# Through-Wall Image Enhancement Using Fuzzy and QR Decomposition

Authors: Muhammad Mohsin Riaz; Abdul Ghafoor
Journal: The Scientific World Journal
 (2014)

---

## Abstract

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.


---

## Body

## 1. Introduction

Mapping of scenes behind obstacles (including building wall, rubbers, grass, etc.) using through-wall imaging (TWI) is an unfolded research domain. Different military and commercial applications (including antiterrorism, hostage rescue and surveillance, etc. [1]) can benefit from TWI. Beside other challenges, minimization of unwanted artifacts (clutters/noise) has enjoyed special importance over last few years [2–13]. These unwanted artifacts significantly decrease target detection and recognition capabilities.Existing TWI image enhancement (clutter removal) techniques include background scene subtraction (only feasible if with and without target images are available) [2], spatial filtering (assuming wall homogeneity at low frequencies) [3], wall modeling and subtraction (requiring complex process for inhomogeneous walls) [4, 5] Doppler filtering (applicable for moving targets only) [6], image fusion (requiring multiple data of the same scene) [7], and statistical techniques [8–13].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.


## 2.  Proposed Image Enhancement Using QRD

Let the input imageM (having dimensions G
×
H) be decomposed into different subspaces (M
cl
⁡, M
tar, and M
no) using singular value decomposition (SVD) as
(1)
M
=
∑
g
=
1
l
1
s
g
u
g
v
g
T
︸
M
cl
⁡
+
∑
g
=
l
1
+
1
l
2
s
g
u
g
v
g
T
︸
M
tar
+
∑
g
=
l
2
+
1
G
s
g
u
g
v
g
T
︸
M
no
,
where U and V are singular vector matrices and S 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,
(2)
M
SVD
=
s
2
u
2
v
2
T
.
Besides the high computational complexity of SVD which is 4
G
2
H
+
8
G
H
2
+
9
H
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 M can be decomposed into an orthogonal unitary matrix Q (having dimensions G
×
H and column vectors q
g) and an upper triangular matrix R (having dimensions G
×
H and row vectors r
g), that is,
(3)
M
=
Q
R
.
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.Table 1
Comparison analysis of QRD algorithms for TWI.
Algorithms
Accuracy
Complexity
Stability
M
-
Q
R
Q
T
Q
-
I
Q
T
M
-
R
M
R
-
1
-
Q
CGS QR
1.091
×
1
0
-
16
1.310
×
1
0
-
10
6.724
×
1
0
-
14
9.913
×
1
0
-
14
2
G
H
2
Unstable
MGS QR
1.075
×
1
0
-
16
4.922
×
1
0
-
13
4.842
×
1
0
-
14
1.083
×
1
0
-
12
2
G
H
2
Stable
HT QR
1.291
×
1
0
-
15
3.795
×
1
0
-
15
3.333
×
1
0
-
16
1.263
×
1
0
-
12
4
G
2
H
+
2
G
H
2
+
2
3
H
3
Stable
Givens QR
7.532
×
1
0
-
16
6.702
×
1
0
-
15
2.711
×
1
0
-
16
1.118
×
1
0
-
12
8
G
2
H
+
2
G
H
2
+
2
3
H
3
StableIdentical to SVD, the first subspaceM
1
=
q
1
r
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 M
tar is
(4)
M
tar
=
∑
g
=
2
G
w
g
q
g
r
g
,
where w
g are weights applied to different subspaces. Fuzzy logic is used for the automatic weight assignment [15].


### 2.1.  Input and Output MFs

Letξ
g
=
∥
r
g
∥ and Δ
ξ
g
=
∥
r
g
∥
-
∥
r
g
+
1
∥ be norms and norm differences, respectively. Note that high value of ξ
g and Δ
ξ
g the corresponding subspace q
g
r
g more likely contains target(s) and is therefore enhanced by applying heavy weights (and vise versa).Three Gaussian membership functions (MFs)ζ
X
x
(
x
1
)
=
exp
⁡
(
-
(
(
c
1
-
c
-
1
(
x
)
)
/
σ
1
(
x
)
)
2
) and (x
∈
{
High
,
Medium
,
Low
}) are defined for ξ
g. Similarly ζ
Y
y
(
x
2
)
=
exp
⁡
(
-
(
(
c
2
-
c
-
2
(
y
)
)
/
σ
2
(
y
)
)
2
) and (y
∈
{
High
,
Medium
,
Low
}) are defined for Δ
ξ
g, where {
c
1
,
c
2
}
∈
[
0,1
], c
-
1
(
x
), c
-
2
(
y
) and σ
1
(
x
), σ
2
(
y
) are means and variances of fuzzy sets, respectively.K-means algorithm [16] is used to adjust the fuzzy parameters. ξ
k and Δ
ξ
h are first clustered into three groups based on respective histograms. The means and variances, respectively, of each group are used as centers c
-
1
(
x
), c
-
2
(
y
) and spreads σ
1
(
x
), σ
2
(
y
) of MFs. Five equally spaced output MFs ζ
Z
z
(
d
)
=
-
(
(
d
-
d
-
(
z
)
)
/
ϱ
(
z
)
)
2
(
z
∈
{
Very
High
,
High
,
Medium
,
Low
,
Very
Low
}), where mean d
-
(
z
) and variance ϱ
(
z
) are used.


### 2.2. Product Inference Engine (PIE)

Gaussian fuzzifier maps the inputξ
g and Δ
ξ
g as
(5)
ζ
X
Y
(
c
1
,
c
2
)
=
exp
⁡
{
-
(
c
1
-
ξ
g
v
1
)
2
}
exp
⁡
{
-
(
c
2
-
Δ
ξ
h
v
2
)
2
}
,
where v
1 and v
2 are parameters used for input noise suppression and are chosen as v
1
=
2
max
⁡
x
σ
1
(
x
) and v
2
=
2
max
⁡
y
3
σ
2
(
y
) [15].Fuzzy IF-THEN rules for image enhancement are the following.Rule 1: IFξ
h is X
High and Δ
ξ
h is Y
High, THEN w
h
PIE is Z
Very
High.
Rule 2: IFξ
h is X
Med and Δ
ξ
h is Y
High, THEN w
h
PIE is Z
High.
Rule 3: IFξ
h is X
High and Δ
ξ
h is Y
Med, THEN w
h
PIE is Z
High.
Rule 4: IFξ
h is X
Med and Δ
ξ
h is Y
Med, THEN w
h
PIE is Z
Med.
Rule 5: IFξ
h is X
High and Δ
ξ
h is Y
Low, THEN w
h
PIE is Z
Med.
Rule 6: IFξ
h is X
Low and Δ
ξ
h is Y
Med, THEN w
h
PIE is Z
Med.
Rule 7: IFξ
h is X
Med and Δ
ξ
h is Y
Low, THEN w
h
PIE is Z
Low.
Rule 8: IFξ
h is X
Low and Δ
ξ
h is Y
High, THEN w
h
PIE is Z
Low.
Rule 9: IFξ
h is X
Low and Δ
ξ
h is Y
Low, THEN w
h
PIE is Z
Very
Low.The output of PIE using individual rule based inference, Mamdani implication, algebraic product fort-norm, and max operator for s-norm [15] is
(6)
ζ
Z
′
(
d
g
)

=

max
⁡
{
x
,
y
,
z
}
⁡

[
sup
⁡
{
c
1
,
c
2
}
ζ
X
Y
(
c
1
,
c
2
)
ζ
X
x
(
c
1
)
ζ
Y
y
(
c
2
)
ζ
Z
z
(
d
g
)
]
.The weightsw
g
PIE are then computed as
(7)
w
g
PIE
=
∑
z
d
-
(
z
)
ϖ
g
(
z
)
∑
z
=
1
ϖ
h
(
z
)
,
where ϖ
g
(
z
) is the height of ζ
Z
′
(
d
g
) in output MFs [15].


### 2.3. 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 w
g
TS is
(8)
IF
ξ
g
is
X
j
1
AND
Δ
ξ
g
is
Y
j
2
THEN
p
(
j
1
+
j
2
-
1
)
=
(
1
1
+
exp
⁡
{
-
ξ
g
}
+
exp
⁡
{
-
Δ
ξ
g
}
)
j
1
+
j
2
-
1
.
Note that the output reduces for large j
1
+
j
2 (which is desirable). The aggregated weights w
g
TS are
(9)
w
g
TS
=
∑
j
1
=
1
3
∑
j
2
=
1
3
p
(
j
1
+
j
2
-
1
)
t
{
ζ
X
j
1
(
ξ
k
)
,
ζ
Y
j
2
(
Δ
ξ
g
)
}
∑
j
1
=
1
3
∑
j
2
=
1
3
t
{
ζ
X
j
1
(
ξ
g
)
,
ζ
Y
j
2
(
Δ
ξ
g
)
}
,
where t represents algebraic product (intersection operator).


## 2.1.  Input and Output MFs

Letξ
g
=
∥
r
g
∥ and Δ
ξ
g
=
∥
r
g
∥
-
∥
r
g
+
1
∥ be norms and norm differences, respectively. Note that high value of ξ
g and Δ
ξ
g the corresponding subspace q
g
r
g more likely contains target(s) and is therefore enhanced by applying heavy weights (and vise versa).Three Gaussian membership functions (MFs)ζ
X
x
(
x
1
)
=
exp
⁡
(
-
(
(
c
1
-
c
-
1
(
x
)
)
/
σ
1
(
x
)
)
2
) and (x
∈
{
High
,
Medium
,
Low
}) are defined for ξ
g. Similarly ζ
Y
y
(
x
2
)
=
exp
⁡
(
-
(
(
c
2
-
c
-
2
(
y
)
)
/
σ
2
(
y
)
)
2
) and (y
∈
{
High
,
Medium
,
Low
}) are defined for Δ
ξ
g, where {
c
1
,
c
2
}
∈
[
0,1
], c
-
1
(
x
), c
-
2
(
y
) and σ
1
(
x
), σ
2
(
y
) are means and variances of fuzzy sets, respectively.K-means algorithm [16] is used to adjust the fuzzy parameters. ξ
k and Δ
ξ
h are first clustered into three groups based on respective histograms. The means and variances, respectively, of each group are used as centers c
-
1
(
x
), c
-
2
(
y
) and spreads σ
1
(
x
), σ
2
(
y
) of MFs. Five equally spaced output MFs ζ
Z
z
(
d
)
=
-
(
(
d
-
d
-
(
z
)
)
/
ϱ
(
z
)
)
2
(
z
∈
{
Very
High
,
High
,
Medium
,
Low
,
Very
Low
}), where mean d
-
(
z
) and variance ϱ
(
z
) are used.


## 2.2. Product Inference Engine (PIE)

Gaussian fuzzifier maps the inputξ
g and Δ
ξ
g as
(5)
ζ
X
Y
(
c
1
,
c
2
)
=
exp
⁡
{
-
(
c
1
-
ξ
g
v
1
)
2
}
exp
⁡
{
-
(
c
2
-
Δ
ξ
h
v
2
)
2
}
,
where v
1 and v
2 are parameters used for input noise suppression and are chosen as v
1
=
2
max
⁡
x
σ
1
(
x
) and v
2
=
2
max
⁡
y
3
σ
2
(
y
) [15].Fuzzy IF-THEN rules for image enhancement are the following.Rule 1: IFξ
h is X
High and Δ
ξ
h is Y
High, THEN w
h
PIE is Z
Very
High.
Rule 2: IFξ
h is X
Med and Δ
ξ
h is Y
High, THEN w
h
PIE is Z
High.
Rule 3: IFξ
h is X
High and Δ
ξ
h is Y
Med, THEN w
h
PIE is Z
High.
Rule 4: IFξ
h is X
Med and Δ
ξ
h is Y
Med, THEN w
h
PIE is Z
Med.
Rule 5: IFξ
h is X
High and Δ
ξ
h is Y
Low, THEN w
h
PIE is Z
Med.
Rule 6: IFξ
h is X
Low and Δ
ξ
h is Y
Med, THEN w
h
PIE is Z
Med.
Rule 7: IFξ
h is X
Med and Δ
ξ
h is Y
Low, THEN w
h
PIE is Z
Low.
Rule 8: IFξ
h is X
Low and Δ
ξ
h is Y
High, THEN w
h
PIE is Z
Low.
Rule 9: IFξ
h is X
Low and Δ
ξ
h is Y
Low, THEN w
h
PIE is Z
Very
Low.The output of PIE using individual rule based inference, Mamdani implication, algebraic product fort-norm, and max operator for s-norm [15] is
(6)
ζ
Z
′
(
d
g
)

=

max
⁡
{
x
,
y
,
z
}
⁡

[
sup
⁡
{
c
1
,
c
2
}
ζ
X
Y
(
c
1
,
c
2
)
ζ
X
x
(
c
1
)
ζ
Y
y
(
c
2
)
ζ
Z
z
(
d
g
)
]
.The weightsw
g
PIE are then computed as
(7)
w
g
PIE
=
∑
z
d
-
(
z
)
ϖ
g
(
z
)
∑
z
=
1
ϖ
h
(
z
)
,
where ϖ
g
(
z
) is the height of ζ
Z
′
(
d
g
) in output MFs [15].


## 2.3. 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 w
g
TS is
(8)
IF
ξ
g
is
X
j
1
AND
Δ
ξ
g
is
Y
j
2
THEN
p
(
j
1
+
j
2
-
1
)
=
(
1
1
+
exp
⁡
{
-
ξ
g
}
+
exp
⁡
{
-
Δ
ξ
g
}
)
j
1
+
j
2
-
1
.
Note that the output reduces for large j
1
+
j
2 (which is desirable). The aggregated weights w
g
TS are
(9)
w
g
TS
=
∑
j
1
=
1
3
∑
j
2
=
1
3
p
(
j
1
+
j
2
-
1
)
t
{
ζ
X
j
1
(
ξ
k
)
,
ζ
Y
j
2
(
Δ
ξ
g
)
}
∑
j
1
=
1
3
∑
j
2
=
1
3
t
{
ζ
X
j
1
(
ξ
g
)
,
ζ
Y
j
2
(
Δ
ξ
g
)
}
,
where t represents algebraic product (intersection operator).


## 3. Simulation and Results

Experimental setup for TWI is constructed using Agilent’s vector network analyzer (VNA) which generates stepped frequency waveforms between2GHz and 3GHz (1GHz band width (BW)) having step size of Δ
f
=
5MHz and step size N
f
=
201. Maximum range is R
max
⁡
=
30
m and range resolution is Δ
R
=
0.15
m.Broadband horn antenna which is mounted on two-dimensional scanning frame (having dimensions2.4
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 enhancement algorithms are simulated in MATLAB and quantitative analysis is performed using MSE, PSNR, IF, FD, MD, and visual inspection:
(10)
MSE
=
1
G
×
H
∑
g
=
1
G
∑
h
=
1
H
(
M
b
s
(
g
,
h
)
-
M
tar
(
g
,
h
)
)
2
,
PSNR
(
dB
)
=
10
log
⁡
10
⁡
1
MSE
,
IF
(
dB
)
=
10
log
⁡
10
⁡
[
P
M
tar
,
t
×
P
M
,
c
P
M
,
t
×
P
M
tar
,
c
]
,
where M
b
s is a reference image obtained by the difference of image, with and without target. P
M
tar
,
t and P
M
tar
,
c are average pixel values of target and clutter in enhanced image, respectively. P
M
,
t and P
M
,
c 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].Figure1 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.Image with two targets. (a) Original image. (b) SVD [11]. (c) Proposed fuzzy QRD (PIE). (d) Proposed fuzzy QRD (TS). (e) Background subtracted reference image.
(a)
(b)
(c)
(d)
(e)Figure2 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.Image with three targets. (a) Original image. (b) SVD [11]. (c) Proposed fuzzy QRD (PIE). (d) Proposed fuzzy QRD (TS). (e) Background subtracted reference image.
(a)
(b)
(c)
(d)
(e)Table2 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.Table 2
MSE, PSNR, IF, MD, and FD comparison.
Scenario
Scheme
MSE
PSNR
IF
MD
FD
Two targets
SVD [11]
0.2726
5.6442
8.1258
1
0
Fuzzy QRD (PIE)
0.1970
7.0553
11.2587
0
0
Fuzzy QRD (TS)
0.1726
7.6296
11.5870
0
0
Three targets
SVD [11]
0.2814
5.5068
7.1265
1
1
Fuzzy QRD (PIE)
0.1933
7.1377
10.8715
0
0
Fuzzy QRD (TS)
0.1824
7.3898
11.0127
0
0


## 4. 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.



---

*Source: 487506-2014-02-23.xml*