4.3 System Implementation
4.3.3 Feature Extraction
The shape of the normalized hand posture images are represented using the proposed DOM based shape descriptors proposed in Chapter 3. The shape descriptors constitute the unique features extracted from the hand posture. Thus, the first set of features extracted are the Krawtchouk and the discrete Tchebichef moments based features.
Based on the review of several shape features presented in Chapter 2, the other robust hand posture de- scriptors considered along with the DOMs for comparative evaluation are the (a) Fourier descriptors (FDs); (b) geometric moments; (c) Zernike moments; (d) Gabor wavelets and the (e) PCA based features.
Given the normalized hand posture shape f (x,y), the extraction of the shape features are explained as follows.
4.3 System Implementation
4.3.3.1 Extraction of moment shape descriptors
The moment based shape descriptors that include the geometric, the Zernike, the Krawtchouk and the discrete Tchebichef moments are region based descriptors. Proper choice of the order of the moments is an important factor in deriving the moment features. The order of the orthogonal moments are chosen based on the reconstruction accuracy.
a) Extracting the proposed DOM based shape descriptors The DOPs namely the WKPs and the DTPs required for approximating the function f (x,y) are derived from (3.50) and (3.64) respectively. The corre- sponding Krawtchouk and the discrete Tchebichef moment features are computed using (3.71) and (3.72) re- spectively. For the Krawtchouk and the discrete Tchebichef moments computed up to order (n+m), the number of moment features obtained are (n+1) (m+1).
b) Extracting the geometric and the Zernike moment descriptors The non-orthogonal geometric moments of order (n+m) representing f (x,y) are derived using (2.11). The order of the geometric moments is chosen experimentally as 14 (n=7 and m=7).
The continuous orthogonal Zernike moment features are computed using the formulation given through (2.12) and (2.13). In the case of Zernike moments, the repetition m is chosen to take only positive integer values. For order up to n and m≥0, the number of Zernike moment features obtained can be easily verified to ben
2+12
if n is even and (n+1)(n+3)4 if n is odd.
An example illustrating the reconstruction accuracy of the Zernike, the Krawtchouk and the discrete Tchebichef moments for various choice of number of moment features is shown in Figure 4.13. The images reconstructed from the moments based approximations at various order are shown in Figure 4.13(b). The reconstruction error computed in terms of the SSIM index and the MHD values are shown in Figure 4.13(c) and 4.13(d) respec- tively. From the plots, we infer that for the given order, the Krawtchouk moments exhibit comparatively higher reconstruction accuracy. The results in Figure 4.13(b) show that the images reconstructed using the Zernike moments are comparatively not well defined. It is noted that the concavities are better defined in Krawtchouk based approximation and the rate of convergence towards the optimal value is faster in the case of Krawtchouk moments. For higher orders, the Zernike moments become numerically unstable resulting in higher reconstruc- tion errors. However, at lower orders, the performance of the Zernike moments is almost close to that of the discrete Tchebichef moments. As the order increases, the performance of the discrete Tchebichef moments is close to that of the Krawtchouk moments.
Based on the analysis on reconstruction error, the order n of the Zernike moments is chosen as 29. Be-
Original shape
(a)
Zernike Krawtchouk discrete Tchebichef
Order: 10
Order: 20
Order: 30
Order: 40
Shapes reconstructed using
(b)
9 25 81 169 289 625 1681
0.6 0.7 0.8 0.9
Number of moments
SSIM
(c)∗X-axis and Y-axis are in log-scale.
9 25 81 169 289 625 1681
0.3 1.22 3.32 9 24.5
Number of moments
MHD
(d)∗X-axis is in log-scale.
Zernike moments Krawtchouk moments discrete Tchebichef moments
Figure 4.13: Illustration of reconstruction of the hand posture shape for different orders of orthogonal moments. (a) Original hand posture shape; (b) Shape reconstructed from orthogonal moments. Comparative plot of (c) SSIM index vs number of moments and (d) MHD vs number of moments.
yond this order, there is only marginal improvement in the reconstruction accuracy. Also, it is known that the computational complexity of the Zernike moments increases with the order. Therefore, as a trade-off be- tween the computational time and the reconstruction accuracy, the order is chosen as n = 29. The optimal choice of the order (n+m) for the Krawtchouk and the discrete Tchebichef moments are chosen as 80 such that n=40 and m= 40. At this order, we observed that both the moments exhibit similar performance with higher reconstruction accuracy.
4.3.3.2 Extraction of non-moment shape descriptors
a) Extracting FDs The FDs representing the hand postures are derived from the boundary of the hand posture images. The number of boundary points representing the shape boundary are normalized to a fixed value. The points on the shape boundary for efficiently representing the hand postures are chosen at uniform intervals and the number of boundary points is experimentally chosen as 255. For these points (x,y)∈B, the FDs are computed using (2.9) and (2.10).
b) Deriving Gabor wavelet features The Gabor wavelet features are derived from the normalized hand posture
4.3 System Implementation
images using the formulation given in (2.26). The number of scales and the orientations are experimentally chosen as 8 and 10 respectively. As per the experimental studies in [4], the optimal value of width of the Gaussian function is chosen asσ=πand the center frequency is chosen asωmax= π2.
c) Computing PCA features Similar to the moments based approach, the PCA based method is a region based shape descriptor. The PCA based shape features are computed through the steps explained in Section 2.3.2.2.
The PCA features are estimated using the definitions in (2.14) to (2.16) and the projected images are computed using (2.17).
The number of eigen components required for computing the transformation matrix is experimentally cho- sen based on the reconstruction accuracy. Given the eigen values {λ1, λ2,· · · , λk} , the eigen values to be retained is computed from the ratio [183],
χeigen =
l
P
i=1
λi
k
P
i=1
λi
; l<k (4.29)
The number of eigenvalues l for which the ratioχeigenis atleast 0.95 are chosen to form the projection matrix.
This ratio implies that 95% of the variance present in the data are retained by the first l number of eigenvalues that are arranged in decreasing order.
Based on (4.29), the number of eigen components representing 99% of the data variance is obtained as 130.
Therefore, the eigenvectors corresponding to the first 130 eigenvalues can be used to form the transformation matrix Wpca. However, the experiments on reconstruction accuracy with respect to the varying l have shown that the number of eigen components required for accurately reconstructing the image from the PCA projections is only 30. An example, illustrating the shape reconstruction accuracy of the PCA projections for varying choice of number of eigen components is shown in Figure 4.14. The reconstruction accuracy is computed using the SSIM index and the MHD value. The results show that for l=30, the reconstruction accuracy in terms of the SSIM index is around 0.99 and the MHD value is around 0.3. Beyond l = 30, the improvement is not very significant and hence, it suggests that the eigenvectors corresponding to the first 30 eigenvalues are sufficient for forming the transformation matrix. For l=30, the ratioχeigenis 0.78 implying that 78% of the data variance are represented by the first 30 eigenvalues.
Original shape
(a)
l= 110 l= 70
l= 30 l= 10
l= 8
(b)
(c)
4 30 50 70 90 110 130
0.98 0.99 1
Number of eigen components (l)
SSIM
(d)
4 30 50 70 90 110 130 0
1 2
Number of eigen components (l)
MHD
(e)
Figure 4.14: Illustration of shape reconstruction with respect to varying number of eigen components (a) Original shape;
(b) Shapes reconstructed from the PCA projections for different number of eigenvalues and (c) the results of binarisation of the reconstructed shapes in (b). The threshold for binarisation is uniformly chosen as 120. Comparative plots of (d) SSIM index vs number of eigenvalues and (e) MHD vs number of eigenvalues, computed between the shape in (a) and the reconstructed binary shapes in (c).