Through-Wall Image Enhancement Using Fuzzy and QR Decomposition

QR decomposition and fuzzy logic based scheme is proposed for through-wall image enhancement. QR decomposition is less complex compared to singular value decomposition. Fuzzy inference engine assigns weights to different overlapping subspaces. Quantitative measures and visual inspection are used to analyze existing and proposed techniques.

In this paper, a TWI image enhancement (clutter reduction) technique using QR decomposition (QRD) and fuzzy logic is presented (preliminary results presented in [13]). Weights are assigned to different QRD subspaces using fuzzy inference engine. Simulation results evaluated using mean square error (MSE), peak signal to noise ratio (PSNR), improvement factor (IF), and visual inspection (based on miss detection (MD) and false detection (FD)) are used to verify the proposed scheme.

Proposed Image Enhancement Using QRD
Let the input image (having dimensions × ) be decomposed into different subspaces ( cl , tar , and no ) using singular value decomposition (SVD) as where and are singular vector matrices and contains singular values. As discussed in [13], conventional SVD for TWI image enhancement assumes that the target is limited to the second spectral component only; that is, Besides the high computational complexity of SVD which is 4 2 +8 2 +9 3 [14], the statement of target containment in the second spectral component is not always true. To cater the above issues, QRD and fuzzy logic based scheme is proposed. The image can be decomposed into an orthogonal unitary matrix (having dimensions × and 2 The Scientific World Journal (3) Table 1 shows the accuracy, stability, and complexity analysis of different QRD algorithms (classical and modified Gram-Schmidt (CGS, MGS), Givens decomposition, Householder transformation (HT), etc. [14]) for TWI. Identical to SVD, the first subspace 1 = 1 1 represents wall clutters and rest subspaces contain targets and noise. Note that due to overlapping boundaries of targets and noise, it is difficult to extract target subspaces accurately. Foregoing in view, a weighting QRD based scheme is proposed to enhance targets. The enhanced image tar is where are weights applied to different subspaces. Fuzzy logic is used for the automatic weight assignment [15].

Input and Output MFs. Let
= ‖ ‖ and Δ = ‖ ‖ − ‖ +1 ‖ be norms and norm differences, respectively. Note that high value of and Δ the corresponding subspace more likely contains target(s) and is therefore enhanced by applying heavy weights (and vise versa).
Three Gaussian membership functions (MFs) are defined for . Similarly -means algorithm [16] is used to adjust the fuzzy parameters. and Δ ℎ are first clustered into three groups based on respective histograms. The means and variances, respectively, of each group are used as centers and variance ( ) are used.

Product Inference Engine (PIE).
Gaussian fuzzifier maps the input and Δ as where V 1 and V 2 are parameters used for input noise suppression and are chosen as V 1 = 2max ( ) 1 and V 2 = 2max 3 ( ) 2 [15]. Fuzzy IF-THEN rules for image enhancement are the following. The output of PIE using individual rule based inference, Mamdani implication, algebraic product for -norm, and max operator for -norm [15] is The The weights PIE are then computed as where ( ) is the height of ( ) in output MFs [15].

Takagi-Sugeno (TS) Inference.
In contrast to PIE, TS inference engine adjusts the output MFs using adaptive and/ or optimization techniques [17]. The TS rule-base (IF-THEN) for computing weights TS is Note that the output reduces for large 1 + 2 (which is desirable). The aggregated weights TS are where represents algebraic product (intersection operator).

Simulation and Results
Experimental setup for TWI is constructed using Agilent's vector network analyzer (VNA) which generates stepped frequency waveforms between 2 GHz and 3 GHz (1 GHz band width (BW)) having step size of Δ = 5 MHz and step size = 201. Maximum range is max = 30 m and range resolution is Δ = 0.15 m.
Broadband horn antenna which is mounted on twodimensional scanning frame (having dimensions 2.4 m × 3 m (width × height) and can slide along cross range and height) operates in monostatic mode with 12 dB gain. Thickness of the wall is 5 cm and relative permittivity and permeability are 2.3 and 1, respectively. The frame is placed 0.03 m away from wall and scanning is controlled by a microcontroller based mechanism. The scattering parameters are recorded at each step and transferred to a local computer for image reconstruction and processing. Received data is converted into time domain and beamforming algorithm is used for image reconstruction. Existing and proposed image Table 2: MSE, PSNR, IF, PSNR (dB) = 10 log 10 1 MSE , where is a reference image obtained by the difference of image, with and without target. tar , and tar , are average pixel values of target and clutter in enhanced image, respectively.
, and , are average pixel values of target and clutter in the original image, respectively.
MD is defined as "target was present in the original image, but was not detected in enhanced image. " FD is defined as "target was not present in the original image, but was detected in enhanced image. " For calculating FD and MD, a threshold is calculated using global thresholding algorithm [18]. Figure 1 shows the original B-scan containing two targets, the background subtracted reference and enhanced images, using existing SVD and proposed QRD based schemes. It can be observed that proposed schemes detect both targets whereas SVD based scheme is unable to locate both targets accurately. Figure 2 shows another example containing three targets. The proposed scheme detects all targets and provides a better target to background ratio compared to SVD based scheme. It is further noted that the proposed TS inference based scheme provides better results compared to PIE. Table 2 shows that proposed fuzzy QRD schemes are better (as compared to the SVD image enhancement scheme) in terms of MSE, PSNR, IF, MD, and FD.

Conclusion
QRD and fuzzy logic based image enhancement scheme is proposed for TWI. Compared with SVD, QRD provides less computational complexity. PIE and TS inference engines are used to assign weights to different QRD subspaces. Simulation results compared on visual and quantitative analysis show the significance of the proposed scheme.