dimensional face
Technology is compared, three-dimensional face identification technology of the invention have the advantages that it is simple, be easily achieved, accuracy of identification it is high, and opposite
Robustness is preferable during portion's expression shape change.
Brief description of the drawings
Fig. 1 is the flow chart of this three-dimensional face identification invention.
Fig. 2 is data format in three-dimensional face file.
Fig. 3 is three-dimensional face and the two-dimension human face after mediation is shone upon.Wherein, it is three-dimensional face to scheme (a), is schemed (b)
For the two-dimension human face after mediation is shone upon, figure (c) is the two-dimension human face after ajusting.
Fig. 4 is six kinds of mediation shape images.Wherein, it is depth map to scheme (a), and figure (b) is curvature figure, and figure (c) refers to for shape
Number figure, figure (d) are schemed for normal vector x, and figure (e) is schemed for normal vector y, and figure (f) is schemed for normal vector z.
Embodiment
The present invention is described in detail below in conjunction with the accompanying drawings.
Referring to Fig. 1, of the invention comprises the following steps that：
First, data set introduction：
In the present invention, tested using BU-3DFE databases.The database includes the 3-D view (56 of 100 people
Position women, 44 non-males).Everyone contain 24 bands, six kinds of expressions (Happy, Disgust, Fear, Angry, Surprise,
Sadness 3-D view), the image of each expression are divided into 4 different grades again.Along with a neutral image, often
Individual amounts to 25 3-D views.Total data set amounts to 2500 images.In the experiment of the present invention, 100 three-dimensional people of neutrality
Face is as baseline sample, and 2400 three-dimensional faces with expression are as test sample.
2nd, data prediction：
The face in a database is given, reads out its data from original " mesh " file first, including
Apex coordinate (vertex), gore (face) and key point (landmark).And new file is preserved into treat
The processing of next step.Its document instance is as shown in Figure 2.
Noise remove, prenasale detection, face cutting, left and right inner eye corner point automatic detection are carried out to three-dimensional face curved surface；
2) reconcile and shine upon：The three-dimensional face curved surface that will have been pre-processed, is shone upon using mediation, and three-dimensional face curved surface is mapped to
On two dimensional surface disk；
The meaning directly perceived shone upon of reconciling is, it is assumed that surface S is made by rubber, then by its areal stretch to plane
Identical element, and fix its border.This mapping nature can reduce the energy (tensile energy) of film, and the mapping that this is minimized is just
It is to reconcile to map.Mediation, which is mapped in, very big application value in engineering, a underlying cause is that it can be in appropriate bar
Differomorphism is provided under part, i.e., mapping now is dijection and smooth.This is also referred to as Rado theorems.
In the present invention, detailed process three-dimensional face curved surface being mapped on two dimensional surface disk D is as follows：Surface is three
Hexagonal lattice dough sheet M, now reconcile and shine upon for discrete situation.For the function being defined within summit
The mediation energy of the function f on summit can be drawn, can be written as：
Wherein, wijFor side [vi, vj] cotangent weight.Given [vi, vj] it is face [vi, vj, vk] and face [vi, vj, vl] adjacent
Side, θkIt is with vkIt is summit in [vi, vj, vk] drift angle in triangle；θlIt is with vlIt is summit in [vi, vj, vl] triangle
Drift angle in shape, now the weight on side be then referred to as：If [vi, vj] it is to be in borderline one
Side, then now unique adjacent triangle only has [vi, vj, vk], then now：Pushed up for all inside
Point, discrete Laplace's equation are changed into：
This is a linear equation system that can be solved, and solution obtains the function f on summit, according on summit
Function f, three-dimensional face curved surface is mapped on two circle of position disks.
3) curvature and normal vector of each apex are calculated：
Calculate the curvature and normal vector of each apex.It is one discrete approx to think the region around the summit
The curved surface containing ginsengIt is bent according to the three-dimensional face pre-processed
Parameter A, B, C, D, E, F and G that the coordinate fitting in face goes out in above formula；Further according to matrixTo its carry out
Characteristic root decomposes, and obtains the i.e. maximum principal curvatures K of characteristic root of maximum1, and minimum characteristic root is minimum principal curvatures K2.So
The maximum principal curve value on summit and minimum principal curve value are just obtained on face curved surface.Next, introduce two seed ginsengs
Number：Shape index Shapeindex and curvature Curvedness.Its calculation formula is respectively：
The minimax curvature value tried to achieve is brought into above-mentioned formula, has just obtained on face curved surface apex
Shape index and curvature.
Also need to calculate the normal direction value on each summit, its calculation formula is：
Wherein, viIt is summit i, vi,jIt is by vertex viAnd vjSide determined by connection.Each top is calculated according to the formula
Normal vector at point.The normal vector is three-dimensional vector, i.e. projection of the normal vector on x, tri- directions of y, z.
In addition, because by three-dimensional face mapping transformation to two-dimension human face coordinate information, the three-dimensional that will have been pre-processed
The third dimension coordinate on each summit parameter new as one, here referred to as depth in face curved surface；
So far, for each summit in three-dimensional face, six kinds of mediation form parameters have been respectively obtained.I.e. six kinds
Mediation form parameter is the projection of shape index Shapeindex, curvature Curvedness, normal vector on x, tri- directions of y, z
And depth.
4) because the face being mapped in step 2) on two dimensional surface disk is all not the form of face, therefore have chosen
Two inner eye corner points and prenasale, according to the coordinate information of these three points, determine a spin matrixThen the spin matrix is multiplied by respectively to all summits, the face after just being ajusted, referring to
Fig. 3；
Six kinds of mediation form parameters on each summit that step 3) is tried to achieve, are respectively embedded in the face after ajusting
In, the two-dimension human face image of six kinds of mediation shapes is obtained, referring to Fig. 4.
5) two-dimension human face image for six kinds of mediation shapes for obtaining step 4) is input to depth convolutional neural networks (vgg
Deep face net) in, using the depth convolutional neural networks trained, extract its depth characteristic；
Using the depth convolutional neural networks trained, the specific process for extracting its depth characteristic is：Choose
" vgg deep face net " networks extract feature.Its network structure is shown in Table 1.
The vgg deep face net network structures of table 1
Before input picture, obtained two-dimension human face image is unified for 214*214*3 size, it is then that image is defeated
Enter to the depth convolutional neural networks, choose the output layer that the 29th layer (conv5-3) is used as depth characteristic.Its feature exported
Vector is the vector of 7*7*512 sizes.It is drawn into 1*25088 vector.Just complete so far based on six kinds of mediations
The depth characteristic extraction of shape two-dimension human face image.
6) face alignment：The depth characteristic of six kinds of mediation shape two-dimension human face images is obtained according to step 5), by sparse
The method of expression, realize the comparison of three-dimensional face.
By the method for rarefaction representation, the detailed process for obtaining the comparison of face is：Given one belongs to different comprising N number of
The data acquisition system of class, the dictionary for defining sparse representation model are D=[d1,d2,...,dN], then for any one image y, there is y
=Dx+ ε, D are the dictionary of sparse representation model, and x is sparse coefficient, and ε is reconstruction error.Wherein, sparse coefficient x can pass through following formula
Solve：
Wherein, L represents the openness of coefficient.Assuming thatIt is the solution of above formula, then minimum reconstruction error vector is
Minimum reconstruction error vector representation image y classification, wherein δiRepresent the indicator function for belonging to i classes.
Minimum reconstruction error vector gives test face y identity, so as to complete the comparison of three-dimensional face.
Specifically, by the method for rarefaction representation, realize that the detailed process of the comparison of three-dimensional face is as follows：
Input using the depth characteristic of obtain six kinds of mediation shape two-dimension human face images as identification.Wherein with expression
Image carries out face alignment using the method for rarefaction representation, obtains various mediations as testing image, neutral expression as benchmark
Discrimination after shape image and its fusion on the whole database.Specific discrimination is shown in Table 2：
Discrimination of the present invention of table 2 on BU-3DFE databases, unit are：%
| HSIg | HSIc | HSIs | HSInx | HSIny | HSInz | HSInxyz | HSIs+nxyz | HSIg+s+nxyz | All |
| 56.33 | 44.00 | 79.63 | 72.42 | 82.79 | 75.96 | 87.04 | 89.38 | 85.42 | 83.33 |
As can be seen from Table 2 except mediation curvature figure, remaining mediation shape image all achieve good discrimination.
Different mediation shapes is merged, finds HSIs+nxyzHighest discrimination, discrimination 89.38% are obtained.Immediately
, for HSIs+nxyzSuch fusion feature, tested on different expression word banks, experimental result is shown in Table 3：
The HSI of table 3s+nxyzDiscrimination on BU-3DFE difference expression word banks, unit are：%
| Happy | Surprise | Fear | Sadness | Anger | Disgust | All |
| 88.8 | 83.0 | 92.0 | 95.8 | 93.5 | 83.3 | 89.38 |
Can be clearly seen that from table 3, the method for the invention robustness in the expression of change is very high, in Fear,
Extraordinary discrimination is achieved in Sadness, Anger class expression.
Shone upon and depth learning technology in place of the main innovation of the present invention in the curved surface in Modern Differential Geometry is reconciled
It is combined, it is proposed that a kind of new three-dimensional face identification technology, and demonstrate the validity of the technology.
Claims (7)
1. a kind of three-dimensional face identification method based on curved surface mediation shape image depth representing, it is characterised in that including following
Step：
1) data prediction：A three-dimensional face curved surface is given, it is pre-processed；
2) reconcile and shine upon：The three-dimensional face curved surface that will have been pre-processed, is shone upon using mediation, and three-dimensional face curved surface is mapped into two dimension
On plane disc；
3) for each summit in three-dimensional face, the curvature and normal vector of each apex is calculated, has respectively obtained six
Kind mediation form parameter, six kinds of mediation form parameters are the projection of shape index, curvature, normal vector on x, tri- directions of y, z
And depth；
4) face that will be mapped on two dimensional surface disk is ajusted, the face after being ajusted；Step 3) is tried to achieve
Six kinds of mediation form parameters on each summit, are respectively embedded in the face after ajusting, have obtained the two of six kinds of mediation shapes
Tie up facial image；
5) two-dimension human face image for six kinds of mediation shapes for obtaining step 4) is input in depth convolutional neural networks, using
Trained good depth convolutional neural networks, extract its depth characteristic；
6) face alignment：The depth characteristic of image is obtained according to step 5), by the method for rarefaction representation, realizes three-dimensional face
Compare.
2. a kind of three-dimensional face identification method based on curved surface mediation shape image depth representing according to claim 1,
Characterized in that, pretreatment detailed process is：Noise remove, prenasale detection, face cutting and left and right inner eye corner point are automatic
Detection.
3. a kind of three-dimensional face identification method based on curved surface mediation shape image depth representing according to claim 1,
Characterized in that, the detailed process that three-dimensional face curved surface is mapped on two dimensional surface disk D is as follows：Surface is triangular mesh
Dough sheet M, now reconcile and shine upon for discrete situation, for the function being defined within summitDraw function f
Mediation energy, be written as：
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Wherein, wijFor side [vi,vj] cotangent weight；Given [vi,vj] it is face [vi,vj,vk] and face [vi,vj,vl] adjacent
Side, θkIt is with vkIt is summit in [vi,vj,vk] drift angle in triangle；θlIt is with vlIt is summit in [vi,vj,vl] triangle
In drift angle, now the weight on side be then referred to as：If [vi,vj] it is to be in borderline one
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Point, discrete Laplace's equation are changed into：
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Curved surface is mapped on two circle of position disks.
4. a kind of three-dimensional face identification method based on curved surface mediation shape image depth representing according to claim 1,
Characterized in that, shape index and curvature obtain especially by procedure below：Calculate the curvature and normal direction of each apex
Amount, it is believed that the region around the summit is a discrete curved surface containing ginsengAccording to the coordinate of the three-dimensional face curved surface pre-processed
Fit parameter A, B, C, D, E, F and G in above formula；Further according to matrixCharacteristic root point is carried out to it
Solution, obtain the i.e. maximum principal curvatures K of characteristic root of maximum1, and minimum characteristic root is minimum principal curvatures K2；Introduce two kinds of parameters：
Shape index Shapeindex and curvature Curvedness, its calculation formula are respectively：
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<mi>K</mi>
<mn>2</mn>
</msub>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
The maximum principal curvatures tried to achieve, minimum principal curvatures are brought into above-mentioned formula, just obtained a summit on face curved surface
The shape index and curvature at place；
Be projected through procedure below of the normal vector on x, tri- directions of y, z obtains：The normal direction value on each summitIt is calculated
Formula is：
<mrow>
<msub>
<mi>n</mi>
<msub>
<mi>v</mi>
<mi>i</mi>
</msub>
</msub>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>n</mi>
<msub>
<mi>v</mi>
<mi>i</mi>
</msub>
<mi>x</mi>
</msubsup>
<mo>,</mo>
<msubsup>
<mi>n</mi>
<msub>
<mi>v</mi>
<mi>i</mi>
</msub>
<mi>y</mi>
</msubsup>
<mo>,</mo>
<msubsup>
<mi>n</mi>
<msub>
<mi>v</mi>
<mi>i</mi>
</msub>
<mi>z</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>k</mi>
</mfrac>
<munderover>
<mi>&Sigma;</mi>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>k</mi>
</munderover>
<mfrac>
<mrow>
<mo>&lsqb;</mo>
<msub>
<mi>v</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>v</mi>
<mi>i</mi>
</msub>
<mo>&rsqb;</mo>
<mo>&times;</mo>
<mo>&lsqb;</mo>
<msub>
<mi>v</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>v</mi>
<mi>i</mi>
</msub>
<mo>&rsqb;</mo>
</mrow>
<mrow>
<mo>|</mo>
<mo>|</mo>
<mo>&lsqb;</mo>
<msub>
<mi>v</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>v</mi>
<mi>i</mi>
</msub>
<mo>&rsqb;</mo>
<mo>&times;</mo>
<mo>&lsqb;</mo>
<msub>
<mi>v</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>v</mi>
<mi>i</mi>
</msub>
<mo>&rsqb;</mo>
<mo>|</mo>
<mo>|</mo>
</mrow>
</mfrac>
</mrow>
Wherein, viIt is summit i, vi,jIt is by vertex viAnd vjSide determined by connection, each apex is calculated according to the formula
Normal vector, the normal vector is three-dimensional vector, i.e. projection of the normal vector on x, tri- directions of y, z.
5. a kind of three-dimensional face identification method based on curved surface mediation shape image depth representing according to claim 1,
Characterized in that, using the third dimension coordinate on each summit in the three-dimensional face curved surface pre-processed the parameter new as one,
Referred to as depth.
6. a kind of three-dimensional face identification method based on curved surface mediation shape image depth representing according to claim 1,
Characterized in that, the detailed process that the face that will be mapped on two dimensional surface disk is ajusted is：Choose two inner eye corner points
And prenasale, according to the coordinate information of these three points, determine a spin matrixThen to institute
The spin matrix is multiplied by some summits respectively, the face after being ajusted.
7. a kind of three-dimensional face identification method based on curved surface mediation shape image depth representing according to claim 1,
Characterized in that, by the method for rarefaction representation, the detailed process for obtaining the comparison of face is：Given one belongs to comprising N number of
Inhomogeneous data acquisition system, the dictionary for defining sparse representation model are D=[d1,d2,...,dN], then for any one image
Y, it is the dictionary of sparse representation model to have y=Dx+ ε, D, and x is sparse coefficient, and ε is reconstruction error；Wherein, sparse coefficient x passes through
Following formula solves：
<mrow>
<mi>m</mi>
<mi>i</mi>
<mi>n</mi>
<mo>|</mo>
<mo>|</mo>
<mi>y</mi>
<mo>-</mo>
<mi>D</mi>
<mi>x</mi>
<mo>|</mo>
<msubsup>
<mo>|</mo>
<mn>2</mn>
<mn>2</mn>
</msubsup>
<mo>,</mo>
<mi>s</mi>
<mo>.</mo>
<mi>t</mi>
<mo>.</mo>
<mo>|</mo>
<mo>|</mo>
<mi>x</mi>
<mo>|</mo>
<msub>
<mo>|</mo>
<mn>0</mn>
</msub>
<mo>&le;</mo>
<mi>L</mi>
</mrow>
In formula, L represents the openness of coefficient；Assuming thatThe solution of above formula, then minimum reconstruction error vector ri(y) it is
<mrow>
<msub>
<mi>r</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>|</mo>
<mo>|</mo>
<mi>y</mi>
<mo>-</mo>
<msub>
<mi>D&delta;</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mo>)</mo>
</mrow>
<mo>|</mo>
<msubsup>
<mo>|</mo>
<mn>2</mn>
<mn>2</mn>
</msubsup>
<mo>,</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>N</mi>
</mrow>
Minimum reconstruction error vector representation image y classification, wherein, δiRepresent the indicator function for belonging to i classes；
Minimum reconstruction error vector gives test face y identity, so as to complete the comparison of three-dimensional face.
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