Chapter 4 Face Recognition Using Orthogonal Transforms

Transcription

1 Chapter 4 Face Recognition Using Orthogonal Transforms Face recognition as a means of identification and authentication is becoming more reasonable with frequent research contributions in the area. In many classification techniques high-dimensional data are involved because large feature vectors are generated to be able to describe complex objects and with high volume of data the storage and real time processing becomes difficult. To avoid these problems the dimension of the feature space is reduced. Assuming that an important structure in the data actually resides in a smaller dimensional space is one method to handle this problem. Under this assumption the goal is to reduce the dimensionality before attempting the classification. The characteristic of image transform is to group low frequency components in few coefficients at the left most corner of the image. These few coefficients are capable of regenerating the image with minimum distortions. In this particular thesis following sinusoidal and non sinusoidal transforms are studied and their performance is checked for both the databases under normal condition, Occluded condition and in the presence of noise addition. 1. Discrete Cosine Transform (DCT) 2. Discrete sine transform (DST) 3. Walsh Hadamard Transform (WHT) 4. Slant transform (ST) 5. Kekre s Transform 6. Discrete Wavelet Transform (DWT)

2 All these transforms have the seperability property i.e. row and column of the image can be transformed separately. Following section 4.1 explains the concept of generating Row / Column Feature Vector in one dimension for the given gray scale image. 4.1 Algorithm for Calculating Row Feature Vectors () and Column Feature Vectors () Preparation of Row Feature Vectors As per the seperability property the image is first transformed row wise and then that intermediate image is transformed again column wise to get complete transform of the image. Here in this chapter the concept of Row feature Vector and Column Feature Vector is introduced. Where in method image is transformed only row wise as shown below in Fig. 4.1 NxN 1xN 1D Transform Row Feature Vector () Fig 4.1 Generation of Row Feature Vector () As shown in Fig.4.1 dimension of row vector is 1xN for image of size NxN. For generation of Row Feature Vectors following steps are followed Step 1. Select a row and take 1D Transform of it to get Row feature Vector () in transform domain. These coefficients are in general descending order of energy. 61

3 Step 2. Substitute these coefficients in the blank array of same size as image and get the row wise transformed image. Store all the images of given training database after processing it according to step 1 or step 2 to get row wise transformed images Comparison of Test Image for Face Recognition The test image is an image of a person either from database or an image of a person present in the database with different pose or gesture or it can be an image out of database. For the given test image the is calculated as explain in step 1 and step 2 of section To save the processing time for the recognition of face the concept of image energy is explored in different aspect. For the transformed image the energy of each row is calculated separately as per following equation. Energy in i th Row N j 1 F(i,j) 2 F i 4.1 Where F i is the energy in the transform of i th row of image I(i,j). To select the coefficients with x% energy M is calculated such that M j 1 2 [ F(i,j) ] xf 4.2 i The row wise energy calculation is as shown pictorially in Fig.4.2 below. Remaining all the coefficients are discarded. (Made equal to zero) 62

4 Select a percentage of energy per row Single row of a transformed image Final Row Feature vector Fig. 4.2 Calculation of Single Row Feature Vector For generation of row wise transformed image for particular energy level the above process is repeated for all the rows of a particular image. Similar procedure is repeated separately for the column of a image to generate Column Feature Vector. Following Fig.4.3 and Fig.4.4 shows the original transformed image and final row wise and column wise transformed image. Row wise transformed image Final transformed image with as per required Percentage of image energy Fig 4.3 Preparation of row wise transformed image for selected percentage of energy 63

5 Select a percentage of energy per column Column wise transformed image Final transformed image with as per required energy Fig. 4.4 Generation of Column transformed image as per selected percentage of energy For the given test image when final transformed image with as per required percentage of image energy is prepared compared the coefficients with the stored coefficients of training set using Euclidean distance as a similarity measures. If the of test image is within the predefine threshold algorithm recognize the image otherwise reject it. Similarly the column transformed image for particular energy level is obtained as explains in Fig Image Transform The image transforms are used extensively in image processing and analysis. In previous section 4.1 the transformed coefficients are obtained in 1D i.e. image is treated only row wise or column wise to get Row Feature Vectors or Column Feature Vectors. In this particular section various transforms are applied on the entire image. First the rows of the image is selected and transformed 64

6 coefficients are obtained and then columns of this intermediate image is selected for transformation as shown below in Fig 4.5 Original Image Original image Row wise transformed image Column wise transformed image Fig 4.5 Image transform applied on full image. The energy of the image is calculated by the following formula M N i 1 j 1 2 F F ( i, j) 4.3 M 1 N1 i 1 j 1 i 2 XF F(i,j) 4.4 M1 and N1 are selected to meet the requirements of eq. 4.4 Where X 1 is a fraction deciding the percentage of image energy. The energy concentration is in the left most corner of the image. So after selecting particular energy level rest all coefficients in the lower part of the image are made zero. Pictorially the transformed image will look like the one shown in Fig. 4.6 Transformed Image Image after energy selection Fig 4.6 Output image after selection of percentage of energy in full image 65

7 In this particular thesis DCT, DST, WHT, Slant Transform, Kekre s Transform and Discrete Wavelet Transform is applied on two distinct databases. The first database is a standard database ORL provided by AT&T lab [118] for research purpose and second database is locally created database without constrain. Both these database are shown in Appendix. 4.3 The Discrete Cosine Transform A Discrete Cosine Transform (DCT) is a sinusoidal orthogonal transform. The DCT has been used in digital signal and image processing and particularly in transform coding systems for data compression/decompression. This transform is real, orthogonal and separable. The DCT is a widely used frequency transform because it closely approximates the optimal KLT transform while not suffering from the drawbacks of applying the KLT. KLT is constructed from the eigen values and the corresponding eigenvectors of a covariance matrix of the data to be transformed, it is signaldependent and there is no general algorithm for its fast computation. The DCT does not suffer from all these drawbacks due to data-independent basis functions. The DCT provides a good trade-off between energy compaction ability and computational complexity. The energy compaction property of DCT is superior to that of any other unitary transform for a wide sense stationary random process with high correlation. This is important because these image transforms pack the most information into the fewest coefficients and yield the smallest reconstruction errors. DCT basis images are image independent as opposed to the optimal KLT which is data dependent. 66

8 All DCTs are separable transforms so the multidimensional transform can be decomposed into a successive application of one dimensional transforms (1-D) in the appropriate directions. The one-dimensional DCT is a technique that converts a spatial domain waveform into its constituent frequency components as represented by a set of coefficients The 1D DCT of a sequence u(n), n N is defined as v( k) ( k) N 1 n u( n)cos (2n 1) k 2N where k N () 1 N ( k) 2 N for 1 k N 1 The inverse transform is given as u ( n) N 1 k ( k) v( k)cos (2n 1) k 2N Where N N DCT transforms the input into a linear combination of weighted basis functions. These basis functions are the frequency components of the input data. For most images much of the signal energy lies at low frequencies and corresponding to large DCT coefficient magnitudes and located to the upper-left corner of the DCT. Conversely the lower-right values of the DCT array represent higher frequencies and turn out to be smaller in magnitude. Similarly for and the low frequency components are arranged in first column and first row of the image respectively. 67

9 4.3.1 Result Analysis and Discussion The proposed algorithm of taking 1D DCT row wise and column wise separately is tested for ORL and local database. Following graph shows that as percentage of energy in image for Row Feature Vector, Column Feature Vector and even for entire image is increased the percentage accuracy also increases Percentage Energy Variation Fig.4.7 Graph of percentage accuracy in Transformed image, Row Feature vector and Column Feature vector for different percentage of energy in local database for DCT. 68

10 Percentage Energy Variation Fig 4.8 Graph of percentage accuracy in transformed image, Row Feature vector and Column Feature vector for different percentage of energy in ORL database in DCT. Observation: The graphs (Fig 4.7-Fig 4.8) show that Row feature Vectors and Column Feature Vectors performed better than the full transformed image in local as well as ORL database. The robustness of algorithm is checked for the different percentage of occlusion and various types of noise like Speckle Noise, Gaussian noise, Salt and Pepper noise introduced on the test image. Following Fig. 4.9 and Fig. 4.1 shows the results for occlusion applied on test images for Local and ORL database. 69

11 x1 2x2 3x3 4x4 5x5 Percentage Occlusion Variation Fig.4.9 Graph of accuracy for different percentage of occlusion introduced as %length X %width of image for Local database in DCT at 9% energy x1 2x2 3x3 4x4 5x5 Percentage Occlusion Variation Fig.4.1 Graph of accuracy for different percentage of occlusion introduced as %length X %width of image for ORL database in DCT at 9% energy. Observation: The graphs show that as occlusion on image increases the accuracy reduces and after 3%x3% of occlusion the algorithm becomes unreliable for and approach and after 7

12 Percentage accuracy 2%x2% for full transform. The algorithm is also tested for various types of noise like Speckle Noise, Gaussian noise, Salt and pepper Noise. Following Graphs shows the performance of algorithm for these noises. Speckle Noise: Percentage Speckle Noise Fig 4.11 Performance of DCT in presence of a Speckle Noise for local database at 9% energy. 71

13 Percentage Speckle Noise Fig 4.12 Performance of DCT in presence of a Speckle Noise for ORL database at 9% energy. Observation: It has been observed clearly from graphs (Fig Fig 4.12) that Column feature vectors withstand Speckle Noise more robustly than other approaches in both databases. After 5% of noise addition the accuracy drops down to less than 5% in and and for full transform noise addition tolerance is upto 3%. 72

14 Gaussian Noise: Percentage Gaussian Noise Fig 4.13 Performance of DCT in presence of a Gaussian Noise for local database at 9% energy Percentage Gaussian Noise Fig 4.14 Performance of DCT in presence of a Gaussian Noise for ORL database at 9% energy. 73

15 Observation: The graphs (Fig 4.13-Fig 4.14) give an idea about the variation of a Gaussian noise and its effect on accuracy for both databases. It is clear from the graphs that ORL database withstands Gaussian noise uoto 2% which is better than local database where only 5% noise addition is advisable after that the accuracy falls down to 5%. Overall column feature vector method is giving better result for both the database. Salt and Pepper Noise: Percentage Salt and Pepper Noise Fig 4.15 Performance of DCT in presence of a Salt and Pepper noise for local database at 9% energy. 74

16 Percentage Salt and Pepper Noise Fig 4.16 Performance of DCT in presence of a Salt and Pepper noise for ORL database at 9% energy. Observation: The graphs(fig 4.15-Fig 4.16) shows that as noise addition in image increases the accuracy reduces and after 5% of noise addition the algorithm becomes unreliable. In ORL database the accuracy of row and column feature vector technique is almost same but in local database the column feature vector gives better result than row feature vector technique. Table 4.1 is an overall performance table for the DCT. The data give for accuracy of algorithm is for 9% image energy. Under occlusion and noise addition conditions the cutoff point is 5% accuracy below which algorithm is not considered. 75

17 Table 4.1 Overall Performance of DCT Method image transform Row Feature Vector () Column Feature Vector () Database Local ORL Local ORL Local ORL Accuracy (9%) %Occlusion withstand Speckle Noise 78% 78% 9% 92% 91% 92% 2%X2% 2%X2% 5%X5% 5%X5% 5%X5% 5%X5% 3% 3% 4% 5% 4% 5% Gaussian 5% 2% 5% 1% 1% 2% Salt & Pepper 3% 3% 3% 4% 5% 5% FAR 3% 1% 5% 4% 1% 5% FRR % % % % % % The Table 4.1 shows overall performance of Discrete Cosine Transform method. The ORL database due to its controlled nature performs better than local database. The Column feature vector method give better result overall compare to Row feature vector and full image transform. The FAR of transform is higher than and approaches. FRR is zero for all the techniques. 4.4 The Discrete Sine Transform (DST) The Discrete Sine Transform (DST) [8] is a real, symmetric and orthogonal transform. The N x N sine transform matrix s is defined as, s ( k, n) 2 N sin 1 ( k, n)( n N 1 1) 4.5 In eq. 4.5, k, n N-1 The Sine transform is real, symmetric and orthogonal ie. 76

18 s * s s T s Thus forward and inverse Sine transforms are identical. The sine transform pair of one dimensional sequence is defined as v(k) 2 N 1 N n 1 u( n)sin ( k 1)( n N 1 1) u(k) 2 N 1 N k 1 v( k)sin ( k 1)( n N 1 1) 4.7 where n N 1 and k N 1 In this particular section the DST is applied on full image and the Row and Column Feature Vectors are generated as explain in section 4.1. The statistical analysis is taken for all these cases in regular conditions as well as under occlusion and noise addition Result Analysis and Discussion The proposed algorithm of taking 1D DST row wise and column wise separately is tested for ORL and local database. Following graph shows that as percentage of energy in image for Row Feature Vector, Column Feature Vector and even for entire image is increased the percentage accuracy also increases. 77

19 Percentage Energy Variation Fig.4.17 Percentage accuracy in Transformed image, Row Feature vector and Column Feature vector for different percentage of energy in local database for DST Percentage Energy Variation Fig.4.18 Percentage accuracy in Transformed image, Row Feature vector and Column Feature vector for different percentage of energy in ORL database for DST. 78

20 Observation: The graphs (Fig 4.17-Fig 4.18) show that as percentage image energy increases the Row feature Vectors and Column Feature Vectors gives better accuracy than the full transformed image in local as well as ORL database. The robustness of algorithm is checked for the different percentage of occlusion and various types of noise like Speckle Noise, Gaussian noise, Salt and Pepper noise introduced on the test image. Following Fig. 4.19, Fig. 4.2 shows the results for occlusion applied on test images for Local and ORL database X1 2X2 3X3 4X4 5X5 Percentage Occlusion Variation Fig.4.19 Graph of accuracy for different percentage of occlusion introduced as %length X %width of image for local database in DST at 9% energy. 79

21 Percentage Occlusion Variation Fig.4.2 Graph of accuracy for different percentage of occlusion introduced as %length X %width of image for ORL database in DST at 9% energy. Observation: The graphs show that as occlusion on image increases the accuracy reduces and after 5% of occlusion the algorithm is not considered. The algorithm is also tested for various types of noise like Speckle Noise, Gaussian noise, Salt and pepper Noise. Following Graphs shows the performance of algorithm for these noises. 8

22 Speckle Noise: Percentage Speckel Noise Fig 4.21 Performance of DST in presence of a Speckle Noise for local database at 9% energy Percentage Speckle Noise Fig 4.22 Performance of DST in presence of a Speckle Noise for ORL database at 9% energy. Observation: It has been observed clearly from graphs(fig Fig 4.22) that Column feature vectors withstand Speckle Noise more 81

23 Percentahe Accuracy robustly than other approaches in both databases. After 5% of noise addition the accuracy drops down to less than 5%. Gaussian Noise: Percentage Gaussion Noise Fig 4.23 Performance of DST in presence of a Gaussian Noise for local database at 9% energy Percentage Gaussion Noise Fig 4.24 Performance of DST in presence of a Gaussian Noise for ORL database at 9% energy. Observation: The graphs (Fig 4.23-Fig 4.24) give an idea about 82

24 the variation of a Gaussian noise and its effect on accuracy for both databases. It is clear from the graphs that databases withstand Gaussian noise up to 5% after that the accuracy falls below 5%. Overall column feature vector method is giving better result for both the database. Salt and Pepper Noise: Percentage Salt and Pepper Noise Fig 4.25 Performance of DST in presence of a Salt and Pepper noise for local database at 9% energy. 83

25 Percentahe Accuracy Percentage Salt and Pepper Noise Fig 4.26 Performance of DST in presence of a Salt and Pepper noise for ORL database at 9% energy. Observation: The graphs (Fig 4.25-Fig 4.26) shows that as Salt and Pepper noise in image increases the accuracy reduces and after 4% of noise addition the algorithm becomes unreliable. The ORL database withstands the noise addition till 5% and local database can withstand noise addition up to 4% Table 4.2 is a overall performance table for the DST. The data given for accuracy of algorithm is for 9% image energy. Under occlusion and noise addition conditions the cutoff point is 5% accuracy below which algorithm is not considered. 84

26 Table 4.2 Overall Performance of DST Method image Row Feature Column Feature transform Vector Vector Database Local ORL Local ORL Local ORL Accuracy (9%) 72% 73% 87% 89% 9% 9% %Occlusion withstand 4%x4% 4%x4% 4%x4% 4%x4% 4%x4% 4%x4% Speckle Noise 3% 3% 5% 5% 5% 5% Gaussian 5% 5% 5% 5% 5% 5% Salt & Pepper 3% 3% 3% 4% 3% 4% FAR 1% 1% 1% 1% 1% 1% FRR % % % % % % The Table 4.2 shows overall performance of Discrete Sine Transform method. The ORL database due to its controlled nature performs better than local database. The Column feature vector method give better result overall compare to Row feature vector and full image transform. FAR is 1% and FRR is zero for all methods. 4.5 Walsh Hadamard Transform (WHT) The WHT [87] is an important fast transform in terms of energy compaction in the field of image processing. It is computationally fast as compared to DFT and DCT since it has entries as +1 or -1. The rows and the columns of the Hadamard matrix is orthogonal. Hadamard matrices of order of 2 n can be recursively generated as n n H (2 ) H(2) H(2 1) 4.8 Where symbol matrix given by indicated Kronecker product and H(2) is seed 85

27 1 1 H (2) The Walsh matrix is sequency ordered Hadamard matrix [88].Walsh matrix is obtained from Hadamard matrix by arranging rows in increasing sequency order where sequency of any Hadamard matrix is obtained by number of sign changes. A new and easy and prompt technique is suggested for generation of Walsh matrix with ordered sequency compared to Hadamard matrix called Kekre s Technique for Ordered Sequency Generation Kekre s Technique for Ordered Sequency Generation The Kekre s Technique for Ordered Sequency Generation algorithm [87] gives the sequence of numbers according to which the Hadamard rows can be arranged so that we obtain Walsh rows. Step 1: Arrange the rows in ascending order. For Hadamard matrix of dimension 16 x 16, ascending ordered rows are,,1,2,3,4,5,6,7,8,9,1,11,12,13,14,15. Step2: Split the row in N/2, the last part is written below the upper row but in reverse order as follows

28 Step3: We get two rows, each of this row is again split in N/2 and inserted below the upper row after reversing. Finally we get one column matrix which is,15,7,8,3,12,4,11,1,14,6,9,2,13,5,1 Entire procedure is tabulated in Table4.3 Table 4.3 Ordered Sequency Generation Result Analysis and Discussion The proposed algorithm of taking 1D WHT row wise and column wise separately is tested for ORL and local database. Following graph shows that as percentage of energy in image for Row Feature 87

29 Vector, Column Feature Vector and even for entire image is increased the percentage accuracy also increases. 88

30 cc Percentage Energy Variation Fig.4.27 Percentage accuracy in Transformed image, Row Feature vector and Column Feature vector for different percentage of energy in local database for WHT Percentage Energy Variation Fig.4.28 Percentage accuracy in Transformed image, Row Feature vector and Column Feature vector for different percentage of energy in ORL database for WHT. 89

31 Observation: The graphs (Fig 4.27-Fig 4.28) show that as percentage image energy increases the Row feature Vectors and Column Feature Vectors gives better accuracy than the full transformed image in local as well as ORL database. The robustness of algorithm is checked for the different percentage of occlusion and various types of noise like Speckle Noise, Gaussian noise, Salt and Pepper noise introduced on the test image. Following Fig. 4.29, Fig. 4.3 shows the results for occlusion applied on test images for Local and ORL database X1 2X2 3X3 4X4 5X5 Percentage Occlusion Variation Fig.4.29 Graph of accuracy for different percentage of occlusion introduced as %length X %width of image for Local database in WHT at 9% energy. 9

32 x1 2X2 3X3 4X4 5X5 Percentage Occlusion Variation Fig.4.3 Graph of accuracy for different percentage of occlusion introduced as %length X %width of image for Local database in WHT at 9% energy. Observation: The graphs shows that as occlusion on image increases the accuracy reduces. The Row and Column feature vector methods can withstand the attack of occlusion better than transform applied on full image. The ORL database performs better than the local database. The algorithm is also tested for various types of noise like Speckle Noise, Gaussian noise, Salt and pepper Noise. Following Graphs shows the performance of algorithm for these noises. 91

33 Speckle Noise: Percentage Speckle Noise Fig 4.31 Performance of WHT in presence of a Speckle Noise for local database at 9% energy Percentage Speckel Noise Fig 4.32 Performance of WHT in presence of a Speckle Noise for ORL database at 9% energy. Observation: It has been observe clearly from graphs(fig 4.31-Fig 4.32) that Column feature vectors withstand Speckle Noise more 92

34 robustly than other approaches in both databases. After 4% of noise addition the accuracy drops down to less than 5% in full transform. Gaussian Noise: Percentage Gaussian Noise Fig 4.33 Performance of WHT in presence of a Gaussian Noise for local database at 9% energy Percentage Gaussian Noise Fig 4.34 Performance of WHT in presence of a Gaussian Noise for ORL database at 9% energy. 93

35 Percentage accuracy Observation: The graphs (Fig 4.33-Fig 4.34) give an idea about the variation of a Gaussian noise and its effect on accuracy for both databases. It is clear from the graphs that ORL database withstands Gaussian noise up to 1% which is better than local database where only 5% noise addition is advisable after that the accuracy falls near to zero. Overall column feature vector method is giving better result for both the database. Salt and Pepper noise: Percentage Salt and Pepper Noise Fig 4.35 Performance of WHT in presence of a Salt and Pepper noise for local database at 9% energy. 94

36 Percentage Salt and Pepper Noise Fig 4.36 Performance of WHT in presence of a Salt and Pepper noise for ORL database at 9% energy. Observation: The graphs (Fig 4.35-Fig 4.36) show that as Salt and pepper noise in image increases the accuracy reduces and after 4% of noise addition the algorithm becomes unreliable. In ORL database the accuracy of row and column feature vector technique is almost same but in local database the column feature vector gives better result than row feature vector technique. Table 4.4 is an overall performance table for the WHT. The data given for accuracy of algorithm is for 9% image energy. Under occlusion and noise addition conditions the cutoff point is 5% accuracy below which algorithm is not reliable 95