Current Trends in the Algebraic Image Analysis: A Survey

Abstract

Survey. The main goal of the Algebraic Approach is the design of a unified scheme for the representation of objects for the purposes of their recognition and the transformation of such representations in the suitable algebraic structures. It makes possible to develop corresponding regular structures ready for analysis by algebraic, geometrical and topological techniques. Development of this line of image analysis and pattern recognition is of crucial importance for automated image mining and application problems solving. It is selected and briefly characterized main aspects of current state of the image analysis algebraization. Special attention is paid to the recent results of the Russian mathematical school.

Descriptive Image Analysis. Foundations and Descriptive Image Algebras

“…(2) the specialization of Zhuravlev's algebra 69 to the case of representation of the recognition-source data in the form of images; (3) a standard language for describing the procedures for analysis and recognition of images (DIA) 19,20,22,25,28 ; (4) the mathematical formulation of the problem of image recognition 13,14 ; (5) mathematical theories of image analysis and pattern recognition 31 ; (6) a model of the process for solving the standard problem of image recognition. 13,14 The main objects and tools of DAIA are as follows:…”

“…The current state and trends of DAIA are determined by the methods, models, and nal and intermediate results obtained during its elaboration. [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] The creation of the DAIA was signi¯cantly in°uenced by the following basic lines of the mathematical theory of pattern recognition:…”

“…DIA was introduced by Gurevich and is now being developed by him and his followers. 19,20,22,25,28 The milestones of image analysis algebraization are as follows:…”

“…(1) von Neumann 50 and Unger 67 (studies of interactive image transformations in a cellular space); (2) Du®, Watson, Fountain, and Shaw 60 (a cellular logic array for image processing); (3) Rosenfeld 57 (digital topology); (4) Minkowski and Hadwiger 32 (pixel neighborhood arithmetics and mathematical morphology); (5) Matheron, Serra, Sternberg 44,58,63,64 (a coherent algebraic theory speci¯cally designed for image processing and image analysis: mathematical morphology); (6) Zhuravlev 69 (algebra of algorithms); (7) Sternberg 63 (the¯rst use of the term \image algebra"); (8) Grenander 12 (pattern theory); (9) Maragos 45,46 (introduced a new theory unifying a large class of linear and nonlinear systems under the theory of mathematical morphology); (10) Pavel 51 (theory of categories techniques in pattern recognition); (11) Davidson 4 (completed the mathematical foundation of mathematical morphology by embedding it into lattice algebra (minimax algebra)); (12) Ritter 55,56 (IA); (13) Gurevich 19,20,22,25,28 (DIA); (14) Haralick [33][34][35]40 , Shapiro [33][34][35]40 , Lee 34,40 , Joo Hyoman 35 , Schafer 46 , Dougherty 5,6 , Sinha 6 , Gader 11 , Khabou 11 , Koldobsky 11 , Radunacu 54 , Grana 54 , Albizuri 54 , Sussner 65,66 , and Soille 62 (development of mathematical morphology and IA 38 ).…”

“…The axiomatic and formal constructions of DAIA are proposed and developed as conceptual and logical bases for extraction of information from images. DAIA includes the following basic tools for image analysis and recognition: a set of methods for analysis and recognition of images, IRRF techniques, a conceptual system of image analysis and recognition, DIM, 17,[24][25][26][27] classes, a DIA language, 19,20,22,25,28 settings of image analysis and recognition problems, and a basic model of image recognition.…”