magnitude information. But for computing the real coefficients, the outputs obtained by the convolution operation with the real part of the Gabor filter only need to be stored. That is why, memory requirement is reduced by half when only the real coefficients are used. In earlier literatures, it was also mentioned that an optimal performance of two-dimensional Gabor filter can be obtained by using real part of the filter. But, there is no concrete experimental validations in this regard. But in our analysis, an experimental evaluation using two-dimensional Gabor wavelet suggests that the real coefficients of Gabor function is sufficient to represent an image for the applications like stereo matching. In this chapter, the performance of local Gabor wavelet features for both overlapping and non-overlapping regions are evaluated. These comparisons are done by considering different window sizes, different number of orientations, and different scales.
Also, performance of these features are analyzed for radiometric changes. The metrics used for performance comparisons are MSE, CC, QI, and SSI. Experimental results show local Gabor wavelet features (overlapping regions) performs better as compared to the local Gabor wavelet features (non-overlapping regions) and global Gabor wavelet features. Additionally, it is shown that the real coefficients of Gabor filter represent an image more accurately as compared to the imaginary coefficients.
List of Publications
Journal Publications
[1] Malathi. T and M.K. Bhuyan, “Estimation of disparity map of stereo image pairs using spatial domain local Gabor wavelet”, IET computer vision, vol. 9, no. 4, pp. 595–602, 2015.
[2] Malathi. T and M.K. Bhuyan, “Asymmetric occlusion detection and filling scheme for the estimation of stereo disparity map”, IET computer vision, DOI 10.1049/iet-cvi.2015.0214, Online ISSN 1751-9640, Available online: April 2016.
[3] Malathi. T and M.K. Bhuyan, “Performance analysis of Gabor wavelet for extracting most infor- mative and efficient features”, Multimedia tools and applications, Springer, DOI 10.1007/s11042- 016-3414-2, Available online: April 2016.
Book Chapters
[4] M.K. Bhuyan and Malathi. T, “Review of the application of matrix information theory in video surveillance”, Matrix information geometry, Springer-Verlag, pp. 293–321, 2013.
[5] N. Mishra, M. K. Bhuyan, T. Malathi, Y. Iwahori and R. J. Woodham, “Pixel-wise background segmentation with moving camera”, Pattern recognition and machine intelligence, Lecture notes in computer science, Springer Berlin Heidelberg, vol. 8251, pp. 423–429, 2013.
[6] Malathi. T and M.K. Bhuyan, “Local Gabor wavelet-based feature extraction and evaluation”, Smart innovations, systems and technologies, Springer-Verlag, vol. 43, pp. 181–189, 2016.
Conference Publications
[7] M.K. Bhuyan and Malathi. T, “Performance evaluation of local Gabor wavelet-based dispar- ity map computation”, in Proc. International conference on electrical, electronics, computer engineering and their applications (EECEA’15), pp. 79–92, 2015.
[8] Malathi. T and M.K. Bhuyan, “Foreground object detection under camouflage using multiple camera-based codebooks”, in Proc. Annual IEEE Indian conference (INDICON’13), pp. 1–6, 2013.
[9] Malathi. T and M.K. Bhuyan, “Multiple camera-based codebooks for object detection under sudden illumination change”, in Proc. International conference on communication and signal processing (ICCSP’13), pp. 310–314, 2013.
[10] Malathi. T and M.K. Bhuyan, “Disparity map estimation using local Gabor wavelet under radiometric changes”, International conference on digital information processing, data mining, and wireless communications (DIPDMWC’15), pp. 148–156, 2015.
A
Appendix
A.1 Detailed explanation to obtain mother wavelet of Gabor filter
Gabor function is a Gaussian modulated complex sinusoidal. The most general 2D complex Gabor function is given by [121]:
ψ(x, y, ξ0, ν0, x0, y0, ρ, θ, σ, β) =
√1
πσβ exp
−
((x−x0) cosθ+ (y−y0) sinθ)
2σ2 +(−((x−x0) sinθ+ (y−y0) cosθ)) 2β2
·exp (i(ξ0(x−x0) +ν0(y−y0) +ρ)) (A.1) In the above equation, the first term in the right hand side (RHS) is the elliptical Gaussian function and the second term is the complex sinuoidal function. The filter is centered at (x = x0, y = y0) in the spatial domain, and at (ξ = ξ0, ν = ν0) in the spatial frequency domain. σ and β are the standard deviations of an elliptical Gaussian along the x andy axes. θis the orientation of the filter, rotated counter-clockwise around the origin. ρis the absolute phase of an individual filter. There are, therefore, eight degrees of freedom in the general Gabor function: ξ0, ν0, θ, ρ, σ, β, x0, y0.
The above equation is simplified by setting the spatial location of the filters center (x0 = 0, y0= 0) and the absolute phase ρ of the filter to 0. The above Equation (A.1) can then be expressed as:
ψ(x, y, ξ0, ν0, θ, σ, β) = 1
√πσβexp − (xcosθ+ysinθ)2
2σ2 +(−xsinθ+ycosθ)2 2β2
!!
·exp (i(ξ0x+ν0y)) (A.2)
The Fourier Transform of the simplified complex-valued Gabor function (Equation (A.2)) is shown below:
ψˆ(ξ, ν, ξ0, ν0, θ, σ, β) = 2p
πσβexp
−1 2
h
((ξ−ξ0) cosθ+ (ν−ν0) sinθ)2σ2+ ((ξ−ξ0) sinθ+ (ν−ν0) cosθ)2β2 i
(A.3) where ξ and ν are the spatial frequencies in radians per unit length along x and y. The number of degrees of freedom can be further reduce by implying constraints onσ,β andθ in terms ofξ0 and ν0 according to the physiological findings. The constraints are as follows:
(i) The aspect ratio βσ of the elliptical Gaussian envelope is 2:1. Hence, Equation (A.2) can be
Appendix
rewritten as:
ψ(x, y, ξ0, ν0, θ, σ) = 1
√2πσexp
− 1 8σ2
h4(xcosθ+ysinθ)2+ (−xsinθ+ycosθ)2i
·exp (i(ξ0x+ν0y)) (A.4)
(ii) The plane wave with frequency (ξ0, ν0) tends to have its propagating direction along the short axis of the elliptical Gaussian. The elliptical Gaussian rotates analogous to the plane wave rotation. Hence, the center frequency (ξ0, ν0) of the filter is related to the rotation angleθof the modulating Gaussian which is given by: ξ0 =ω0cosθ and ν0 =ω0sinθ where ω0 = p
ξ02+ν02. Inposing this constraint into Equation (A.4) yields:
ψ(x, y, ξ0, ν0, θ, σ) = 1
√2πσexp
− 1 8σ2
h
4(xcosθ+ysinθ)2+ (−xsinθ+ycosθ)2i
·exp (i(xω0cosθ+yω0sinθ)) (A.5)
(iii) The half-amplitude bandwidth of the frequency response is about 1 to 1.5 octaves along the optimal orientation. The relationship between σ and ω0 can be expressed as:
σ= κ
ω0 (A.6)
where,κ=√ 2 ln 2
2φ+1 2φ−1
. Here,φis the bandwidth in octaves. Incorporating this constrain in Equation (A.5) gives:
ψ(x, y, ω0, θ) = ω0
√2πκexp
−ω02 8κ2
h
4(xcosθ+ysinθ)2+ (−xsinθ+ycosθ)2i
·exp (i(xω0cosθ+yω0sinθ)) (A.7)
where, θ = arctanνξ00 and κ is fixed for Gabor wavelets of a particular bandwidth. The whole family can be translated to any spatial position (x0, y0). In order to make the Gabor filters into admissible wavelets, we need to introduce the following constraint.
(iv) Admissible wavelets are functions having zero mean. The sine component of the complex-valued Gabor filter has zero mean, but its cosine component has nonzero mean (DC response). The DC response can be computed from its Fourier transform Equation (A.3), withξ = 0 andν= 0
given by:
ψˆ(ξ= 0, ν = 0, ν0) =√
8πσexp
− κ2
2
(A.8) A family of admissible 2D Gabor wavelets can be obtained by subtracting this DC response (Equa- tion (A.8)) from the Gabor filter (Equation (A.7)),
ψ(x, y, ω0, θ) = ω0
√2πκexp
−ω02 8κ2
h4(xcosθ+ysinθ)2+ (−xsinθ+ycosθ)2i
·
exp (i(xω0cosθ+yω0sinθ))−exp
− κ2
2
(A.9)
Each of these two families of Gabor wavelets can be generated by rotation and dilation (affine group) of the mother Gabor wavelet which is as follows:
ψ(x, y) = 1
√2πexp
−1
8 4x2+y2
·
exp (iκx)−exp
− κ2
2
(A.10)