## Integral equation-wavelet collocation method for geometric transformation and application to image processing.(English)Zbl 1474.65459

Summary: Geometric (or shape) distortion may occur in the data acquisition phase in information systems, and it can be characterized by geometric transformation model. Once the distorted image is approximated by a certain geometric transformation model, we can apply its inverse transformation to remove the distortion for the geometric restoration. Consequently, finding a mathematical form to approximate the distorted image plays a key role in the restoration. A harmonic transformation cannot be described by any fixed functions in mathematics. In fact, it is represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the geometric restoration. In this paper, a novel wavelet-based method is presented, which consists of three phases. In phase 1, the partial differential equation is converted into boundary integral equation and representation by an indirect method. In phase 2, the boundary integral equation and representation are changed to plane integral equation and representation by boundary measure formula. In phase 3, the plane integral equation and representation are then solved by a method we call wavelet collocation. The performance of our method is evaluated by numerical experiments.

### MSC:

 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 65T60 Numerical methods for wavelets
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### References:

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