PNC in 2D Curve Modeling: Interpolation and Extrapolation

Introduction

The problem of multidimensional data modeling appears in many branches of science and industry. Image retrieval, data reconstruction, object identification or pattern recognition are still the open problems in artificial intelligence and computer vision. The chapter is dealing with these questions via modeling of high-dimensional data for applications of image segmentation in image retrieval and recognition tasks. Handwriting based author recognition offers a huge number of significant implementations which make it an important research area in pattern recognition. There are so many possibilities and applications of the recognition algorithms that implemented methods have to be concerned on a single problem: retrieval, identification, verification or recognition. This chapter is concerned with two parts: image retrieval and recognition tasks. Image retrieval is based on probabilistic modeling of unknown features via combination of N-dimensional probability distribution function for each feature treated as random variable. Handwriting and signature recognition and identification represents a significant problem. In the case of biometric writer recognition, each person is represented by the set of modeled letters or symbols. The sketch of proposed Probabilistic Features Combination (PFC) method consists of three steps: first handwritten letter or symbol must be modeled by a vector of features (N-dimensional data), then compared with unknown letter and finally there is a decision of identification. Author recognition of handwriting and signature is based on the choice of feature vectors and modeling functions. So high-dimensional data interpolation in handwriting identification (Marti & Bunke, 2002) is not only a pure mathematical problem but important task in pattern recognition and artificial intelligence such as: biometric recognition (Nosary, Heutte & Paquet, 2004), personalized handwriting recognition (Djeddi & Souici-Meslati, 2010 & 2011), automatic forensic document examination (Van, Vuurpijl, Franke & Schomaker, 2005; Schomaker, Franke & Bulacu, 2007), classification of ancient manuscripts (Siddiqi, Cloppet & Vincent, 2009). Also writer recognition (Garain & Paquet, 2009) in monolingual handwritten texts (Ozaki, Adachi & Ishii, 2006) is an extensive area of study (Chen, Lopresti & Kavallieratou, 2010) and the methods independent from the language (Chen, Cheng & Lopresti, 2011) are well-seen (Bulacu, Schomaker & Brink, 2007). Proposed method represents language-independent and text-independent approach because it identifies the author via a single letter or symbol from the sample. The method of Probabilistic Nodes Combination (PNC) enables interpolation and modeling of two-dimensional curves using nodes combinations and different coefficients γ: polynomial, sinusoidal, cosinusoidal, tangent, cotangent, logarithmic, exponential, arc sin, arc cos, arc tan, arc cot or power function, also inverse functions. This probabilistic view is novel approach a problem of modeling and interpolation. Computer vision and pattern recognition are interested in appropriate methods of shape representation and curve modeling. PNC method represents the possibilities of shape reconstruction and curve interpolation via the choice of nodes combination and probability distribution function for interpolated points. It seems to be quite new look at the problem of contour representation and curve modeling in artificial intelligence and computer vision. Proposed method, called Probabilistic Nodes Combination (PNC), is the method of 2D curve interpolation and extrapolation using the set of key points (knots or nodes). Nodes can be treated as characteristic points of data for modeling and analyzing. The model of data can be built by choice of probability distribution function and nodes combination. PNC modeling via nodes combination and parameter γ as probability distribution function enables value anticipation in risk analysis and decision making. Two-dimensional curve is extrapolated and interpolated via nodes combination and different functions as discrete or continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function. Novelty of this book consists of two generalizations: generalization of previous MHR method with various nodes combinations and generalization of linear interpolation with different (no basic) probability distribution functions and nodes combinations.