Optical excitation fractional Fourier transform (FrFT) based enhanced thermal-wave radar imaging (TWRI)

Fei Wang; Yonghui Wang; Junyan Liu; Yang Wang

doi:10.1364/OE.26.021403

1. Introduction

In the past decades, infrared thermography nondestructive testing (IRT-NDT) modality has been gained more attentions in the industrial field, because it has the merits of whole field, non-contact and being applicable to a wide range of materials. Generally, IRT-NDT approach can be divided into passive and active thermography. In active thermography, a modulated external stimulation imposed to the object produces a similar thermal perturbation over the object surface and induces a diffusive thermal-wave nearer to the surface. Thermal-wave propagation generates a surface temperature difference or contrast due to local subsurface inhomogeneous thermal diffusion, which can be captured by infrared camera and further used for subsurface discontinuity or inhomogeneity analysis.

Recently, IRT-NDT has made considerable progress by the introduction of various waveform stimulation formalisms accompanied by suitable post processing methods along with conventional pulse and lock-in approaches. In pulsed thermography [1,2], a short-duration high peak power is imposed on the test objective surface, and the subsequent temporal temperature of the tested surface is captured and processed by an IR imaging system located on opposite to the sample surface. Lock-in thermography (LIT) [3,4] uses of a single frequency, low peak energy power, periodic waveform stimulation and analyzes thermal response either the amplitude or phase based methods. Being less sensitivity to non-uniform heating and surface emissivity variation, the phase image of LIT is often used to detect the subsurface defect instead of amplitude image. Pulsed phase thermography (PPT) makes use of phase analysis with the excitation heat flux similar to PT, and fast Fourier transform (FFT) over thermal profile is applied to separate the phase details in frequency domain [5].

Thermal-wave radar imaging (TWRI) can overcome the problems associated with conventional approaches, likely high peak power used in PT and repetition experimentation with LIT, by combined the linear frequency modulation (LFM) continuous wave radar and frequency-domain photothermal radiometry physics, and TWRI extends the detection depth/depth-resolution dynamic range [6,7]. The first report on LFM of thermal-wave and application of correlation and spectral analysis to photothermal-wave system using frequency chirps were introduced by Mandelis et.al in a series of articles in 1986 [8–10]. Subsequently, Mulaveesala and Tuli et.al also presented a similar approach using frequency modulated thermal-wave imaging (FMTWI) and its digital counterpart (DFMTWI), and the amplitude and phase angle of corresponding frequency component obtained by fast Fourier transform (FFT) were used for the detection of subsurface defects with different depths [11,12]. Furthermore, Tabatabaei and Mandelis (thermal-wave radar) utilized the cross-correlation (CC) technique that is a special match-filter to obtain the instantaneous characteristics of photothermal-wave radar signal in time-domain (i.e. CC signal main peak, CC signal peak delay time, CC amplitude and CC phase), and it was recognized to break through the maximum detection defect depth/depth resolution of limitations of conventional lock-in thermography (LIT) [13,14]. More recently, Melinkov and Mandelis reported on the detection of the presence of subsurface cracks in green automotive parts with a camera-based thermal-wave radar imaging and the specifications [15].

As the application of the match filter, the pulse compression (i.e. linear frequency modulation) played an important role to make the width of CC main lobe narrower and its peak higher (or better signal-noise-ratio SNR) as the total energy must be conserved [13]. Fast Fourier transform (FFT) is one of the most frequently used tools for filtering a desirable frequency component in frequency domain, and it can be employed for depth-resolution anomaly detection of TWRI by using a chirped heat stimulus of a band of suitable frequencies. Both the cross-correlation (CC) and FFT can deal with the thermal-wave radar signal only in the time domain or frequency domain, and additionally, Hilbert transform and chirp based Z transform techniques are also used for the improvement of anomaly detectability and depth-resolution for TWRI and FMTWI [16,17]. However, as subsurface defect depth resolution is related to the modulated probing frequencies of the external stimulation and their analysis through post processing, inadequate frequency resolution will result in missing some subsurface details. Additionally, lacking of accurate theoretical prediction to determine the experimentation time, unknown depth of subsurface anomalies, frame rate of IR camera, and limited frequency resolution provided by the FFT will limit the depth resolution capability of TWRI or FMTWI [16]. Hence, the attempt of novel signal analysis technique is required for the enhancing defect detectability and the improvement of depth resolution in TWRI or FMTWI application.

It is the intention of this paper to completely demonstrate that the fractional Fourier transform (FrFT) based technique can be employed for obviously enhancing the subsurface defect detectability by use of TWRI, and to introduce a dual-directional chirp modulation optically stimulus thermography imaging and FrFT amplitude, FrFT phase, FrFT rotation angle at the peak of FrFT power spectra as the contrast parameter exhibiting higher sensitivity.

2. Theory

2.1 Thermal-wave radar signal

In this case of thermal-wave radar imaging (TWRI), the incident heat flux was modulated by a dual-directional (down then up) chirp signal, and the heat flux was expressed as,

q (t) = {\begin{matrix} \frac{q_{0}}{2} {1 - \cos [2 π (f_{0} + \frac{f_{e} - f_{0}}{2 T_{s}} t) t]} \begin{matrix} 0 \leq t \leq T_{s} \end{matrix} \\ \frac{q_{0}}{2} {1 + \cos [2 π (f_{e} - \frac{f_{e} - f_{0}}{2 T_{s}} (t - T_{s})) (t - T_{s})]} \begin{matrix} T_{s} \leq t \leq 2 T_{s} \end{matrix} \end{matrix}

here, q(t), q₀, f₀, f_e, T_s are the surface heat flux, the maximum heat flux, chirp stating frequency, chirp ending frequency, chirp sweep period, and t is the time.

In TWRI method, the incident heat flux consists of dc and ac components from Eq. (1). As it is desirable to derive the thermal-wave field (i.e. the ac oscillation component of the temperature only), and the temporal mean component of the temperature will be removed. In our previous publications, the thermal-wave signal was derived by subtracting the temporal mean component (the fitted item) from the surface temperature [17]. In this case, the temporal mean component of the temperature is obtained by polynomial fitting of variable time t based on the least-square method, and then, the ac oscillation component of the temperature which is commonly named as thermal-wave signal can be obtained by subtracting the temporal mean component (the fitted item) from the surface temperature.

T_{d c} (t) = \sum_{m = 0}^{M} A_{m} t^{m}

T_{a c} (t) = T_{s u r} (t) - T_{d c} (t)

here, T_dc and T_ac are the temporal mean component and the ac oscillation component of the temperature, T_sur is the surface temperature of sample, and m, A_m and M represent the polynomial indicating subscript, polynomial coefficient and polynomial order number, respectively.

The heat flux in TWRI flows across the solid with a finite thickness, a part of it will be reflected to the front surface from the rear surface, and the other part will be transmitted through the back surface. The chirp modulated heat flux given to the object surface induces similar thermal-waves inside the test object. These propagating thermal waves reflect back from the boundary of the anomaly and contribute for a temperature rise over the object surface. Figure 1 shows the thermal-wave diffusion process in a solid with a finite thickness L.

Fig. 1 One-dimensional thermal-wave diffusion diagram.

Download Full Size | PDF

Under the conductive and radiative heat transfer boundary condition, the surface temperature dependence of time will be obtained by solving the one dimensional heat equation in a rectangular coordinate system (seen in Fig. 1) given by

\frac{\partial^{2} T (z, t)}{\partial z^{2}} = \frac{ρ c}{k_{z}} \frac{\partial T (z, t)}{\partial t}

here, T(z, t) is the instantaneous temperature at time t corresponding to the depth z, and k_z, ρ, c represent the thermal conductivity, the density and specific heat, respectively. From Fig. 1, based on the one dimensional heat equation Eq. (3), the surface temperature in different regions dependence of time will be solved by means of the inverse Fourier transform and Green function method, and given by [17],

T_{d} (0, t) = \int_{- \infty}^{\infty} \frac{Q (ω)}{k σ - h} \frac{r e^{2 σ H} + 1}{r^{2} e^{2 σ H} - 1} e^{j ω t} d t + T_{0}

T_{h} (0, t) = \int_{- \infty}^{\infty} \frac{Q (ω)}{k σ - h} \frac{r e^{2 σ L} + 1}{r^{2} e^{2 σ L} - 1} e^{j ω t} d t + T_{0}

with,

r = \frac{k σ + h}{k σ - h}

σ = \frac{(1 + j)}{Λ}; Λ = \sqrt{\frac{2 k}{ρ c ω}} = \sqrt{\frac{k}{ρ c π f (t)}}

here, T_d (0, t), T_h(0, t) represent the surface temperature in the defective region and healthy region as shown in Fig. 1, σ is the complex wave number, Q(ω) is the Fourier transform of heat flux q(t), h is the heat transfer coefficient, Λ is called the thermal diffusion length, and f(t) is the modulated frequency as function of time, and here it can be obtained from Eq. (1), and T₀ is the surrounding temperature.

Assuming the carbon fiber reinforced polymer (CFRP) laminate material is given, the thermal conductivity along thickness k_z was set to 0.6 W·m⁻¹·K⁻¹, the density of CFRP ρ was equal to 1500 kg·m⁻³, the specific heat of CFRP c was selected as 1200 J·kg⁻¹·K⁻¹, and the CFRP laminate thickness L was set to 5mm, an artificial flat bottom hole as the subsurface defect and the depth H was equal to 1mm. Under the condition of q₀ = 20 W/m², h = 10 W·m⁻²·K⁻¹ and T₀ = 26°C, the surface temperatures in the defective and healthy regions were approximately calculated by using Eqs. (4a) and (4b) at chirp starting frequency f₀ = 0.2Hz, ending frequency f_e = 0.02Hz, chirp period T_s = 50s. The surface temperature and the corresponding thermal-wave signal are presented in Fig. 2.

Fig. 2 The calculated results, (a) surface temperature, and (b) thermal-wave signal.

Download Full Size | PDF

From Fig. 2, it can be found that the thermal-wave signal is similar to the heat flux formalism. As the depth of defective region is less than the healthy region, and the magnitude of thermal-wave signal in defective region is larger than the one in healthy region.

2.2 Fractional Fourier transform of thermal-wave radar signal

In the most recent years, the fractional Fourier transform (FrFT) has attracted attention in many areas such as optics, quantum mechanics, pattern recognition, time-frequency representation and signal processing [18–20], and it can be interpreted as a rotation of signals in the continuous time-frequency plane and serves as an orthonormal signal representation for chirp signals [21]. Besides being a generalization of Fourier transform, the FrFT has been proved to relate to other time-varying signal analysis tools (i.e. Wigner-distribution, short-time Fourier transform, STFT, and wavelet-transform, WT). The FrFT is a special case of time-frequency analysis filter technique that, for a thermal-wave radar signal can be defined as [22],

\begin{array}{l} F_{α} [u, s T W R (t)] = A_{α} \exp [j π (\cot α - \csc α) u^{2}] \\ \begin{matrix} \times \end{matrix} \int_{- \infty}^{\infty} \exp [j π \csc α {(u - t)}^{2}] \exp [j π (\cot α - \csc α) t^{2}] s T W R (t) d t \end{array}

A_{α} = \frac{\exp [- j π sgn (\sin α) + \frac{j α}{2}]}{\sqrt{| \sin α |}} \begin{matrix} α \equiv \frac{p π}{2} \end{matrix}, 0 < | p | < 2.

Where, sTWR(t), F_α[u, sTWR(t)] represent the thermal-wave radar signal and the fractional Fourier transform, α, p and u are co-called the fractional ration angle, the fractional order and FrFT domain, respectively. In order to implant the FrFT of thermal-wave radar signal, the sampling signal in time domain representation is reconstructed by dimensionless normalization scaling (DNS) method [22]. Firstly, the thermal-wave radar signal in time-domain representation is approximately assumed to confine to the interval [-Δt/2, Δt/2] and its frequency-domain representation interval [-Δf/2, Δf/2]. From the above assumption, it indicates a sufficiently large percentage of the thermal-wave radar signal energy is confined to these intervals. Furthermore, for a given class of functions, this can also be ensured by choosing Δt and Δf sufficiently large. The time-bandwidth product is crudely defined as N≡Δt × Δf, and it is always greater than unity because of Heisenberg-Gabor uncertainty relation (H-G inequality). A scaling factor S with the dimension of time and the scaled coordinates x = t × S⁻¹ and v = f × S are introduced, and the time and frequency domain representations will be confined to intervals of length Δt × S⁻¹ and Δf × S under these new coordinates. Choosing the scaling factor S = (Δt/Δf)^1/2, the lengths of both intervals are now equal to the dimensionless quantity (Δt × Δf)^1/2 and it can be presented by Δx. In the newly defined coordinates, the thermal-wave radar signal will be constructed in both domains with N = Δx² samples spaced Δx⁻¹ = N^-1/2 apart. The expression of integration calculation in Eq. (5a) can be thought as a convolution computation between the chirp modulation of thermal-wave radar signal and another chirp signal, and the schematic diagram of discrete FrFT algorithm is shown in Fig. 3.

Fig. 3 Thermal-wave radar signal FrFT processing block diagram.

Download Full Size | PDF

The FrFT modulus or power spectra distribution of the calculated thermal-wave radar signal in Fig. 2(b) as function of the fractional ration angle α or the fraction order p and FrFT domain u using the FrFT processing algorithms shown in Fig. 3 is demonstrated in Fig. 4, here, the fraction order p was the range from 0 to 1. From Fig. 4, the FrFT power spectra exists a peak value, and this implies that the FrFT of TWR signal expresses a centralized feature of energy spectral density in FrFT domain. The FrFT at the fraction order p = 0, is the thermal-wave radar signal in time domain, for the fractional order p = 1, the FrFT becomes an ordinary Fourier transform. Furthermore, the FrFT modulus peak values of TWR signals in defective and healthy region are different and located on the fractional order p = 0.9499 and p = 0.9323, respectively. Hence, the fractional order of FrFT modulus peak value is related to the subsurface defect depth.

Fig. 4 The FrFT modulus distribution of calculated TWR signal, (a) TWR signal of defective region, and (b) TWR signal of healthy region.

Download Full Size | PDF

3. Experimental-setup and thermal-wave radar imaging

The experimental setup is presented in Fig. 5(a), it consists of a mid-infrared camera (FLIR SC7000, 320 × 256 pixel active elements, spectral range of 3.6–5.1μm, and frame rate 100 Hz for full window size), two same continuous-wave fiber-coupled 50-W/808 nm NIR diode lasers (JENOTIK, Germany) and a signal generation/acquisition device (National instrument, NI-6229). The laser beam is spread and homogenized by engineering microlens forming a square illumination area with uniform intensities. The laser power can be modulated either sinusoidally at a fixed frequency (Lock-in thermography, LIT) or a chirp waveform (TWRI) to induce the thermal waves inside the sample. The laser power intensity was selected as 8W/cm² in the following experiments. The data acquisition /signal processing program is designed in Labview environment to determine the amplitude and phase images in the case of LIT, and also to carry out the cross-correlated match filter to determine the CC signal peak, CC delay time, CC amplitude and CC phase images in the case of TWRI. In the case of FrFT based enhanced TWRI, a post signal program is performed in the Matlab environment to implant the discrete FrFT algorithm based on Eqs. (5a) and (5b), the thermal-wave radar signal is firstly reconstructed by DNS method, and subsequently, the discrete FrFT of reconstructed thermal-wave radar signal is calculated by discrete FrFT given in Fig. 3.

Fig. 5 (a) Experimental setup and (b) CFRP sheet specimen with artificial FBHs.

Download Full Size | PDF

In order to experimentally study on the capabilities of FrFT based enhanced TWRI method, a carbon fiber reinforced polymer (CFRP) sheet specimen with artificial flat bottomed holes (FBHs) was designed and prepared as shown in Fig. 5(b), and it was 135mm long, 100mm wide and 10mm thick. The defect depth refers to the distance between the sound face and the top surface of defect, and FBHs consist of 4.0, 6.0, 10.0 and 12.0mm diameters at defect depths of 0.5, 1.0, and 2.0mm.

Figure 6 shows the thermal behavior responses of a carbon fiber reinforced polymer (CFRP) specimen in the selective defective region and healthy region (seen in Fig. 5) by a dual-directional (down then up) chirp (or linear frequency modulation, LFM) modulated heat stimulus, and here, the selective defect depth is equal to 1 mm.

Fig. 6 The responses of dual chirp modulated heat stimulus, chirp parameters: 0.2Hz-0.02Hz-0.2 Hz, frequency sweep rate of 0.0018 Hz/s, (a) surface temperature, and (b) thermal-wave radar signals.

Download Full Size | PDF

From Fig. 6, the surface temperatures and the corresponding thermal-wave radar signals of defective region and healthy region are different and consistent with the theoretical computational results given in Fig. 2, and the thermal-wave radar signal belongs to a typical class of non-stationary and time-frequency vary signal similar to the chirp reference/modulation signal.

The FrFT modulus or power spectra distribution of thermal-wave radar signal in Fig. 6(b) as function of the fraction order p (varied from 0 to 1) and FrFT domain u is shown in Fig. 7(a).

Fig. 7 (a) The FrFT modulus distribution of TWR signal in the healthy region with the fractional order range of 0<p<1, (b) the FrFT modulus peak comparisons at the fractional order p = 0.9323, and (c) the FFT modulus of TWR signal. Chirp parameters: 0.2Hz-0.02Hz-0.2Hz, frequency sweep rate of 0.0018Hz/s.

Download Full Size | PDF

From Fig. 7(a), the FrFT power spectra exists a peak value at fractional order p = 0.9323. Figure 7(b) presents the FrFT modulus comparisons of TWR signal between defective region and healthy one (p = 0.9323). It can be found that the FrFT modulus peak value of defective region is much higher than the healthy one. Figure 7(c) shows the FFT modulus of TWR signal in defective region and healthy one (p = 1.0), respectively, and there exist obvious differences in the range of chirp modulated frequencies. Additionally, the peak value in FrFT domain is increased compare to the FFT, and it indicates the FrFT modulus peak can improve the signal-noise-ratio (SNR) and enhance the detectability of much deeper subsurface defects compare to LIT based on FFT processing.

Therefore, the main peak value (defined as Am_peak) that is the maximum value of the FrFT modulus and the main peak value centered on the corresponding fractional rotation angle α or order p and the FrFT domain u (defined as the p_peak and u_peak) are given by,

A m_{p e a k} = \max (| F_{α} (u) |)

(p = p_{p e a k}, u = u_{p e a k}) \Rightarrow {\begin{matrix} \frac{\partial | F_{α} (u) |}{\partial p} =0 \\ \frac{\partial | F_{α} (u) |}{\partial u} =0 \end{matrix}

here, Am_peak means the peak value of FrFT modulus, and p_peak, u_peak present the corresponding fraction order and FrFT domain to the FrFT modulus peak value.

Similar to CC of TWRI or LIT, the amplitude of FrFT peak strongly depends on the amplitude of TWR signal, its location on the fractional rotation angle is linked to the depth of signal source. Hence, One can further develop the corresponding concept likely CC or LIT by calculating the FrFT phase, (defined as Ph_peak), and it is given by,

P h_{p e a k} = a \tan {\frac{Im [F_{α} (u)] |_{| F_{α} (u) | = \max (| F_{α} (u) |)}}{Re [F_{α} (u)] |_{| F_{α} (u) | = \max (| F_{α} (u) |)}}}

here, Re and Im are the real and imaginary part of FrFT, respectively.

Similarly, the FrFT modulus peak value of thermal-wave radar signal, the corresponding fractional order p, FrFT domain u and the FrFT peak phase at each pixel can be calculated, and then the characteristic images are formed by assembling each pixel.

4. Results and discussion

Figure 8 shows the amplitude and phase images obtained from FrFT peak and FFT at the component frequency f = 0.046Hz corresponding to the maximum FFT amplitude shown in Fig. 7(c). It is readily to find from Figs. 8(a) and 8(b), that only the #12 defect is invisible in both FrFT peak amplitude and phase images. However, all the defects of depth 2.0mm are invisible in FFT amplitude image from Fig. 8(c) and all the defects of diameter 4.0mm are not clear in FFT phase image from Fig. 8(d). It can be seen that FrFT based imaging not only resolves the defect depths but also shows more defects clearly compare to FFT based. These advantages are contributed to the fact that FrFT process (through time-frequency plane rotation) puts most signal energy under its narrow peak (increasing depth resolution) while significantly improving the SNR (maximizing detection depth).

Fig. 8 The amplitude and phase images from FrFT and FFT based, (a) FrFT peak amplitude (b) FrFT peak phase, (c) FFT amplitude, and (d) FFT phase, chirp parameters: 0.2Hz-0.02Hz-0.2Hz, frequency sweep rate of 0.0018Hz/s.

Download Full Size | PDF

Figure 9 illustrates the amplitude and phase horizontal profiles of both FrFT and FFT based processing by taking the profiles along the line passing through the centers of defects as seen all given lines in Fig. 8(a). From Fig. 9, it can be seen that the amplitude and phase contrasts of FrFT and FFT based processing between defective regions and healthy ones are decreased with the increase of defect depth, as the fact that FrFT or FFT is still a typical lineal transformation, and the thermal wave diffusion will lead to the amplitude decayed and the phase delay increased with increasing the defect depth.

Fig. 9 Profiles along the lines passing through the centers of defects, (a) FrFT amplitude, (b) FrFT phase, (c) FFT-amplitude and (d) FFT-phase.

Download Full Size | PDF

Differential normalized distribution method [23] from our previous work used to estimate the sizes of the artificial defects (here, #2, #5, #8 and #11 FBHs) in the CFRP specimen, as listed in Table 1, indicates that the FrFT based processing (integrated FrFT-amplitude and FrFT phase) is providing the closest size of the subsurface defect than the contemporary FFT based approach.

Table 1. Estimated the size of defect using differential normalized distribution method

View Table

In order to compare the defect detectability of CFRP sheet in amplitude and phase images from FrFT and FFT based, SNR of defect was defined as [17],

S N R = \frac{| m e a n [I {(x, y)}_{x, y \in R_{D}}] - m e a n [I {(x, y)}_{x, y \in R_{H}}] |}{σ (R_{H})}

with,

σ (R_{H}) = s t d [I {(x, y)}_{x, y \in R_{H}}]

where, I(x,y) denotes the characteristic value of TWR signal at each pixel, R_D, R_H present the defective area and healthy region, and σ(R_H) stands for the standard deviation of healthy region.

In this case, four defects (#4, #2, #8 and #6) were used for the SNRs calculation and comparison study, for #4 defect (FBH diameter D = 10mm, depth H = 0.5mm, aspect ratio r = D/H = 20) and #2 defect (FBH diameter D = 12mm, depth H = 1.0mm, aspect ratio r = D/H = 12) defects, the H₁ was used as the healthy region, and for #8 defect (FBH diameter D = 6mm, depth H = 1.0mm, aspect ratio r = D/H = 6) and #6 defect (FBH diameter D = 10mm, depth H = 2.0mm, aspect ratio r = D/H = 5) defects, the H₂ was selected as healthy area. The defective area used includes 15 × 15 pixels around the center of FBH, and the healthy region selected consists of 20 × 20 pixels. Figure 10 illustrates the measured SNRs of defects selected in both amplitude and phase images from FrFT and FFT based, Fig. (8). From Fig. 10(a), the FFT amplitude has higher SNR for the shallow defect or the larger aspect ratio defect (depth H = 0.5mm, aspect ratio r = 20) than FrFT peak amplitude, and for other defects (depth H = 1.0mm, 2.0mm, aspect ratio r = 10, 6, and 5), FrFT peak amplitude expresses much higher SNRs than FFT based amplitude. In contrast, from Fig. 10(b), the FFT phase has higher SNR for the deep defect or the smaller aspect ratio defect (depth H = 2.0mm, aspect ratio r = 5) than FrFT peak phase, but for other defects (depth H = 0.5mm, 1.0mm, aspect ratio r = 20, 10, and 6), FrFT peak phase has much higher SNRs than FFT based phase. Nevertheless, in terms of SNR, the FrFT peak amplitude channel is significantly stronger than the FrFT peak phase channel for the deep defect or the smaller aspect ratio defect (depth H = 2.0mm, aspect ratio r = 5) and therefore the FrFT peak amplitude image should always be used to complement the information obtained from the FrFT peak phase image.

Fig. 10 The SNRs comparisons, (a) the SNRs of FrFT and FFT based amplitudes, and (b) the SNRs of FrFT and FFT based phases.

Download Full Size | PDF

Figure 11 shows the results involving the FrFT peak rotation angle or the fractional order of FrFT modulus peak, and it presents the FrFT modulus peak rotation angle image and horizontal profiles, respectively.

Fig. 11 (a) FrFT peak ration angle (α = p_peakπ/2) image, and (b) the horizontal profiles of FrFT peak ration angle.

Download Full Size | PDF

It can be observed that the FrFT modulus peak rotation angle image is much less sensitive to the non-uniformity of absorption or emissivity, and its value is linked to the true defect depths as they maintain the same value over the FBHs of diameter D = 10mm and 12mm.

It should be note that the FrFT of thermal-wave response at surface is a complex quantity from Eqs. (5a) and (5b), here, the real part of FrFT as function of the fractional rotation angle α at the peak modulus in the FrFT domain is given by,

A m_{R} (α) = r e a l {[F_{α} (u)]}_{| F_{α} (u) | = \max | F_{α} (u) |}

A m_{I} (α) = i m a g {[F_{α} (u)]}_{| F_{α} (u) | = \max | F_{α} (u) |}

here, Am_R(α), Am_I(α) represent the real and the imaginary part of FrFT for a given fractional rotation angle, respectively.

As the fractional rotation angle is linked to the depth of signal source, in the FrFT based analysis, a defect will appear at a fractional rotation angle or fraction order corresponding to its depth.

Figure 12 explores this fact and exhibits the depth scanning capability using the real part, the imaginary part and phase unscrambling provided by FrFT based processing corresponding to the CFRP specimen. Defects at the same depth are appearing in a real part of FrFT or the FrFT phasegram at a corresponding fraction order. However, in this case, for the given fraction order, the real part of FrFT has a relative high defect detectability compare to the imaginary part of FrFT seen in Fig. 12(a) and 12(b). Shallower defects are appearing at a relatively lower fractional order than their deeper counterparts.

Fig. 12 Defect depth scan using FrFT based analysis, (a) the real part of FrFT, (b) the imaginary part of FrFT and (c) FrFT phase.

Download Full Size | PDF

5. Conclusions

Fractional Fourier transform based time-frequency plane rotation methodology has been introduced to deal with the chirp class thermal-wave signal. The FrFT aggregation properties of power spectra enhanced the defect detectability and subsequent depth resolution capability for thermal-wave radar imaging is experimentally validated. The FrFT of TWR signal expresses the centralized feature of energy spectral in FrFT domain, and the FrFT peak amplitude, FrFT peak phase and FrFT peak rotation angle images are formed and used for the detection of subsurface defects. Defect detectability is quantified in terms of signal-noise-ratio and verified that FrFT based processing methodology visualized subsurface features in details with high depth resolution. Experiment using CFRP specimen with dual directional chirp modulated stimulation and subsequent post processing using FrFT method provided the evidence that FrFT based has the merit of the enhancement of defect detectability and the improvement of accuracy in defect size estimation compare to FFT based or LIT. It is concluded that enhanced depth resolution for varied fractional orders in FrFT is providing better subsurface analysis in thermal-wave radar imaging.

Funding

Foundation for Innovative Research Groups of the National Nature Science Foundation of China under Grant No.51521003; the National Natural Science Foundation of China under Contract No.61571153, No.51173034; Self-planned Task of State Key Laboratory of Robotics and System (HIT) and the Programme of Introducing Talents of Discipline of Universities (grant No.B07108).

References and links

1. N. P. Avdelidis, B. C. Hawtin, and D. P. Almond, “Transient thermography in the assessment of defects of aircraft composites,” NDT Int. 36(6), 433–439 (2003). [CrossRef]

2. X. P. V. Maldague, Theory and Practice of Infrared Thermography for Nondestructive Testing. (NewYork: John Wiley&Sons, Inc., 2001), P368.

3. G. Busse, D. Wu, and W. Karpen, “Thermal-wave imaging with phase sensitive modulated thermography,” J. Appl. Phys. 71(8), 3962–3965 (1992). [CrossRef]

4. J. Y. Liu, Y. Wang, and J. M. Dai, “Research on thermal wave processing of lock-in thermography based on analyzing image sequences for NDT,” Infrared Phys. Technol. 53(5), 348–357 (2010). [CrossRef]

5. X. P. V. Maldague and S. Marinetti, “Pulsed phase infrared thermography,” J. Appl. Phys. 79(5), 2694–2698 (1996). [CrossRef]

6. N. Tabatabaei and A. Mandelis, “Thermal-wave radar: a novel subsurface imaging modality with extended depth-resolution dynamic range,” Rev. Sci. Instrum. 80(3), 034902 (2009). [CrossRef] [PubMed]

7. R. V. Hernandez, A. Melnikov, A. Mandelis, K. Sivagurunathan, and M. E. R. Garcia, “Non-destructive measurements of large case depths in hardened steels using the thermal-wave radar,” NDT Int. 45(1), 16–21 (2012). [CrossRef]

8. A. Mandelis, “Frequency modulated (FM) time delay photoacoustic and photothermal wave spectroscopies. Technique, Instrumentation, and detection. Part I: Theoretical,” Rev. Sci. Instrum. 57(4), 617–621 (1986). [CrossRef]

9. A. Mandelis, L. M. L. Borm, and J. Tiessinga, “Frequency modulated (FM) time delay photoacoustic and photothermal wave spectroscopies. Technique, Instrumentation, and detection. Part II: Mirage effect spectrometer design and performance,” Rev. Sci. Instrum. 57(4), 622–629 (1986). [CrossRef]

10. A. Mandelis, L. M. L. Borm, and J. Tiessinga, “Frequency modulated (FM) time delay photoacoustic and photothermal wave spectroscopies. Technique, Instrumentation, and detection. Part III: Mirage effect spectrometer, dynamic range, and comparison to pseudo-random-binary-sequence (PRBS) method,” Rev. Sci. Instrum. 57(4), 630–635 (1986). [CrossRef]

11. R. Mulaveesala and S. Tuli, “Theory of frequency modulated thermal wave imaging for nondestructive subsurface defect detection,” Appl. Phys. Lett. 89(19), 191913 (2006). [CrossRef]

12. R. Mulaveesala and S. Venkata Ghali, “Coded excitation for infrared non-destructive testing of carbon fiber reinforced plastics,” Rev. Sci. Instrum. 82(5), 054902 (2011). [CrossRef] [PubMed]

13. N. Tabatabaei, A. Mandelis, and B. T. Amaechi, “Thermophotonic radar imaging: An emissivity-normalized modality with advantages over phase lock-in thermography,” Appl. Phys. Lett. 98(16), 163706 (2011). [CrossRef]

14. N. Tabatabaei and A. Mandelis, “Thermal Coherence Tomography Using Match Filter Binary Phase Coded Diffusion Waves,” Phys. Rev. Lett. 107(16), 165901 (2011). [CrossRef] [PubMed]

15. A. Melnikov, K. Sivagurunathan, X. Guo, J. Tolev, A. Mandelis, K. Ly, and R. Lawcock, “Non-destructive thermal-wave-radar imaging of manufactured green powder metallurgy compact flaws (cracks),” NDT Int. 86, 140–152 (2017). [CrossRef]

16. B. Suresh, S. Subhani, A. Vijayalakshmi, V. H. Vardhan, and V. S. Ghali, “Chirp Z transform based enhanced frequency resolution for depth resolvable non stationary thermal wave imaging,” Rev. Sci. Instrum. 88(1), 014901 (2017). [CrossRef] [PubMed]

17. J. Gong, J. Liu, L. Qin, and Y. Wang, “Investigation of Carbon fiber reinforced polymer (CFRP) sheet with subsurface defects inspection using Thermal-wave radar imaging (TWRI) based on the Multi-Transform technique,” NDT Int. 62, 130–136 (2014). [CrossRef]

18. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21(12), 842–844 (1996). [CrossRef] [PubMed]

19. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

20. W. X. Cong, N. X. Chen, and B. Y. Gu, “Recursive algorithm for phase retrieval in the fractional Fourier transform domain,” Appl. Opt. 37(29), 6906–6910 (1998). [CrossRef] [PubMed]

21. L. B. Almeida, “Fractional Fourier Transform and Time-Frequency Representation,” IEEE Trans. Signal Process. 42(11), 3084–3091 (1994). [CrossRef]

22. H. M. Ozaktas, O. Ankan, M. A. Kutay, and G. Bozdaki, “Digital Computation of Fractional Fourier Transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996). [CrossRef]

23. J. Y. Liu, Q. J. Tang, X. Liu, and Y. Wang, “Research on the quantitative analysis of subsurface defects for non-destructive testing by lock-in thermography,” NDT Int. 45(1), 104–110 (2012). [CrossRef]

Defect	Actual defect size /mm (depth H = 1.0mm)	Estimated size /mm
Defect	Actual defect size /mm (depth H = 1.0mm)	FrFT-amplitude	FrFT-phase	FFT-amplitude	FFT-phase
#2	12	12.05	11.69	12.36	17.25
#5	10	9.38	11.28	11.14	14.40
#8	6	6.34	6.59	5.07	7.53
#11	4	4.30	3.68	4.26	—

Optical excitation fractional Fourier transform (FrFT) based enhanced thermal-wave radar imaging (TWRI)

Abstract

1. Introduction

2. Theory

2.1 Thermal-wave radar signal

2.2 Fractional Fourier transform of thermal-wave radar signal

3. Experimental-setup and thermal-wave radar imaging

4. Results and discussion

5. Conclusions

Funding

References and links

Cited By

Figures (12)

Tables (1)

Equations (17)

Optics Express