Weighted Quasi-Interpolant Spline Approximations of Planar Curvilinear Profiles in Digital Images

• Published in: Mathematics
• Main Authors: Andrea Raffo, Silvia Biasotti
• Format: Article
• Language: English
• Published: MDPI AG 2021-11-01
• Subjects: spline functions; quasi-interpolation; image processing; <i>G</i>¹ continuity
• Online Access: https://www.mdpi.com/2227-7390/9/23/3084

The approximation of curvilinear profiles is very popular for processing digital images and leads to numerous applications such as image segmentation, compression and recognition. In this paper, we develop a novel semi-automatic method based on quasi-interpolation. The method consists of three steps: a preprocessing step exploiting an edge detection algorithm; a splitting procedure to break the just-obtained set of edge points into smaller subsets; and a final step involving the use of a local curve approximation, the Weighted Quasi Interpolant Spline Approximation (wQISA), chosen for its robustness to data perturbation. The proposed method builds a sequence of polynomial spline curves, connected <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>0</mn></msup></semantics></math></inline-formula> in correspondence of cusps, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>G</mi><mn>1</mn></msup></semantics></math></inline-formula> otherwise. To curb underfitting and overfitting, the computation of local approximations exploits the supervised learning paradigm. The effectiveness of the method is shown with simulation on real images from various application domains.