Gauss' Law in Image Processing and Analysis via Fast Numerical Calculation of Vector Fields

This paper describes a numerical approach to the expedite calculation of vector fields in two-dimensional (2D) spaces and how it has allowed the effective application of Gauss' law in image analysis and computer vision. The adopted technique for calculation of vector fields consists of decomposing than in terms of two scalar subfields, which can then be obtained in terms of convolutions of the image with large templates, an operation that is effectively performed by the fast Fourier transform. The performance of the technique has been quantified in terms of RMS error and execution time, yielding encouraging results. The application of Gauss' law to digital images has also been addressed. Effects implied by the spatial quantization inherent to digital images, which can severely constrain the representation of the circulation path considered in the Gauss' law, were experimentally investigated, and it has been verified that biased results can be obtained for short circulation paths. A process for generating calibration curves helping to circumvent such a problem has also been described. The application of the developed framework in image analysis, more specifically for construction of quadtrees, is described, assessed, and illustrated with respect to real images. The extension of the techniques to general circulation paths is also briefly outlined.

[1]  Gayle F. Miner.  Lines and electromagnetic fields for engineers , 1996 .

[2]  Irene Gargantini,et al.  Translation, Rotation and Superposition of Linear Quadtrees , 1983, Int. J. Man Mach. Stud..

[3]  Yee-Hong Yang,et al.  Multiresolution skeletonization an electrostatic field-based approach , 1994, Proceedings of 1st International Conference on Image Processing.

[4]  Charles W. Therrien,et al.  Discrete Random Signals and Statistical Signal Processing , 1992 .

[5]  Rangachar Kasturi,et al.  Machine vision , 1995 .

[6]  Kenneth Steiglitz,et al.  Operations on Images Using Quad Trees , 1979, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[7]  Milan Sonka,et al.  Image Processing, Analysis and Machine Vision , 1993, Springer US.

[8]  H. M. Schey.  Div, grad, curl, and all that , 1973 .

[9]  Yee-Hong Yang,et al.  Electrostatic field-based detection of corners of planar curves , 1993, Proceedings of Canadian Conference on Electrical and Computer Engineering.

[10]  Roberto Marcondes Cesar Junior,et al.  Piecewise Linear Segmentation of Digital Contours in O(N. Log(N)) Through a Technique Based on Effective Digital Curvature Estimation , 1995, Real Time Imaging.

[11]  P. Morse,et al.  Methods of theoretical physics , 1955 .

[12]  Hanan Samet,et al.  The Quadtree and Related Hierarchical Data Structures , 1984, CSUR.

[13]  R. Bracewell.  Two-dimensional imaging , 1994 .

[14]  Roberto Marcondes Cesar Junior,et al.  Towards effective planar shape representation with multiscale digital curvature analysis based on signal processing techniques , 1996, Pattern Recognit..

[15]  Bruno O. Shubert,et al.  Probabilistic models in engineering sciences , 1979 .