Camera Calibration Using Neural Network for Image-Based Soil Deformation Measurement Systems Zhao, Honghua Ge, Louis Civil, Architectural, and Environmental Engineering, University of Missouri-Rolla, (Received 16 July 2006; accepted 3 August 2007 Published : 1 October 2007 Geotechnical Testing Journal (GTJ) Vol31 No.2 M97G0102 姚芳德

Outline Introduction Camera Calibration Neural Network Calibration Model Neural Network Model Validation Application Conclusion

Introduction This calibration algorithm provides a highly accurate prediction of object data points from their corresponding image points. The experimental setup for this camera calibration algorithm is rather easy, and can be integrated into particle image velocimetry (PIV) to obtain the full-field deformation of a soil model. The performance of this image-based measurement system was illustrated with a small-scale rectangular footing model. This fast and accurate calibration method will greatly facilitate the application of an image-based measurement system into geotechnical experiments.

Camera Calibration Camera calibration is an important component of an image-based measurement system. Its goal is to link a point P X,Y in the object coordinate system to the corresponding point p u,v in the image plane coordinate system.

Neural Network Calibration Model The neural network trained in this study is a three-layer, feed-forward neural network (2–15–2 NN). Input data are the image points p u,v obtained from the extraction of the corner points. The target data are the corner points P X,Y from the cali-bration plane. Input Layer Hidding Layer Output Layer . P(u,v) P(X,Y)

Neural Network Calibration Model The neural network training algorithm used was a back- propagation function, which updates the weight and bias values according to Levenberg-Marquardt optimization method. The hidden layer maps from input vector to a vector of output n3=2 by a tangent sigmoid transfer function tansig 倒傳遞網路 activation 活躍 Activation function Q is the number of outputs, N is the total number of data sets Dgh is the set of target data, Xout ,gh is the set of network output data

Neural Network Calibration Model Levenberg-Marquardt Optimization The Levenberg–Marquardt algorithm (LMA) provides a numerical solution to the problem of minimizing a function, generally nonlinear, over a space of parameters of the function. The primary application of the Levenberg–Marquardt algorithm is in the least squares curve fitting problem: given a set of empirical data pairs of independent and dependent variables, (xi, yi), optimize the parameters β of the model curve f(x,β) so that the sum of the squares of the deviations empirical 根據經驗的 Deviations 偏差 becomes minimal.

Neural Network Calibration Model Levenberg-Marquardt Optimization In each iteration step, the parameter vector, β, is replaced by a new estimate, β + δ. To determine δ, the functions are approximated by their linearizations Where is the gradient of f with respect to β. Linearizations 線性化 Deviations 偏差 where I is the identity matrix, giving as the increment, δ, to the estimated parameter vector, β.

Neural Network Calibration Model Levenberg-Marquardt Optimization Fit the function y = acos(bX) + bsin(aX) using the LMA parameters a=100, b=102 used in the initial curve Linearizations 線性化 Deviations 偏差 the function cos(βx) has minima at parameter value and

Neural Network Model Validation The proposed neural network calibration model was verified through the testing data, which were not used in the training process.

Comparison with Others Comparison with the linear and second-order polynomial calibration algorithms

Comparison with Others Angle ErrorAnalysis First the calibration plane was closely aligned to the image sensor plane; then, the calibration plane was rotated to different angles

Application Images were taken every 30 seconds. The proposed neural network camera calibration algorithm was implemented into MatPIV A small-scale wood block (6.35 cm by 7.62 cm by 15.88 cm) was placed on top of the sand in a model container (35.56 cm by 6.60 cm by 35.56 cm), as shown in Fig. 9. The height of the sand was 18.92 cm 7.45 in. , and a loading frame was used to apply the vertical load to the foundation. The camera was set up to take im- ages of the soil below the footing during the loading. The loading rate was set to 2.54 mm/min 0.1 in./min .

Conclusion A three-layer back-propagation neural network calibration algorithm was developed for camera calibration in an image-based measurement system. This algorithm was compared with the linear calibration and second-order polynomial calibration algorithms. The neural network calibration model will give a very accurate result independent of the angle between the image plane and the object plane. This fast and accurate calibration method will greatly facilitate for the application .