In the application of the standard Levenberg-Marquardt training process of neural networks, error oscillations are frequently observed and they usually aggravate on approaching the required accuracy. In this paper, a modified Levenberg-Marquardt method based on variable decay rate in each iteration is proposed in order to reduce such error oscillations. Through a certain variation of the decay rate, the time required for training of neural networks is cut down to less than half of that required in the standard Levenberg-Marquardt method. Several numerical examples are given to show the effectiveness of the proposed method.

Current research on dynamic displacement measurement based on computer vision mostly requires professional high-speed cameras and an ideal shooting environment to ensure the performance and accuracy of the analysis. However, the high cost of the camera and strict requirements of sharp image contrast and stable environment during the shooting process limit the broad application of the technology. This paper proposes an improved vision method to implement multi-point dynamic displacement measurements with smartphones in an interference environment. A motion-enhanced spatio-temporal context (MSTC) algorithm is developed and applied together with the optical flow (OF) algorithm to realize a simultaneous tracking and dynamic displacement extraction of multiple points on a vibrating structure in the interference environment. Finally, a sine-sweep vibration experiment on a cantilever sphere model is presented to validate the feasibility of the proposed method in a wide-band frequency range. In the test, a smartphone was used to shoot the vibration process of the sine-sweep-excited sphere, and illumination change, fog interference, and camera jitter were artificially simulated to represent the interference environment. The results of the proposed method are compared to conventional displacement sensor data and current vision method results. It is demonstrated that, in an interference environment, (1) the OF method is prone to mismatch the feature points and leads to data deviated or lost; (2) the conventional STC method is sensitive to target selection and can effectively track those targets having a large proportion of pixels in the context with motion tendency similar to the target center; (3) the proposed MSTC method, however, can ease the sensitivity to target selection through in-depth processing of the information in the context and finally enhance the robustness of the target tracking. In addition, the MSTC method takes less than one second to track each target between adjacent frame images, implying a potential for online measurement.