The Determination of Distances between Images by de Rham Currents Method

The goal of the paper is to develop an algorithm for matching the shapes of images of objects based on the geometric method of de Rham currents and preliminary affine transformation of the source image shape. In the formation of the matching algorithm, the problems of ensuring invariance to geometric image transformations and ensuring the absence of a bijective correspondence requirement between images segments were solved. The algorithm of shapes matching based on the current method is resistant to changes of the topology of object shapes and reparametrization. When analyzing the data structures of an object, not only the geometric form is important, but also the signals associated with this form by functional dependence. To take these signals into account, it is proposed to expand de Rham currents with an additional component corresponding to the signal structure. To improve the accuracy of shapes matching of the source and terminal images we determine the functional on the basis of the formation of a squared distance between the shapes of the source and terminal images modeled by de Rham currents. The original image is subjected to preliminary affine transformation to minimize the squared distance between the deformed and terminal images.


Introduction
Analysis and matching of image shapes of objects is an important problem in pa ern recognition [1], image registration [2], biometrics [3], computational anatomy [4]. e determination of distances for matching the shapes of objects is one of the methods for analyzing shapes in pa ern recognition. Known distances used in pa ern recognition are: Hausdor , Frechet, Procrustes, Wasserstein and others [5]. One of the most e ective methods for matching the shapes of objects is the LDDMM method (Large deformation di eomorphic metric mapping [6]), in which the distance between the shapes is determined by the minimized functional consisting of the integral of the deformation energy of the original image and the terminal and the sum squared of deviations between the resulting deformable and terminal image. e traditional methods of matching image shapes in pa ern recognition problems have the following disadvantages. Firstly, the lack of invariance of methods in a ne transformations of the shapes of images of objects; secondly, the requirement of bijective correspondence between image segments; thirdly, the lack of accounting of the orientation of the shapes of the source and terminal images; fourthly, the lack of accounting of the functional dependence of image segments.

Problem statement
Purpose of this paper is to develop an algorithm for matching the image shapes of objects, which is devoid of the above disadvantages. An algorithm for matching shapes based on the geometric de Rham current method [7] and preliminary a ne transformation of the original image form is proposed. e method of currents can be used to represent and analyze forms of various nature: point landmarks, curves, surfaces, signals. If Ω ( ) is the space of continuous di erential -forms in ∈ ℝ , then the space of de Rham -currents Ω * ( ) is the dual space to the space Ω ( ); -current (⋅) ∈ Ω * ( ) is a linear functional mapping a di erential -form ∈ Ω ( ): → ( ) ∈ ℝ. For any hypersurface ∈ ℝ we can associate such current (⋅) ∈ Ω * that [7]: In the formation of the matching algorithm, the following problems were solved: ensuring the invariance to geometric image transformations, ensuring the absence of a bijective correspondence requirement between image segments [8][9][10]. Using the de Rham current algorithm allow us to increase the accuracy of matching by taking into account the orientation of the segments of the image shape. e algorithm for matching shapes based on the current method is stable when changing the topology of the shapes of objects and changing parameterization. e problem of correctly determination the distance between currents that decode the shapes of objects is solved by imbedding the space of de Rham currents in RKHS (reproducing kernel Hilbert spaces) [11]. e study of the shapes of objects is proposed to be carried out by forming test vector elds. Since the de Rham current is not a scalar, for working with currents it is necessary to use vector-valued RKHS [12,13].
When analyzing the data structures of an object, not only the geometric shape is important, but also the signals associated with this shape with functional dependence. Signals can include structures that are more complicated than real numbers; e.g. vector, tensor signals, quaternions, etc. To take these signals into account, it is proposed to expand de Rham currents with an additional component corresponding to the signal structure. e results of a di eomorphic matching of the shapes of objects with an extension of the LDDMM algorithm to the case of metamorphosis, in which there may be no bijective correspondence between the segments of the source and nal images, are presented in the article [14]. In this case, a functional is formed that corresponds to the image deformation and determines the distance between the shapes of the initial and terminal images. In order to increase the accuracy of matching the shapes of the source and terminal images in this paper, we determine the functional on the basis of the formation of the squared distance between the shapes of the source and terminal images modeled by de Rham currents. e source image undergoes a preliminary a ne transformation formalized by Lie groups to minimize the squared distance between the two shapes. e minimization of the functional of the squared distance between the image shapes constructed using de Rham currents is based on the QPSO algorithm.

Hamiltonian mechanics of image points
Representation of an image a er a di eomorphic transformation can be considered as an evolution of point landmarks of an image based on the principles of Hamilton mechanics. Consider the parameterization of the image by particles. Let ( ); = 1, … , be the position vector of the particle and ( ) ; = 1, … , be the corresponding momentum vector in time . If we assume that the moments and velocities of particles are interconnected by the relation: =  ⋅ , where  is an invertible linear operator, then the inverse operator  −1 : =  −1 ⋅ =  . For an operator  = id − ∇ 2 in space ℝ 2 , the inverse operator  =  −1 can be approximated by the Gauss function: We construct a functional 0 corresponding to the deformation of the image represented by a set of points: 3. Matching the shapes of objects e theory of currents was developed by G. de Rham [7]. e denomination "current" is chosen by analogy with electromagnetism. For example, in accordance with the law of induction of M. Faraday, the intensity of the current in the wire loop caused by a change in the magnetic eld is proportional to the change in the ux of this magnetic eld through the surface bounded by the loop. is means that if you measure the current strength in the wire for all possible changes in the magnetic eld, you can get the loop geometry. In the works [15][16][17] presents the concept of currents for the formation of a measure of the di erence between simplicial complexes, which does not imply a bijective correspondence between the structures of objects. e concept of using currents is to study the shape of objects by forming test vector elds.

Example 2
Let us consider an example of a di eomorphic deformation of the image shape of a symbol of an inde nite shape into an image shape of the shape of number 2 (Fig. 1), number 7 (Fig. 2) and number 8 (Fig. 3). e evolution of deformations of a di eomorphic shape was determined based on the solution of equations (1). e functional is minimized by values using the QPSO algorithm (see Appendix 2, [18] ). In g. 1, 2, 3 shows intermediate shapes of images for times: = 0 (source image shape), = 0, 5 (intermediate image shape), = 1 (terminal image shape). In this case, the values of the squared distance between the source image and the terminal shape 2 , ′ , determined from relation (2) with = 1, are: • for the case of deformation of the shape of the symbol in the shape of numbers 2: 2 , ′ = 78, 6; • for the case of deformation of the shape of the symbol in the shape of the number 7: 2 , ′ = 78, 0; • for the case of deformation of the shape of the symbol in the shape of the gure 8: 2 , ′ = 16, 8. erefore, the algorithm recognizes the character as the number 8. It should be noted that during deformation of the shape of the symbol into the shape of the gure 8, the topological genus of the shape changes from 0 to 1, that is, the deformation is not a di eomorphism, but a metamorphosis.

Normalization of images based on a ne transformations
To improve the accuracy of matching of source and terminal images, these images should be normalized. Below we propose such a normalization method, in which the original image undergoes a ne transformation and the functional between the converted original and terminal images is minimized. A er that, the normalized original image undergoes a di eomorphic transformation, while the distance (2) between the converted and terminal images is reduced, which will increase the accuracy of the matching.
An a ne transformation is a special case of a di eomorphic transformation. An a ne transformation can be represented in the form [19]: where ∈ ℝ × is an invertible matrix, ∈ ℝ , , are vectors in an a ne space ∈ ℝ . In the case of an a ne transformation of a curve (surface) point approximating the shape of a deformable object, it can be represented as: → ⋅ + , = 1, … , . As the minimized functional, we choose the square of the distance between the points of the source and nal images: ( , ) = , ′ 2 , where , ′ 2 it is determined in accordance with (2), is the current corresponding to the initial shape of the object, ′ is the current corresponding to the shape of the deformable object a er a ne transformation. Let be the parameters of the a ne transformation: ∈ Ξ; = 1, … , , where Ξ, is the set of matrix components and vector components . e values of the parameters of the particle can be found using the QPSO algorithm (quantum particle swarm optimization, see Appendix 2, [18]) to minimize the functional (Ξ). We denote the value of the minimized functional on the set: , ∈ Ξ: = 1, , … , , , where is the iteration step number, and ∈ [1 … ] is the particle number. Let , be the values of the parameters that provide the smallest value of the functional for the particle a er the -th iteration, and be the values of the parameters that provide the smallest value of the functional for all particles a er the -th iteration. We choose the values of the best values of the parameters from the relation: Before the a ne transformation, the value , ′ (see (2)) is equal , ′ = 8, 2. A er carrying out the a ne transformation and minimizing the distance , ′ , we obtain the required components of the matrix ∶ = 1,2 −0,26 0,38 0,8 , and the vector ∶ = ( 0 0 ) . Preliminary a ne transformation reduced the distance to , ′ = 0, 67.
Conclusion e paper considered an algorithm for matching image shapes, based on the de Rham currents method and preliminary a ne transformation of the source image shape. e de Rham current method can be used to represent shapes of various nature: point landmarks, curves, surfaces, signals. Using the proposed matching algorithm allows us to solve the problem of ensuring invariance to geometric transformations of images and ensuring the absence of a bijective correspondence requirement between image segments. e algorithm for matching shapes based on the current method is stable when changing the topology of the shapes of objects and changing parameterization. An application of the method of reproducing kernel Hilbert space (RKHS) to obtain metrics of the shape of an object is proposed.
To increase the accuracy of matching the shapes of the source and terminal images, it is proposed that the source image be subjected to preliminary a ne transformation. e problem of invariance to geometric transformations of images (translation, rotation, scaling, skew) is solved. e minimization of the functional of the squared distance between the image shapes is based on the QPSO algorithm. e results of a di eomorphic matching of the shapes of objects with the extension of the LDDMM (large deformation di eomorphic metric mapping) algorithm to the case of metamorphosis, in which there may be a bijective correspondence between the segments of the source and terminal images, are presented. To improve the accuracy of matching the shapes of the source and terminal images, we determine the functional on the basis of the formation of a squared distance between the shapes of the source and terminal images modeled by de Rham currents.