Temperature Distribution Measurement Based on ML-EM Method Using Enclosed Acoustic CT System

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Sensors & Transducers 2013 by IFSA http://www.sensorsportal.

1 Sensors & Transducers 2013 by IFSA Temperature Distribution Measurement Based on ML-EM Method Using Enclosed Acoustic CT System Shinji Ohyama, Masato Mukouyama Graduate School of Science and Engineering, Tokyo Institute of Technology , O-Okayama, Meguro-Ku, Tokyo, Japan Tel.: Received: 23 January 2013 /Accepted: 19 March 2013 /Published: 29 March 2013 Abstract: In this paper, a measurement method for cross-sectional temperature distributions is considered. A novel method based on an acoustic computed tomography (CT) technique is proposed. Specifically, the temperature distributions are estimated using the time of flight (TOF) of multidirectional ultrasonic propagation paths. Maximum Likelihood Expectation Maximization (ML-EM) method and the Median Filter are introduced to reconstruct the temperature distributions in a square area properly. The effectiveness of the proposed reconstruction method is confirmed experimentally. Copyright 2013 IFSA. Keywords: Acoustic tomography, ML-EM, Temperature distribution measurement. 1. Introduction The computed tomography (CT) technique is a well-known means of measuring the density distribution in a plain. X-ray CT, for example, can be applied to estimate cross-sections of human bodies in a medical field. This CT technique can be expanded to many other fields. Among them, ultrasonic CT system is studied and applied to temperature distribution measurement. In the ultrasonic CT system, the projection data are obtained from the time of flight (TOF) between transmitters and receivers. It is based on the fact that TOF is depending on the temperature distribution on the propagation path. In this report, the method to improve accuracy in the practical application of ultrasonic CT system is discussed. In the practical application case, the measurement plain is square and TOF measurement between not all of the ultrasonic transducers is possible because of its directivities; therefore, conventional reconstruction technique; Filtered back projection that requires complete projection can t reconstruct properly in the corner of the measurement plain. For this reason new reconstruction technique has been newly introduced, and effectiveness of the proposed method has been confirmed. First, a principle of the ultrasonic CT technique, especially about a separation technique of temperature and wind velocity effects from measured time of flight data and CT reconstruction method, is explained. Next, by numerical experiments, measurement precision of the temperature distribution will be confirmed. Finally, by using fabricated experimental system, measurement of the temperature distribution is executed, and performance of proposed method will be evaluated. Article number P_

2 2. Principle of Ultrasonic CT 2.1. Ultrasonic CT System and its Projection Data An arrangement of multiple ultrasonic transducers is shown in Fig. 1. Transducers are arranged on a circumference of the square measurement plain. In the ultrasonic CT system, TOF between all pairs of transmitters and receivers are measured. Equation (1) expresses mathematically-modeled TOF t mn between transmitter m and receiver n as shown in Fig. 1 based on straight line propagation path model shown in Fig. 2. (1) determined by absolute temperature T, specific heat ratio, gas constant R and molecular weight of air M as expressed in eq. (2). Therefore, distribution of c T corresponds to temperature distribution. (2) Equation (1) shows that TOF include not only components of temperature c T but also components of wind velocity v x, v y therefore, to reconstruct only temperature distribution, the components of temperature c T should be separated from TOF by using TOF of both-way t mn, t nm and eq. (4) expressed by r-s coordinate as shown in Fig. 1. In this research, line integral in outside of the measurement plain is assumed to be zero. (3) (4) The components of temperature c T separated from TOF expressed in eq. (4) are regarded as the projection data of ultrasonic CT. In order to obtain temperature distribution, reconstruction technique is applied for these projection data. Fig. 1. Configuration of Ultrasonic CT Reconstruction Technique Concerning the CT technique, many researches have been investigated in terms of image reconstructions from projection data. Filtered Back projection (FBP) is used most extensively for image reconstruction, which is based on the relations shown in Fig. 3. Fig. 3 shows relations among scalar distribution (x, y) corresponding to the 1/ c T for ultrasonic CT, 2-Dimensional Fourier Transform of f(x, y), and projection data p(r, ). Fig. 2. Straight line propagation path model. In this propagation model, the line between a transmitter and a receiver is assumed as a propagation path and only tangential component of the wind vector to the propagation path is assumed as effects of the wind velocity. It is based on following assumption; c T >> v x, v y. Assuming that c T represents a sound velocity without effects of wind velocity v x, that is Fig. 3. Relationship among objective distribution, projection data and k -space on projection slice theorem. 52

3 Equation (5) shows the relation based on the inverse-fourier transformation and eq. (6) shows the relation based on the projection slice theorem. (5) (6) incomplete projection; Maximum Likelihood Expectation Maximization (ML-EM), introduced by L. A. Shepp and Y. Vardi [3]. Unlike Algebraic Reconstruction Technique that can be also applied to incomplete projection, this method does not require large matrix operation therefore it saves computational cost. Equation (10) shows projection data discretized spatially based on concept of ML-EM. (10) From these equations, eq. (7) can be obtained, it shows that f(x, y) is reconstructed by the inverse-fourier transformation of a product of and a Fourier transformation of p(r, ). (7) Therefore, the objective distribution (x, y) is calculated from eq. (8) in a spatial-domain. (8) where q(r, ) means convolution between h(r) (which is spatial-domain expression of ) and p(r, ), that is given by the following equation., (9) As expressed in eq. (8), reconstruction based on FBP requires complete projection (which means that a pixel is configured in the measurement plain is covered with the propagation paths from all directions). However, in the ultrasonic CT system complete projection cannot be obtained all over the area because transducers are fixed on the circumference of measurement plain and actual transducers have directivities. Fig. 4 shows the area assured complete projection under the condition that transducers are arranged at equal distances and directed center of measurement plain, and TOF measurement can be done on the condition that transducers are within 45 degrees from direction of the strongest directivity characteristics each other. Outside of the area surrounded by bold line in Fig. 4, specifically in the corner of the measurement plain, is not assured complete projection and cannot be reconstructed properly based on FBP. For this reason, we focus on one of the reconstruction techniques that can be applied to Fig. 4. The area covered with the propagation paths from all directions. Each parameter in eq. (10) is denoted as follows: i: suffix of propagation path {i i=1, 2, 3,, M} j: suffix of pixel configured in measurement plain {j j=1, 2, 3,, N} p i : projection data of propagation path i {p i p i p} i : propagation time per unit distance in pixel j { i i } C ij : length of propagation path i in pixel j { C ij C ij C} ML-EM tries to improve the estimated result to a solution iteratively. As described in Ref. [3], this algorithm is based on the update equation as shown in Fig. 5. Assuming that k represents number of iterations of reconstruction procedure, C represents length of propagation path which can be calculated from arrangement of transducers. Fig. 5. The update equation of the ML-EM algorithm. 53

4 Fig. 6 shows a transition of data in the implemented reconstruction procedure based on ML-EM. In the practical application case, measurement errors are included in the projection data and they cause spike-like artifacts. Therefore we adopt the filtering by Median Filter as shown in Fig. 6 to reduce these artifacts. Actual procedure of reconstruction based on ML-EM is shown as follows; (0) Let k pre-specified initial distribution (k=0), e.g., uniform. (1) Calculate forward projection pˆ from k estimated in the last iteration. (2) Calculate weighted back projection of ratio between measured projection data p and calculated projection data pˆ. (3) Normalize effect of weight. (4) Update the distribution (k=k+1) and filter by Median Filter. (5) Iterate the procedure from (1) to (4) until convergence of k. 3. Numerical Experiments In order to confirm the effectiveness of ML-EM to the reconstruction of temperature distributions in the corner of the measurement plain, we set the numerical models of ultrasonic CT system as shown in Fig. 7 and temperature distribution as shown in Fig. 8, calculate the forward projection from numerical model, and reconstruct the temperature distribution from calculated forward projection data. In the numerical model of ultrasonic CT system, 80 transducers are arranged at equal distance and directed to the center of the measurement plain, and TOF measurement can be done on the condition that transducers are within 45 degrees from direction of the strongest directivity characteristics each other. To verify the effectiveness of the adoption of ML-EM instead of FBP, reconstructed temperature distributions are compared with each other. Fig. 6. Implemented reconstruction procedure based on ML-EM including Median Filter. Fig. 7. Configuration of Experimental system. Fig. 9 shows temperature distribution reconstructed by FBP from projection data calculated from numerical temperature distribution model shown in Fig. 8. Deformation of heat source caused by incomplete projection is shown in Fig. 9. Moreover, the peak of reconstructed temperature distribution (22.93 degree) is less than that of numerical temperature distribution model (25.0 degree) by 2.07 K, also, it is confirmed that in the corner of the measurement plain FBP cannot reconstruct the objective distribution with a high degree of accuracy. Fig. 10 shows temperature distribution reconstructed by introduced method; ML-EM from projection data calculated from numerical temperature distribution model shown in Fig. 8. The peak of reconstructed temperature distribution (24.99 degree) is close to that 54

5 of numerical temperature distribution model (25.0 degree) and deformation of heat source is much smaller than temperature distribution reconstructed by FBP. In addition, to evaluate reconstructed results quantitatively, Sum of Square Difference (SSD) between numerical temperature distribution model and reconstructed temperature distributions are calculated. The SSD of temperature distribution reconstructed by ML-EM is less than 30 % of that by FBP as shown in Fig. 11. Furthermore, Root Mean Square Error of the temperature distribution reconstructed by ML-EM is 0.09 K, and sufficiently small for assumed system. From these results, possibility of improvement in the reconstruction accuracy caused by adaptation of ML-EM is confirmed. Fig. 8. Numerical experimental conditions. Fig. 9. Reconstructed results by means of FBP. Fig. 10. Reconstructed results by means of ML-EM. 55

characteristics and result of fundamental experiments, we assume TOF measurement can be done on the condition that transducers are within 45 degrees from direction of the strongest directivity

Experiments In order to confirm the effectiveness of ML-EM in the actual experimental system, we set some conditions and reconstruct the temperature distribution from ultrasonic TOF measured by

6 characteristics and result of fundamental experiments, we assume TOF measurement can be done on the condition that transducers are within 45 degrees from direction of the strongest directivity characteristics each other. Fig. 11. SSD of reconstructed temperature distributions. 4. Experiments In order to confirm the effectiveness of ML-EM in the actual experimental system, we set some conditions and reconstruct the temperature distribution from ultrasonic TOF measured by experimental system. Fig. 13. Photograph of ultrasonic transducer Experimental Setups Fig. 12 shows a photograph of the measurement plain and transducer array. In the experimental system, 80 transducers (20 transducers per side) are arranged on circumference of the square area that is 100 cm on a side. Fig. 14. Directivities of transmitter module and receiver module. Fig. 12. Photograph of the measurement plain and transducer array. Fig. 13 shows a photograph of the transducer employed in the experimental system. A transmitter module (MA40S4S produced by Murata Manufacturing Co., Ltd.) and a receiver module (MA40S4R produced by Murata Manufacturing Co., Ltd.) are contained in the transducer. Fig. 14 shows the directivities of transmitter module and receiver module. From these Fig. 15 shows the two configurations of the heat source. In the configuration (1), the hair drier used as heat source is arranged at center of the measurement plain and 15 cm below the measurement plain. In the configuration (2), the hair drier is arranged in the corner of the measurement plain to verify the effectiveness of the adoption of ML-EM instead of FBP Experimental Results The temperature distribution reconstructed from TOF measured in configuration (1) is shown in Fig. 16. In addition, to verify the precision of the temperature measurement, profiles (y=0) of the reconstructed temperature distributions are compared 56

7 with referential temperature data obtained by thermistors as shown in Fig. 17. Fig. 16 shows that there are few differences in re-construction accuracy between FBP and ML-EM, in the case of reconstruction at the center of the measurement plain in which complete projection is assured. Given set condition as shown in Fig. 15, both reconstructed temperature distributions are likely estimated results. Whereas, in comparison with the temperature distribution reconstructed by ML-EM, the temperature distribution reconstructed by FBP includes the line artifact as shown in Fig. 16 (b). Fig. 15. Arrangement of heat source. Fig. 16. Temperature distributions reconstructed from TOF measured in the configuration (1). Fig. 17. Profiles of reconstructed temperature distributions (y=0). The temperature distribution reconstructed from TOF measured in configuration (2) is shown in Fig. 18, and profiles (yy =40) of reconstructed results are shown in Fig. 19. Given set condition as shown in Fig. 15, the temperature distribution reconstructed by ML-EM is likely estimated result. By contrast, in the temperature distribution reconstructed by FBP deformation of heat source is included. Besides, in comparison with the temperature distribution reconstructed by FBP, the temperature distribution reconstructed by ML-EM is close to the referential data obtained by thermistors. As a result, by using proposed method, a temperature distribution in the corner of the measurement plain can be reconstructed more accurately. To conclude, the certain advantages of the proposed method have been confirmed experimentally. 57

Fig. 18. Temperature distributions reconstructed from TOF measured in the configuration (2). Fig. 19. Profiles of reconstructed temperature distributions (y = 40). 5.

8 Fig. 18. Temperature distributions reconstructed from TOF measured in the configuration (2). Fig. 19. Profiles of reconstructed temperature distributions (y = 40). 5. Conclusions In this paper, the novel method to measure two dimensional temperature distribution with enclosed square type ultrasonic CT is proposed. In the square type ultrasonic CT, not all of the area is assured complete projection because of directivities of transducers, specifically corner of the area. Therefore, conventional reconstruction technique; FBP cannot reconstruct temperature distribution in such area properly. For this reason, ML-EM is introduced as a reconstruction technique that is applicable for incomplete projection and the effectiveness of adaptation of ML-EM is confirmed by both numerical and actual experiments. In the future work, measurement range of temperature should be expanded and scale of measurement plain should be expanded. References [1]. Shinji Ohyama, Junya Takayama, Yuuki Watanabe, Tetsuya Takahoshi and Kazuo Oshima, Temperature distribution and wind vector measurement using ultrasonic CT based on the time of flight detection, Sensors and Actuators A: Physical, Vol. 151, Issue 2, 2009, pp [2]. L. A. Shepp and B. F. Logan, The Fourier Reconstruction of a Head Section, IEEE Trans. Nucl. Sci, Vol. 21, 1974, pp [3]. L. A. Shepp and Y. Vardi, Maximum likelihood reconstruction for emission tomography, IEEE Trans. Med. Imaging, Vol. MI-1, 1982, pp [4]. K. Lange and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Computer Assisted Tomography, Vol. 8, 1984, pp [5]. Murata Manufacturing Co., Ltd., Product catalog, Copyright, International Frequency Sensor Association (IFSA). All rights reserved. ( 58

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