Fractional Discrete Cosine Transformation Based ...

Proceedings of 2015 RAECS UIET Panjab University Chandigarh 21-22nd December 2015

Fractional Discrete Cosine Transformation Based Reduced Set of Coefficients for Face Recognition V.P. Vishwakarma

Kumud Arora Inderprastha Engineering College,Ghaziabad, INDIA [email protected]

Abstract— In this paper an attempt is made to explore the effect of using reduced set of Discrete Fractional Cosine Transformation based features on the face recognition accuracy. Input image feature set is transformed from spatial domain to spatial frequency domain using FRDCT. The large number of coefficients of fractional order spectrum of the face images obtained by the application of 2D FRDCT is scaled down by classical data dimensionality reduction technique LDA approach. Reduced feature set is then classified by the use of nearest neighbor classifier. The effectiveness of the proposed approach is demonstrated through the simulation on the benchmark database (AT&T). Experimental results also show that unlike DCT, which preserves strong information packing capability, FRDCT also preserves this capability with varying rotation orders. Keywords—FRDCT(Fractional Discrete Cosine Transformation); LDA(Linear Discriminant Analysis); PCA (Principal Component Analysis)

I.

INTRODUCTION

Over the past two decades the demand for improving the accuracy of face recognition is there. The inherent nature of high-dimensions allied to the face images along with the presence of redundant and noisy dimensions vitiate the accuracy of classification performance [1]. Effective envision of the structure of high-dimensional data requires the images of various subjects to be mapped in a way that images of one subject are close to each other and dissimilar to the images of other subjects. In the quest to generate the reliable feature vectors, transformation coding is used abundantly in research literature for more than half century. Various transformations (DFT, DCT, DWT etc.) are applied both globally and locally for transforming the correlated pixels of image to the frequency domain [10]. However, due to better information packing capability and better approximation capability from few coefficients, Discrete Cosine Transform (DCT) has emerged as the de-facto image transformation in most visual systems from the last two decades. (DCT) represents finite sequence of input data points in terms of a sum of cosine functions at different frequencies [13]. Out of four standards of DCT variant, in particular the DCT-II, is often used in face recognition problem because of its strong "energy compaction" property [7]. The application of fractional version of discrete cosine transformation came up just few years back [3] whereas the fraction version of the root of this

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Poonam Garg

GGSIP University, New Delhi INDIA [email protected]

Institute of Management and Technology, Ghaziabad, INDIA [email protected]

transformation (Discrete Fraction Fourier Transformation) can be traced up to five decades back [8]. In case of automatic face recognition, one of the important pattern classification problem , the accurate recognition is heavily dependent upon the accuracy with which the features are extracted for face pattern representation as well as the performance of classifier used to distinguish between faces [1]. Till date many variants of application of Discrete Cosine Transformation (DCT) for global and local transformation has been reported. FRDCT introduced by Pei and Cariolaro [2-3] is based on fractional orders of Eigen values of DCT II Kernel. FRDCT kernel represents spatial domain as well as frequency domain information simultaneously for the extracted feature set. Shekhar et al. [19] extracted the facial features using FRDCT and classified it using Nearest Neighbor Classifier. However they used all the transformed coefficients for classification. This paper proposes to reduce the dimensionality of coefficients by LDA approach which reduces the dimensionality of data from N to K-1 where N=K*number of Training images used and K=number of unique classes present in the database. The reduced data set is further reduced by eliminating certain percentage of high order frequency components to give further boost in reducing the processing time. This paper aims to explore the utilization of discrete cosine transformation (DCT) domain with different rotation orders and varying fractions of coefficients. Section 2 revisits the preliminary concepts associated with FRDCT and Face recognition. In section 3 describes the proposed approach for facial feature representation using FRDCT, reducing the feature set dimensionality using FRDCT and final stage being the classifier for classification. Section 4 cites the simulation studies as well as experimental result analysis. In section 5, paper concludes with conclusion and further work that needs to be done. II.

REVIEW OF FRDCT & FACE RECOGNITION

In research literature various fractional transform algorithms are defined in the field of signal and image processing like fractional Fourier transform (FrFT) [4-5], fractional Cosine transform (FrCT) [2], fractional Sine transform (FrST) [3] and fractional wavelet transform [18]. The problems associated with conventional discrete Fourier transform is handled by fractional discrete Fourier transform

Proceedings of 2015 RAECS UIET Panjab University Chandigarh 21-22nd December 2015 [3]. By the motivation of FrFT, Pei and Cariolaro introduced Fractional Discrete Cosine Transform [2]. The fractional discrete Cosine transform is used in many applications like face recognition [19], image compression and watermarking [11, 12]. Fractional DCT, generalized form of conventional DCT with an additional free parameter and using this free parameter FRDCT makes its applications in all areas where DCT is used.

The same way the G2 block diagonal matrix is defined with angle η. Among the choices for θ and η we consider here θ(α)=2πα and η(α) = πα, where ‘α’ is the fractional order of the transform in time frequency plane. The value of α varies between 0&1 both inclusive i.e. 0 < α < 1 . A. Properties of FRDCT The fundamental properties of FRDCT kernel matrix: Real and Orthogonal i.e. Fα , N ∗ F 'α , N = I . FRDCT kernel unlike

DCT kernel is also parametric additive as well as parametric associative i.e. Fα,N*Fβ,N = F(α+β),N . If α=0.3 and β=0.2 then Fα= FRDCT(8,0.3)

Fig. 1. Fractional DCT

Mathematically, the 2D-FRDCT of an image with size dimensions [R, C] having rows and columns, with fractional order (α, β) is given by: R

X α , β ( m, n) =

0.8795

0.1821 -0.1558 -0.0424

0.0165

-0.0271

0.8965

0.3021

0.0898

0.1603 -0.0385 -0.0021 -0.2629

0.3112 -0.2238

0.7843

0.0262 -0.3027 -0.3705 -0.0900 -0.0145 0.7521

0.0084 -0.2779

0.1597

0.1067

0.8862 -0.0638 -0.3192 -0.0549

-0.1517 -0.1075

0.3587

0.4526 -0.0736

0.0862 -0.1476

0.2021 -0.4388

0.2240

-0.2136

0.1622

0.2035 -0.1997

0.9444 -0.0410

(1)

(2)

where m, n are 1,2,….N and ⎧ 1 if m = 0 o r m = N ⎪ Km = ⎨ 2

where U is unitary matrix with columns of eigen vectors u n, D is diagonal matrix with diagonal entries of eigen values λ n α raised by their α th power. A block diagonal matrix D is computed by: 0 ⎞ α ⎛G1N/2 (θ(α)) D =⎜ (4) ⎟ G 2 ( η ( α )) 0 ⎝ ⎠ N/2

where G1 and G2 are block diagonal matrix defined by

G1N / 2 (θ (α )) = ⎛⎜

cos(θ (α ))

⎝ − sin(θ (α ))

sin(θ (α )) ⎞ cos(θ (α ))

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⎟ ⎠

(5)

0.0554

0.7307 -0.3299 0.4315

0.7951

0.1789 -0.0091

0.1239 -0.0108

0.2081

0.9524

0.2072

0.0642

0.2037 -0.1712

0.9007

0.0437 -0.1988 -0.2640 -0.0769 -0.0308

0.1256 -0.0669 -0.1330

0.8855 -0.0083 -0.2251

0.0613 -0.1602

0.0614

0.1331 0.2587

0.3058 -0.0347 0.1728

-0.1645

0.0985

0.1262 -0.1250

0.1129

0.2993 -0.1941

0.9028 -0.0342

0.1284 -0.3119

0.1009

0.0180 -0.1688

0.9476 -0.0285 -0.2166 -0.0579

0.0519 -0.0867

0.1266

0.0581

0.8759 -0.2510 0.2981

0.9055

F0.3*F0.2 = F (0.5), where F0.3 and F0.2 are the kernels of one dimensional FRDCT obtained by (1). FRDCT kernel obtained above is parametric periodic with 2 i.e. Fα+2,N = Fα,N. In fact, (Krnl) α+2 =U*(D ) α+2 *UT , where

α+2

The fractional DCT is defined through the real powers of DCT Kernel matrix. The Eigen decomposition of Kernel matrix can be expressed as: (Krnl)a =U (D )a UT (3)

0.1225

0.7895 -0.0115 0.2108

0.1125 -0.1319 -0.0512 -0.0043

-0.1248 -0.0566

D

⎪⎩ 1 f o r o t h e r v a l u e s o f m

0.1158

0.4119 -0.2983

Fβ= FRDCT(8,0.2)

The transformation kernel Krnl (α,β) of Fractional DCT is defined by determining the fractional powers of the Eigen values of DCT kernel first row wise then column wise. For N point DCT, the kernel is defined as:

2 ( 2 m + 1) nπ × K m × cos( ) N 2N

0.3076

0.1076 -0.2393

p =1 q =1

K rn l =

0.0070

0.2030 -0.0953 -0.2191

C

∑∑ X ( p, q) ∗ Krnl (α , β )( p, q, m, n)

0.2689

=

(

G1N/2 (2π (α + 2))

0

0

G2N /2 (η(π (α + 2))

)

(6)

Block diagonal matrix (D)α is not symmetric for α) ≠2 α and α) ≠ α. FRDCT is reality preserving. For the range of α, FRDCT is partially like the Cosine transform and partially like the identity operation. When α =1, the FRDCT becomes the Discrete Cosine transform. When α =0, the FRDCT becomes the identity operation. For Face Recognition, it is desirable to extract features space with accuracy from the training images even though some sort of distortion or deformation (up to certain extent) is present. To reduce the huge dimensionality of data, approaches like PCA (Eigen face approach) and LDA (fisher face approach) are used [16-17]. LDA creates a linear combination of independent features which yields the largest mean differences between the desired classes. The basic idea of LDA is to find a linear transformation such that feature clusters are most separable after the transformation achieved through scatter matrix analysis [1, 16]. The goal of LDA is to maximize the

Proceedings of 2015 RAECS UIET Panjab University Chandigarh 21-22nd December 2015 between-class scatter matrix measure while minimizing the within-class scatter matrix measure [8]. LDA attempts to search for a linear transformation achieved through scatter matrix analysis such that the feature clusters are most separable after the transformation. LDA gives better discrimination analysis along with data dimension reduction than PCA but when the number of classes is less. After the original image has been transformed and discriminated, unlike DCT, the transformed coefficients reflect the spatial-frequencies that range from high to low. The very first coefficient refers to the image’s lowest frequency (DC-coefficient) which usually carries the majority of the most representative information from the original signal. The last few coefficients refer to the higher frequencies. These higher frequencies generally represent more detailed or fine information of signal [10] and are more susceptible to noise influence. The middle frequency coefficients carry different information levels of the original signal. III.

Image Database

PROPOSED APPROACH

Fractional Row order & Column order spectrum

In this section we present our methodology for using one of the appearance-based approaches LDA (“fisherface” approach) to the transformed feature vectors obtained from the application of FRDCT to the datasets. Fig. 1 shows the overview of the proposed method. It is composed of the following steps: a. Transformation of face image features from spatial domain to spatial-frequency domain using FRDCT with rotation order (0.5) using eqns. (1-3) for both the dimensions of image. b. Discriminant Analysis of input feature set using LDA approach. c. Truncation of the high order frequency components. d. Reduced set of transformed coefficients are classified by classifier (K- Nearest Neighbor classification).

FRDCT components in ZIGZAG order

Discrimination Analysis & Dimension Reduction using LDA

Quantization of High Frequency Components

Classifier

Error Rate (%)

Fig. 2. Block Diagram of Proposed Approach

IV. EXPERIMENTAL ANALYSIS In this section, the utility of the above described approach of using reduced set of FRDCT coefficients for face recognition is established. For this purpose, the experiments have been conducted on AT&T face database. In the present study, the performance of the proposed approach has been evaluated using appearance based discrimination analysis & dimension reduction approach LDA with nearest neighbor classification. To check the robustness of the proposed approach, for every experimental run the average of ten randomly selected training & testing image sets selected among AT&T database containing forty images of ten subjects with each image of 112x92 has been used. All images were cropped to include only the face. In fractional DCT, the geometric structure of the spatial domain is displayed by the magnitude of the fractional power of cosine transform. From the experiments, it was observed that FRDCT preserve both the magnitude and phase of the re-transformed image (Fig. 3). Upon retransform, the transformed images are restored back into the approximately correct spatial domain.

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Fig.3 Images Retrieved at Various Fractional Orders [0:0.1:1]

For investigating the performance of the proposed approach, we have used the nearest neighbor classification. This simple classification approach is used to establish the claim that classification performance of fractional order spectrum coefficients is comparable to that of DCT coefficients. The below given performance graph (Fig. 4) indicates the classification error rate achieved with the complete image DCT coefficients and the FRDCT coefficients obtained at fractional order 0.5. It has been found that with FRDCT coefficients after discrimination analysis, gives better

Proceedings of 2015 RAECS UIET Panjab University Chandigarh 21-22nd December 2015 classification accuracy than DCT coefficients. DCT coefficients however perform better when there is only one image in the training set.

Error Rate Vs Number of Images on AT&T Database 30 Error Rate with complete FRDCT Coefficients Error Rate with reduced low frequency(eight) FRDCT coefficients 25

Error Rate Vs Number of Images on AT&T Database Clas s ific ation E rror Rate

25 Error Rate with FRDCT Error Rate with DCT 20

Error Rate

15

20

15

10

5

10 0

5

0

2

3

4

5 6 Number of Images

7

8

9

Fig.6. Classification Error (% ) with complete set of FRDCT and DCT coefficients 2

3

4

5 6 7 Number of Training images

8

9

Fig.4. Classification Error (%) with complete set of FRDCT and DCT Coefficients

Taking advantage of the fact that high frequency don’t have much energy, we found that even after truncating 15% of FRDCT coefficients from every image of Database (Fig. 5), the classification rate is almost near to what we achieved with full coefficients.

To check the performance of proposed approach, we also compared the classification error rate obtained with the PCA based eigen-features (Fig. 7). Error rate obtained for various principal components retained is compared with that of obtained with LDA. Around 50 to 55 principal components are sufficient enough to represent more than 99% of variance [1]. s PCA Vs LDA with FRDCT Coefficients

30

30 Principal Components 40 Principal Components 50 Principal Components 25

Error Rate Vs Number of Images on AT&T Database

60 Principal Components 70 Principal Components

25


Complete FRDCT Coefficients Error Rate with 15% reduced FRDCT coefficients

20


20

120 Principal Components LDA

Clas s ific ation E rror Rate

15

15 10

10

5

0

5

2

3

4

5

6

7

8

Number of Training Images

0

Fig. 7. Classification error rate with various Principal components Vs. LDA FRDCT coeficients 1

2

3

4

5 6 Number of Images

7

8

9

10

Fig. 5. Classification Error (%) with complete set of FRDCT and reduced set of FRDCT coefficients

Performance graph (Fig.6) indicates the effect of truncating lower order coefficients. This performance graph indicates that truncating lower order FRDCT coefficients affect adversely their correct classification performance.

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It was found experimentally with 60 principal components the classification error rate is very close to that of obtained with LDA where the cumulative variance retained is 99.6%. As indicated in graph(Fig.7) the classification error rate obtained with PCA is less than LDA when the number of training images is less however this trend reverses as the number of images in the train data set increases.

Proceedings of 2015 RAECS UIET Panjab University Chandigarh 21-22nd December 2015 V.

CONCLUSION

In this paper face recognition using reduced set of Fractional Discrete Cosine Transformation has been presented. Experiments were conducted using transformed coefficients analyzed by LDA for dimension reduction and discrimination analysis. Reduced transformed coefficient set is fed to Nearest Neighbor for classification. From the experimental results it was observed that face recognition rate using FRDCT reduced coefficient set with LDA as dimension reduction approach varies between 88-96 % and using PCA as dimension reduction approach varies between (78.44% - 94%). Also the performance of FRDCT coefficient set is not satisfactory at very small order of rotation (0