A new formula for bivariate Hermite interpolation on variable step grids and its application to image interpolation

Area of Science:

• Numerical Analysis
• Computer Vision
• Medical Imaging

Background:

• Bivariate Hermite interpolation (BHI) is crucial for reconstructing functions from scattered data.
• Existing BHI formulas can be computationally intensive and complex.
• Efficient image interpolation is vital for medical imaging and computer vision applications.

Purpose of the Study:

• To introduce a novel, algebraically simpler formula for bivariate Hermite interpolating polynomials.
• To adapt and apply this formula for image interpolation tasks, particularly on regular grids.
• To evaluate the performance of the proposed method against existing interpolation techniques.

Main Methods:

• Derivation of a generalized BHI polynomial formula for variable step grids.
• Simplification of the BHI formula for unit-step grids relevant to image interpolation.
• Approximation of image partial derivatives using compact finite differences.
• Experimental assessment on synthetic and real-world medical images.

Main Results:

• The proposed BHI formula is algebraically simpler and more efficient than existing methods.
• The BHI method, using up to second-order partial derivatives, surpasses convolution-based and generalized interpolation methods in most tested scenarios.
• Performance was validated on diverse datasets, including medical images from various modalities.

Conclusions:

• The novel BHI formula offers a more efficient and effective approach to image interpolation.
• The method demonstrates superior performance, especially when incorporating higher-order partial derivatives.
• This work provides a valuable tool for image processing in fields like medical imaging.