Method of Tree Radar Signal Processing Based on Curvelet Transform

Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 7, 243 - 250, 2016

doi:10.21311/001.39.7.30

Method of Tree Radar Signal Processing Based on Curvelet Transform Zhongliang Xiao, Jian Wen*, Lin Gao, Xiayang Xiao, Weilin Li and Can Li School of Technology, Beijing Forestry University, Beijing 100083, China *Corresponding author(E-mail:[email protected]) Abstract Ground penetrating radar has been widely applied on geological disaster surveys, road investigations, bridge health monitoring, and other subsurface surveys. Many studies introduced this method on tree non-destructive detection. However, as for the complicated internal structure of tree trunk, the radargram of tree always have the problems of complicated recognition, low signal noise ratio (SNR) and images hardly to be explained. To get a clear photo and a high signal noise ratio, this paper applied the Curvelet transform to wipe out the noise to extract the characteristics of signal and to improve the signal noise ratio. As a comparative, the wavelet transform is also used to de-noise as well. The SNR of Curvelet transform is large than wavelet transform both in simulated data and real tree radar data. And the image of the one which was processed by Curvelet transform is clearer than processed by wavelet transform. The result shows that the Curvelet transform is suitable for processing the tree radar image with many curve characteristics. And this study provides an important foundation of further radar image processing. Key words: Ground Penetrating Radar, Tree Radar, Curvelet Transform, Wavelet Transform, Signal Noise Tatio. 1. INTRODUCTION With the advancement in networking and multimedia technologies enables the distribution and sharing of Ground penetrating radar (GPR) as an effectively noninvasive and non-destructive technique has been extensively applied in geological disaster surveys (Tang and Sun, 2011), road investigations (Janne, Kari and Herronen, 2012), bridge health monitoring (Alani and Kruk, 2012), concrete detection (Kalogeropoulos and Kruk, 2013) and other subsurface surveys. However, many researchers tentatively applied the GPR to detect the timbers or living trees. The research of Vega Pérez-Gracia used the GPR to analyze the wattle trees, pines and dried timbers to investigate the relative permittivity. The study of Dayakar Devaru and Sandeep Pyakurel applied GPR to pinpoint the exact location of embedded metals in logs and detect the knots and rots of saw mills. In the research of Camilla Colla and R. Martínez-Salasil, GPR has been employed for detailed investigation of a historic timber beam from a roof structure to the study of the physical properties and dielectric anisotropy of the timber. (Colla, 2010; Martínez-Sala and Rodríguez-Abad, 2013) However, as for the complexity of tree trunk, almost of above studies have the problems of complicated recognition, low signal noise ratio and images hardly to be explained. Those problems restricted the application of GPR in the detection of trees. However, Curvelet transform is an effective method which has been widely used in GPR images in many other domains except tree area. S.Cieszczyk presents the application of Curvelet transform to the analysis of GPR images received from synthetic and experimental data of concrete. (Cieszczyk and Lawicki, 2013). Bao Qianzong used the Curvelet transform to de-noise the GPR images of pipes which are buried in clean sand (Qianzong and Qingchun et al, 2014). Andreas Tzanis applied Curvelet transform to extract the characteristics of the GPR data from archaeological, geotechnical and hydrogeological surveys (Tzanis, 2012). Those above studies proved that Curvelet transform is an effective and powerful method to process an analysis of noisy and complex GPR data. Therefore, this paper tries applying Curvelet transform to de-noise and extract the characteristics from tree GPR images. Both the simulated data and real tree B-scan GPR data will be presented in the following sections of this paper. 2. METHOD AND MATERIALS 2.1. GPR Principle GPR emits electromagnetic waves from a transmitter antenna into the objective to be measured. These pulses are reflected from the interfaces between materials with different dielectric constants and received by the receiver antenna. The reflected waves are then recorded, processed, and analyzed to measure the travel time and amplitude of the direct and reflected electromagnetic waves. The analysis of the recorded signals can give physical and geometrical information about the measured materials (Xianghui and Lihai, 2013). 243


2.2. The First Generation of Curvelet Transform The first generation of Curvelet transform is based on ridgelet transform. It is like the limit of the calculus. The curves can be regarded as a straight line in a sufficiently small scale. Then the curves can be approximated with many straight lines. So the Curvelet transform has been called the integral of Ridgelet transform. The basic scale S of the single scale Ridgelet transform is fixed. However, Curvelet transform decomposes in all scale S(S ≥ 0) as much as possible. The Ridgelet transform has a wonderful detection of straight lines. But, almost of all lines in images is bending. Curvelet transform divided an image into blocks, then regarding curves in blocks as straight lines. The first generation of Curvelet transform is a multi-scale ridgelet transform essentially. 2.3.The Continuous Curvelet Transform The continuous Curvelet transform smoothly divided the frequency domain into rings with different angles with a couple window functions. The radial window W(r) and angular window V(t), which must be smooth, nonnegative and real-valued. The support of W is r∈ (1/2, 2) and it must obey the equation (1a); the support of V is t∈(-2π，2π] and the corresponding admissibility condition must obey the equation (1b). 

W

2

(2 j r )  1 r  (3 / 4,3 / 2)

(1a)

(t  l )  1 t  (1/ 2,1/ 2)

(1b)

j 



V

2

t 

For each j≥j0, the window function Uj is defined as j/2 2   U j (r , )  23 j / 4 W (2 j r )V ( ) 2

(2)

The ⌊j/2⌋ is denoting the integer part of j/2. Uj is a kind of wedge window in polar coordinate. It is showed in Fig (1a). The support spacing of Uj , which is limited by the window W(2-jr) and window V(2⌊j/2⌋θ) is a wedge area. The area obeys the parabolic scaling relationship: width≈length2 Evidently, W(2-jr) isolates ω-values in the corona (2j-1, 2j+1).Likewise, V(2⌊j/2⌋θ) isolates ω-values in the angular sectors (-2-⌊j/2⌋π，2-⌊j/2⌋π). Define the mother ^

Curvelet as φj(x) and its Fourier transform as Φj (ω) = Uj (ω). The complete Curvelet family in the ω-domain is generated from Φj,0,0 by dilation, rotation and translation according to: Φ j,l,k (r , θ )  Φ j ,0,0 ( Rθ j ,l ξ )  e

 i xk( j,l ) , ξ

(3)

Where, θl =2π·2⌊-j/2⌋·l (l=0,1,2…,0≤θl≤2π)) is a sequence of equi-spaced rotation angles, X k( j ,l )  Rθl1 (k1  2 j , k2  2 j / 2 )

(4)

The Rθl is acquired by the rotation of the θl.  cos  Rθl     sin 

sin   1  , R  R cos  

(5)

K= (k1, k2), (k1, k2∈Z2) and it represents the translation parameters. The annular which is approximately around the zero has not been covered in the ω-plane, so a complete covering of it must define a low-pass window Wj0 which is supported on the unit circle and is isotropic. This window obeys W j0 (r )   W (2 j r )  1 2

2

(6)

j 0

So that coarsest scale Curvelet will be Φ j0 (ω)  2 j0 W j0 (2 j0 ω )

(7)

The x–domain representation of the form  j0 , k ( x)   j0 ( x  2 j0 k )

(8) 244


Based on the above construction principles, the Curvelet can be defined as: c( j,l,k ) = f , Φ j,l,k   2 f ( x)Φ j,l,k ( x)dx

(9)

R

And his frequency ω-domain form is 1 1 c( j,l,k ) = fˆ (ω)Φˆ j,k,l (ω)dω  2  2  2   2 

 fˆ (ω)U

 j,l 

j

( Rθl ω)e

i xk

,ω

(10)

dω

Figure 1. (A) The schematic diagram of continuous Curvelet transform in frequency domain. (B) The schematic diagram of continuous Curvelet transform in space domain. It is a conclusion that Curvelet is the same as wavelet comprises the coarsest and detail coefficients. The coarsest coefficients are non-directional. As a result of the parabolic scaling, the Curvelet transform is an optimally sparse representation for functions along curves. 2.4. The Discrete Curvelet Transform The discrete curevelet transform smoothly divides the frequency domain into annular rows with different angles in polar coordinate. However, this division is not suitable for the 2-D Cartesian coordinate system. Thus, the inventor of Curvelet transform proposed discrete formulations of Curvelet transform. The discrete Curvelet transform (DCT) replaced the circular coronae with rectangular and replaced the rotations with shearing. The shearing result is showed in Fig 2. In Cartesian coordinate system, the radial window W is W j2  ω  Φ2j 1  ω -Φ2j  ω , j  0

(11)

Where Φ (ω) = φ (2-jω1) ·φ (2-jω2) and φ is a low-pass one-dimensional window that with the admissibility condition 0≤ φ ≤ 1，ω ∈ [-2, 2]. With those definitions and limitations, the radial window obeys the equation (12) in Cartesian coordinate and it can hold for all ω. W

2 j0

 ω   W 2j  ω  1

(12)

j> j0

The angular window V is now taken to be  j/2 ω  V j  ω   V  2  2  ω1  

(13)

  ω j j 1  j/2  2  2 j / 2   ω1, ω2  : 2  ω1  2 , 2 ω  1 

(14)

Defining a set of slops with the same interval, tan θι  l  2

  j / 2 

l  2

  j / 2 

,..., 2

  j / 2 

1

(15)

The window Uj is replaced as 245




U j,l  ω  W j  ωV j Sθl ω



(16)

Where  1 Sθl     tan θl

0 1 

(17)

is a shear matrix.

Figure 2. The schematic diagram of continuous Curvelet transform in frequency domain. So the discrete Curvelet transform is defined as

 



φ j,k,l  x   2-3j / 4 φ j SθTl x  Sθ-Tl b , b   k1  2-j , k2  2 j 

(18)

With above definitions and conditions, the Curvelet transform in Cartesian coordinates is





T

i S c  j,k,l    fˆ  ωU j Sθl1ω e θl

b,ω

dω

(19)

The cut block S-Tθl b is not a standard rectangle, in order to use the fast Fourier algorithm, the discrete transform rewrite as





T

i b, S c  j,k,l    fˆ  ωU j S θT ω e θl l

ω

dω

(20)

At this condition it can use the partial Fourier transform to implement it. 3. EXPERIMENT AND DATA ANALYSIS 3.1. The Curvelet Transform Applied in Simulated Tree GPR data To test the effectiveness of the Curvelet transform, this method was applied to a synthetic model. This synthetic model was generated by Gprmax software and it was shown in Figure 3. It mainly falls into three layers (bark in the top, trunk in the mid, cavity in the bottom). In the trunk section, there is a round knot with the radius of 5 centimeters and a square rot area with the side length of 10 centimeters. All those compositions in this model are with different dielectric constants. Then the forward image of this simulated tree was generated by the Gprmax software and it was presented in Figure 4(A). In the above GPR image, the detected target of circular knot and square rot were presented as parabola with different radians. And the interface between trunk and cavity consists of many crossed curves. To simulate the real tree GPR scan image, the white Gaussian noise is added into it. The image with noise was showed in Figure 4(B). It is found that the parabola of knot and the rot region almost has been submerged by noise. Then this model with noise was dealt with Curvelet transform and the result was compared with the wavelet transform. The wavelet in the toolbox of Matlab, the data was decomposed by wavelet transform, de-noised by the soft threshold method and reconstructed by inverse wavelet transform. The result was showed in Figure 4(C). Little parabola of scammer and decay are extracted from the noise images. As well, the curves of the interface of the trunk and cavity have a characteristic of a little repellent. The original smooth curves performed as zigzag. As same, Curvelet transform which is comprised in Curvelab package (downloaded from the Curvelet official website) was used to shear the GPR data images and 246


the coefficients (the shading in figure 2) which contain the signal information or characteristics selected to reconstruct the signal images. The result is presented in Figure 4(D). The parabola of the knot and rot is extracted clearly from the noise image and the curves of interface are also as smooth as the original forward image. Comparing the effects of the two ways, it is obvious that the effect of Curvelet transform is better than wavelet transform. Because the interfaces between trunk and decay are curve edges, the round knot and curb rot perform as parabolas.

Figure 3. The synthetic model of an abnormal tree with a knot, rot and cavity.

Figure 4. (A) The forward image of this simulated tree without noise. (B) The forward image of this simulated tree model with white Gaussian (C) The forward image of simulated tree de-noised with wavelet transform. (D)The forward image of this simulated tree de-noised with Curvelet transform. Furthermore, to illustrate the effectiveness of Curvelet transform, the digital index of mean square error (MSE), peak signal noise ratio (PSNR) and signal noise ratio (SNR) (Terrasse and Nicolas et.al., 2015)[17] were counted by equation (21), (22), (23). An image, which was de-noised, with little MSE, large PSNR and SNR states a better de-noised effect. The result is showed in table 1. The MSE of wavelet transform is 3.2186 and Curvelet transform is 2.9889. The PSNR of wavelet transform is 74.9234 and the Curvelet transform is 75.2449. The SNR of wavelet transform is 40.2014 and the SNR of Curvelet transform is 40.8441. The result of Curvelet has a less MSE and larger SNR and SNR. It is also explain that Curvelet transform has an advantage in processing the radargram.

   pdata i, j   rdata i, j   i =N j =M

2

MSE =

2

i =1 j =1

M N 2   max  rdata  i, j     PSNR =20  log10   MSE    

(21)

(22)

247

Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 7, 243 - 250, 2016 i=N j  M  rdata 2  i, j    SNR  20  log10    2 2 pdata i, j  rdata i, j       i 1 j 1 

(23)

Table 1. The digital result of simulated model by the two methods. MSE

PSNR

SNR

Wave-T

3.2186

74.9234

40.2014

Curve-T

2.9889

75.2449

40.8441

3.2. The Curvelet Transform Applied to Real Tree GPR Data The Curvelet transform achieved relatively good result in simulated data. To further verify whether the Curvelet transform has effective effect in real tree B-scan data. There are two willow examples that will be presented as follow. Example 1 The first willow has seventy-five percent decays of the entire trunk. The out contour of it is irregular. Its largest diameter is 65 cm and shortest diameter is 45cm. The TRU TM (Tree Radar Unit) System was used around the trunk to collect the GPR data and the data was showed as B-scan GPR image in Figure 5(A). Due to the large area of decays, the image is complex and it has random noise caused by environment and fractures caused by synthetic improper operation. The same as the simulated data, the Curvelet transform was applied to real data and the result compared with the result of wavelet transform. The processed images of two methods are presented in Figure 5(B) and Figure 5(C) respectively. It can be seen from the images that the Curvelet transform has a better curve features preservative than wavelet transform and the fracture has been nearly eliminated in the Curvelet transform processed one.

Figure 5. (A) The raw data image of real tree. (B) The data image of real tree tree de-noised with wavelet transform. (C) The data image of real tree de-noised with Curvelet transform And the MSE, PSNR, and PSNR also have been counted and the result is showed in table 2. The MSE of wavelet transform is 1.6808 and the Curvelet transform is 1.4070. The PSNR of wavelet transform is 89.7862 and the Curvelet transform is 90.5737. The SNR of wavelet transform is 53.1227 and the Curvelet transform is 54.6997. The result of Curvelet transform meets the expectation. Table 2. The digital results of real tree data of two methods. MSE

PSNR

SNR

Wavelet-T

1.6808

89.7862

53.1227

Curvelet-T

1.4070

90.5737

54.6997 248


Example 2 The Curvelet transform was applied to another willow tree with a few decays in trunk. Its largest diameter is 35cm and shortest diameter is 30cm. The GPR data image is displayed in Figure 6(A). Because the tree hasn’t been badly defected, the GPR data image is less complex than the experiment one. The same as the experiment one, the Curvelet transform was used to denoise and extract the curve characteristics from radargram and the wavelet transform was used to process the data image as a comparison. Both processed images were showed in Figure 6(B) and Figure 6(C). And the result informed us that the Curvelet transform has a better processed effect with more curve information preservation and less fracture. It also can be informed from the digital results and it was presented in table 3. It also proves that the Curvelet transform achieved a better processing target with less MSE and larger PSNR and SNR. Thus it may conclude that the image processed by Curvelet transform is better than that by wavelet transform in processing tree GPR data image.

Figure 6. (A) The raw data image of real tree. (B) The data image of real tree de-noised with wavelet transform. (C) The data image of real tree de-noised with Curvelet transform. Table 3. The digital results of real tree data of two methods. MSE

PSNR

SNR

Wavelet-T

0.1285

100.9508

66.9120

Curvelet-T

0.0515

104.9271

74.8668

4. CONCLUSION AND DISSCUSION This article presents the application of Curvelet transform to analysis of GPR images of trees received from forward model and real tree experimental data. Curvelet transform as an effective and powerful method to process an analysis noisy and complex GPR data has been used in many other fields. Such as Economou, N and Tzanis, A (Economou and Kritikakis et al., 2016; Tzanis, 2015). However, the superiority of Curvelet transform is that it has the optimally sparse representation of bivariate functions with singularities on 2-D differentiable curves. Compared with other domains, the application in tree domain may have an advantage. Because the external contours of trees are irregularity rounds. So if there are decays in tree trunk, the edge of decay is a 2-D differentiable curve. Therefore it is possible that the Curvelet transform is more effective in processing the tree GPR data image than the GPR data of other areas. And in this paper the Curvelet transform is applied to processing the complicated tree GPR data images and the wavelet transform is used as a comparison. Wavelet transform is seen as the microscope in the field of image processing for the wonderful processed effect. However, the wavelet transform far exceeds its ability in curve edges with singularities and Curvelet transform is apt at processing curve edges. Both the visual effects and numerical calculations of the processed results of the two methods prove that the Curvelet transform is better than wavelet transform in processing images with abundant curve characteristics.

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So it is can conclude that Curvelet transform is an effective method in processing radargram of tree. However, the shapes of detected target have migrations. So the further study may major in the migration technique to restore the original shape of detected targets in trunk. ACKNOWLEDGEMENT This work was supported by the National Natural Science Foundation of China (Grant No. 31600589) REFERENCES Alani A, Kilic G, Aboutalebi M. (2012) “Applications of Ground Penetrating Radar in Bridge Health Monitoring Using Different Frequency”, Antennae Systems. Egu General Assembly, 14:12179. Bao, Qian Zong, Q. C. Li, and W. C. Chen. (2014) ““GPR data noise attenuation on the Curvelet transform”, Journal of Applied Geophysics, 11(3), pp.301-310. Colla, C, (2010) ““Gpr of a timber structural element”, IEEE International Conf on Ground Penetrating Radar, pp.1-5. Cieszczyk, S., Lawicki, T., & Miaskowski, A. (2013) “The Curvelet transform application to the analysis of data received from gpr technique”, Electronics & Electrical Engineering, 19(6), pp.99-102. Economou, N., Kritikakis, G., Economou, N., & Kritikakis, G. (2016). “Attenuation analysis of real gpr wavelets: the equivalent amplitude spectrum (eas)”, Journal of Applied Geophysics, 126, pp.13-26. Janne Poikajärvi, Kari Peisa, Tomi Herronen, Per Otto Aursand, Pekka Maijala, & Anita Narbro (2012) “GPR in road investigations equipment tests and quality assurance of new asphalt pavement”, Nondestructive Testing & Evaluation, 27(3), pp.293-303. Kalogeropoulos, A., Kruk, J. V. D., Hugenschmidt, J., Bikowski, J., & Brühwiler, E. (2013) “Full-waveform gpr inversion to assess chloride gradients in concrete”, NDT&E International, 57(6), pp.74-84. Martínez-Sala, R., Rodríguez-Abad, I., Barra, R. D., & Capuz-Lladró, R. (2013) “Assessment of the dielectric anisotropy in timber using the nondestructive gpr technique”, Construction & Building Materials, 38, pp.903-911. Tang, X., Sun, T., Tang, Z., Zhou, Z., & Wei, B. (2011) “Geological disaster survey based on Curvelet transform with borehole ground penetrating radar in tonglushan old mine site”, Environmental Sciences, 23(11), pp.78–83. Tzanis, Andreas (2015) “The Curvelet Transform in the analysis of 2-D GPR data: Signal enhancement and extraction of orientation-and-scale-dependent information”, Journal of Applied Geophysics, 115(2C), pp.145-170. Terrasse, G., Nicolas, J. M., Trouvé, E., & Émeline Drouet (2015) “Application of the Curvelet transform for pipe detection in gpr images”, IEEE, IGARSS 2015, pp.4308-4311 Tzanis, A. (2013) “Detection and extraction of orientation-and-scale-dependent information from two-dimensional GPR data with tuneable directional wavelet filters”, Journal of Applied Geophysics, 89, pp.48-67. Xiang-Hui, D. I., & Wang, L. H. (2013) “Study on the feasibility about gpr applied on wood nondestructive testing”, Nondestructive Testing, 35(11), pp.51-54.

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