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Graph matching using conformal module
EURASIP Journal on Image and Video Processing volume 2019, Article number: 26 (2019)
Abstract
Graph matching and classification play fundamental roles in computer vision. The computational complexity of the conventional method based on a spectrum method is high, which prevents it from handling large graphs in practice. This work proposes a novel framework for tackling the challenge by using conformal module. We apply the classical Hodge theory from differential manifold to the graph setting and compute the combinatorial conformal invariant of the graph, called as conformal module, which can be used as the fingerprint for the graph. The method is applicable for viewpoint classification and posture detection. The experimental results demonstrate the efficiency and efficacy of the proposed method.
1 Introduction
Graph matching has been widely used to solve a large variety of problems in computer vision. Fundamentally, graph matching aims at finding correspondences between two sets of features extracted from images. When matching or recognizing structural objects, graph matching does well on considering the pairwise node interactions [1]. Furthermore, when there are correspondences between two images, graph matching can finish many vision tasks such as object categorization [2], feature tracking [3], symmetry analysis [4], and posture recognition [5, 6].
In the past thirty years, extensive research has been carried out on graphing matching [7], however, there still exists several challenges in current methods. For example, almost all matching algorithms require high time complexity, so it is difficult to deal with large graphs. No methods have been provided to deal with graphs in 3D vision field, which means that graphs embedded in high genus surfaces cannot be matched using current approaches. Many techniques are proposed to solve graph matching in terms of rigid transformation, which can hardly handle the circumstance where two graphs to be matched have large nonrigid transformations. Although the third issue is considered in [8], the other two issues are still open. Our main contributions conclude as follows: (1) the proposed method can discover the combinatorial invariant of a given graph, (2) the proposed method can tolerate nonrigid transformation on a given graph, (3) the proposed method can solve the matching problem in linear time, (4) the proposed method can be applied on not only planar graphs, but also those embedded in high genus surfaces.
The main mathematical theory in our method is based on combinatorial Hodge theory. There is some work on surface registration [9, 10] and shape analysis [11] by the combinatorial Hodge theory. In the textbook [12], Gu et al. introduced the detailed theoretic treatment on combinatorial Hodge theory. These works focus on computing the conformal invariants and conformal mappings induced by the Riemannian structure of the input surfaces. By contrast, current work focuses on computing the conformal modules induced solely by the combinatorial structure of the graphs. The process is that we apply combinatorial Hodge theory to compute the combinatorial harmonic functions on graphs. The solution of harmonic functions is equivalent to solving combinatorial Laplace matrices. The combinatorial harmonic 1forms can give the combinatorial conformal invariant of a given graph, called as conformal module, which can be used as the fingerprint of a given graph. So the conformal module can be used to deal with some matching problems. In this paper, we compute the conformal modules by the combinatorial structure of the given graphs for viewpoint classification and posture detection.
2 Previous works
Graph matching problem has been an active field in computer science and mathematics. There is a large literature for this topic in the past thirty years, so a thorough survey is beyond the scope of the current work. In the following, we only review the most related work. For more detailed treatment, we refer readers to [7]. There usually exists two categories of graph matching methods: exact matching and inexact matching.
Exact matching aims at graph isomorphism, where the interest is in an exact matching between nodes and edges. Most of approaches to find the graph isomorphism are based on Tree Search. Due to the brute Tree Search algorithm, some improvements are developed. Ullmann proposes one of the most important and popular tree search algorithms [13]. He prunes the unfruitful matches which are not consistent with current partial matching. Cordella suggests another interesting approach based on Tree Search [14, 15]. He provides a heuristic based on analysis of the sets of nodes adjacent to the ones already considered in current partial matching. Except for Tree Search, some other techniques are proposed. According to group theory, an approach was proposed by McKay [16]. He defines equivalence classes for graphs in terms of isomorphism and does the equality verification in O(n^{2}). There are other methods based on machine learning techniques to speed up the matching against a large library of graphs [17, 18]. But they are used for filtering out the unmatched candidates not for matching.
Inexact matching focuses on mapping between graphs with weighted attributes on nodes and edges. The solution is a quadratic assignment problem and the optimization is NPhard [19]. Looking for better optimization strategies is important for major research in graph matching. According to the relaxations of the constraints on a onetoone mapping F between the nodes in both graphs [8], methods can be generally classified into several categories. The first is to approximate the constraints as an orthogonal one F^{T}F = I, which is called as spectral method. Under the orthogonal constraint, optimizing graph matching can be solved in closed form as an eigen value problem [20,21,22]. Furthermore, Leordeanu and Hebert relax F to be of unit length \( {\left\Vert F\right\Vert}_2^2=1 \) for handling more complex problems in computer vision [23]. The work in [24, 25] proposes a spectral method to decompose the graph into subgraphs, where the process can be cast into a hierarchical framework and suitable for parallel computation. The second group of methods relaxes F to be a doubly stochastic matrix, the convex hull of F. Under this constraint, optimizing graph matching can be solved as a nonconvex quadratic programming problem and a local optimization could be done within several proposed strategies. For instance, Gold and Rangarajan propose the graduated assignment algorithm for iteratively solving a series of linear approximations of the cost function in the form of Taylor expansion. The work of [26] proposes a pathfollowing algorithm. Besides the optimizationbased work, probabilistic frameworks are also useful for interpreting and solving graph matching [27]. Several work aims at higherorder tensor factorization [8, 28, 29], where higher order geometrical relations make graph matching invariant to rigid transformation. Due to a small increment in the order of relations induces a combination explosion of data, it is only suitable for very sparse graphs. Furthermore, this method cannot deal with nonrigid transformations.
3 Method
This section briefly introduces the theoretical background for combinatorial conformal module. We apply algebraic topology method to consider our matching problem. Algebraic topology focuses on studying topological problems by algebraic methods and uses simple linear algebraic methods to solve these problems. The detailed description of algebraic topology methodology can be seen in [12], where the authors consider these methods on surfaces with Riemann structure. In this paper, we apply algebraic topology method to the graph setting and consider combinatorial Hodge theory on graphs. Combinatorial Hodge theory can deduce a unique combinatorial conformal module for each graph, so it can be viewed as the fingerprint of a given graph.
We consider all objects on a threeconnected planar graph G which is embedded on a topological surface S. A planar graph means that there are no edge crossings on G and different edges of the graph can only intersect at the endpoints. So each face of G can be viewed as a topological disk. We usually use a triple G = (V, E, F) to represent a graph, where V is the vertex set, E is the edge set and F is the face set. At the first step, we need to introduce some tools about computational topology. These similar concepts are defined on surfaces with Riemann structure in [12]. Here we use them on graphs.
The vertex set is denoted as
Each oriented edge e is denoted as
where v_{i}, v_{j} are its two endpoints.
A face f is represented as
where \( {v}_{i_k} \) is its vertex and (i_{1}, i_{2}, ⋯, i_{k}) is any cyclic permutation of (1, 2, ⋯, k).
We call the vertices of G as 0dimensional cells, the oriented edges of G as 1dimensional cells and the faces of G as 2dimensional cells. Their linear combinations can form a linear chain space. We will give some definitions on the chain space of a graph.
The kdimensional chain group of a graph G = (V, E, F) is given by
where \( {\varepsilon}_k^i \) denotes the ith kdimensional cell of G_{.} We call each element of C_{k}(G, Z) as a kchain of G.
Definition 1: The kdimensional boundary operator ∂_{k} : C_{k}(G, ℤ) → C_{k − 1}(G, ℤ) on graph G is defined as
The boundary operator has the linear property. In this paper, we only consider the low dimensional boundary operator, for instance,
The kchain in the kernel of ∂_{k} satisfies
We call each element in Ker ∂_{k} as a closed chain. The closed chains are closed loops and the boundaries of any 2dimensional patches are called as boundary loops.
The kchain in the image of ∂_{k + 1} satisfies
Each element of Img ∂_{k + 1} is called as an exact chain.
Property 1: The boundary of a boundary is empty:
The proof can be seen in [12].
From the above property, we can easily obtain that Img ∂_{k + 1} is a sublinear space of Ker ∂_{k}. The complementary space is called as the homology group of the graph.
Definition 2: Suppose ∂_{k} : C_{k}(G, ℤ) → C_{k − 1}(G, ℤ) is the kdimensional boundary operator on a graph G. Ker ∂_{k} = {ε_{k} ∈ C_{k}(G, ℤ) ∂_{k}ε_{k} = 0} is the kernel of ∂_{k}. Img ∂_{k + 1} = {ε_{k} ∈ C_{k} ∃ε_{k + 1} ∈ C_{k + 1}, ε_{k} = ∂_{k + 1}ε_{k + 1}} is the image of ∂_{k + 1}. The kdimensional homology group of the graph G is defined as the quotient group, satisfying
Each element in H_{k}(G, Z) is called a kdimensional homologous class. A closed kchain ε_{k} ∈ ker ∂_{k} represents a homologous class, denoted as [ε_{k}] ∈ H_{k}(G, Z). Each element in the homologous class can form the basis of homology group. The similar algorithm in [12] is used to compute the homology basis of a discrete surface. Figures 1 and 2 show the homology basis of two graphs with different topology.
In order to do computation on graphs, we define the cohomology group which is the dual of the homology group. In fact, cohomology group is the space of linear functions defined on homology group. The following discussions are considered in parallel because all the objects are dual of the above counterparts such as cochain and exterior differential operator.
The kdimensional cochain space of the graph G is a linear function defined on kdimensional chain space and given by
We call each element in C^{k}(G, R) as a kform.
The dual of the boundary operator ∂ is called as the exterior differential operator d.
Definition 3: The exterior differential operator d_{k} : C^{k}(G, ℝ) → C^{k + 1}(G, ℝ) is defined as follows:
where ω^{k} ∈ C^{k}(G, R), ε_{k + 1} ∈ C_{k + 1}(G, Z).
Corresponding to closed chain, we have that any kform in the kernel of d_{k} satisfies
We call each element in Ker d_{k} as a closed kform.
Similarly, any kform in the image of d_{k} satisfies
We call each element in Ker d_{k} as an exact kform.
From Property 1, we obviously obtain the following property.
Property 2: The exterior differential operator d_{k} : C^{k}(G, ℝ) → C^{k + 1}(G, ℝ) is a linear operator with the property
According to property 2, we have that exact forms are closed. Therefore, Img d_{k − 1} is a sublinear space of Ker d_{k}. The complementary space is defined as the cohomology group of the graph.
Definition 4: Suppose the linear operator d_{k} : C^{k}(G, ℝ) → C^{k + 1}(G, ℝ) is the exterior differential operator on a graph G. Ker d_{k} = {ω^{k} ∈ C^{k}(G, ℝ) d_{k}ω^{k} = 0} is the kernel of d_{k}. Img d_{k − 1} = {ω^{k} ∈ C^{k}(G, R) ∃ω^{k − 1} ∈ C^{k − 1}(G, R), ω^{k} = d_{k − 1}ω^{k − 1}} is the image of d_{k − 1}. The kdimensional cohomology group of the graph G is defined as the quotient group, satisfying
A graph G embedded on a topological surface S can induce a CWcell decomposition. The Poincare dual of G is denoted as \( \overline{G} \) which is also a graph embedded on S such that each vertex v_{i} of G corresponds to a face \( {\overline{v}}_i \) of \( \overline{G} \) and each face f_{j} of G corresponds to a vertex \( {\overline{f}}_j \) of \( \overline{G} \). Furthermore, each oriented edge e of G corresponds to an oriented edge \( \overline{e} \) on \( \overline{G} \) as follows: if e = [v_{i}, v_{j}], \( \overline{e}={\overline{v}}_i\cap {\overline{v}}_j \). Furthermore, if the left face of e is f_{l} and the right face of e is f_{r}, \( \overline{e}=\left[{\overline{f}}_r,{\overline{f}}_l\right] \).
Definition 5: Suppose a graph and its dual are denoted as G and \( \overline{G} \) respectively. The Hodge star operator \( {}^{\ast }:{C}^k\left(G,\mathrm{\mathbb{R}}\right)\to {C}^{2k}\left(\overline{G},\mathrm{\mathbb{R}}\right) \) is a linear operator, satisfying
where ω ∈ C^{k}(G, R), ε ∈ C_{k}(G, Z), \( \overline{\varepsilon}\in {C}_{2k}\left(\overline{G},Z\right) \), and \( {}^{\ast}\omega \in {C}^{2k}\left(\overline{G},R\right) \).
Combining definition 3 and definition 5, we can obtain another linear operator σ_{k} : C^{k}(G, R) → C^{k − 1}(G, R), satisfying
The operator σ_{k} can be viewed as the adjoint of the exterior differential operator, called as the codifferential operator. Combining operator d_{k} and σ_{k}, we can define the Laplace operator.
Definition 6: Suppose d_{k} : C^{k}(G, ℝ) → C^{k + 1}(G, ℝ) is the exterior differential operator on a graph \( G.{}^{\ast }:{C}^k\left(G,\mathrm{\mathbb{R}}\right)\to {C}^{2k}\left(\overline{G},\mathrm{\mathbb{R}}\right) \) is the Hodge star operator on G. σ_{k} = ^{∗}d_{x}^{∗} : C^{k}(G, R) → C^{k − 1}(G, R) is the codifferential operator on G. The Laplace operator Δ : C^{k}(G, R) → C^{k}(G, R) is defined as
Obviously, the Laplace operator is symmetric, that is, for any ξ, η ∈ C^{k}(G, R),
Furthermore,
We mainly consider functions whose values are equal to zero with the action of the Laplace operator. These functions have inspiring mathematical and physical properties. In this paper, we will compute these functions on the graph setting. Based on the combinatorial structure of a graph, the solution of harmonic functions is equivalent to solving combinatorial Laplace matrices on the graph. Finally, these combinatorial harmonic functions can give out the combinatorial conformal invariant of a given graph, called as conformal module, which can be used as the fingerprint of a given graph. So, the conformal module can be applied into matching applications on graphs.
Definition 7: Suppose d_{k} : C^{k}(G, ℝ) → C^{k + 1}(G, ℝ) is the exterior differential operator on a graph G. σ_{k} = ^{∗}d_{x}^{∗} : C^{k}(G, R) → C^{k − 1}(G, R) is the codifferential operator on G. Δ = σ_{k + 1}d_{k} + d_{k − 1}σ_{k} : C^{k}(G, R) → C^{k}(G, R) is the Laplace operator on G. A kform ω ∈ C^{k}(G, ℝ) is called to be harmonic, if
Since (Δω, ω) = d_{k}ω^{2} + σ_{k}ω^{2} = 0, we can easily have the following property.
Property 3: A harmonic kform ω ∈ C^{k}(G, ℝ) satisfies
From the physical viewpoint, the exterior differential operator d_{k} is the curl operator and the codifferential operator σ_{k} is the divergence operator. So a harmonic kform means that it is both curlfree and divergencefree.
The group of all the harmonic kforms is denoted as \( {H}_{\Delta}^k\left(G,\mathrm{\mathbb{R}}\right) \), namely
According to Hodge theory, any a kform can be decomposed into an exact form, a coexact form and a harmonic form.
Theorem 1: Suppose ω is a kform. d_{k − 1} : C^{k − 1}(G, ℝ) → C^{k}(G, ℝ) is the exterior differential operator on a graph G. σ_{k} = ^{∗}d_{x}^{∗} : C^{k}(G, R) → C^{k − 1}(G, R) is the codifferential operator on G. Then there is a unique (k − 1)form α, (k + 1)form β and a harmonic kform γ, such that
moreover, such kind of decomposition is unique.
The detailed proof can be checked in [12].
For example k = 2, suppose G is embedded on a closed surface with genus g.{γ_{1}, γ_{2}, ⋯, γ_{2g}} is the generator of 1dimensional homology group H_{1}(G, Z). Let ω_{1}, ω_{2} be two harmonic 1forms with respect to the homology equivalence class, namely
Due to the linear property, we have
So ω_{1} − ω_{2} is harmonic and \( {\int}_{\gamma_k}\left({\omega}_1{\omega}_2\right)={\int}_{\gamma_k}{\omega}_1{\int}_{\gamma_k}{\omega}_2=0.\left({\omega}_1{\omega}_2\right) \) is exact. There is a function f : G → R such that ω_{1} − ω_{2} = df. According to the maximum principle and the empty boundary, f has no extremal value and is constant. That means the uniqueness holds: ω_{1} − ω_{2} = df = 0.
Suppose ω ∈ H^{1}(G, R) is closed and f : G → R is a smooth function. So ω + df and ω lie in the same cohomology class. The solution of σ(ω + df) = 0 is equivalent to the solution of Poisson equation:
In order to compute harmonic functions on a graph, we need define the combinatorial Laplace matrix.
Definition 8: Suppose G = {v_{1}, v_{2}, ⋯, v_{n}} is a planar graph. The combinatorial Laplace matrix Δ = (δ_{ij}) on G is defined as
where v_{i}~v_{j} means two vertices are adjacent and d(v_{i}) is the degree of v_{i} in the graph. According to theorem 1, the following Corollary is straightforward.
Corollary: The group of all harmonic kforms on G is isomorphic to the kdimensional cohomology group, that is
Suppose ω_{1} and ω_{2} are cohomological kforms, then there is a (k − 1)form η, such that ω_{1} = ω_{2} + dη. Assume the Hodge decomposition of ω_{1} is
dω_{1} = 0, dδβ_{1} = 0, so δβ_{1} = 0. Directly, we get
That means two cohomological forms share the same harmonic form. Therefore, each cohomological class in H^{k}(G, R) corresponds to a unique harmonic form in \( {H}_{\Delta}^k\left(G,R\right) \), which establishes the isomorphism.
Combinatorial Conformal Invariants: Suppose a graph G is embedded on a genus g surface S. ω is a combinatorial harmonic 1form on G and γ is a closed loop. We can choose a tubular neighborhood of γ,
We can periodically embed N(γ) to the plane. Each period is represented as a complex number, denoted as ∫_{γ}ω.
We can choose a canonical homology group basis in H_{1}(G, Z) as {a_{1}, b_{1}, a_{2}, b_{2}, ⋯, a_{g}, b_{g}}. Namely, they satisfy the following conditions
where a ⋅ b represents the algebraic intersection number. Suppose a closed differential 1form is ω. The integral of ω on a_{i}
is called as the aperiod of ω. Similarly,
is called the bperiod of ω.
For all a_{i}, slicing the surface S to get S_{i}, the boundary of S_{i} is \( \partial {S}_i={a}_i^{+}{a}_i^{} \). We construct a function f_{i} such that
and denote α_{i} = df_{i}.
For all b_{j}, slicing the surface S to get S_{j}, the boundary of S_{j} is \( \partial {S}_j={b}_j^{+}{b}_j^{} \). We construct a function g_{j} such that
and denote β_{j} = dg_{j}.
The combinatorial period matrix of G is as following:
For genus one case, the graph G can be embedded into the plane as a parallelogram. The combinatorial period matrix Mod(G) represents the shape of the parallelogram. For high genus graph G, the combinatorial period matrix is the generalization. The combinatorial period matrix can be used as the fingerprint of the embedded graph G.
One face f_{0} on the graph G is selected as the unique exterior face. Four vertices on the boundary of f_{0} are chosen, denoted as {v_{0}, v_{1}, v_{2}, v_{3}}, sorted counter clockwisely. The graph G with four corner vertices is called as a topological quadrilateral. The corner vertices divide the boundary of the exterior face into four segments, denoted as left, top, right, and bottom segments, such that v_{0} is the left lower corner. Then we can compute a harmonic function u : G → ℝ with Dirichlet boundary condition, namely
In fact, the harmonic 1form ω is the exterior differential of u, ω = du.
Theorem 2: Suppose G is a topological quadrilateral which is a planar graph with an exterior face f_{0} and four corner vertices {v_{0}, v_{1}, v_{2}, v_{3}}. The harmonic 1form maps the topological quadrilateral to a rectangle with width w and height h, then the conformal module of the topological quadrilateral is
This theorem can be proven by an extremal length theory. More detailed proof is shown in [30].
So, there is a conformal mapping ϕ which maps a topological quadrilateral into a canonical rectangle and maps four vertices {v_{0}, v_{1}, v_{2}, v_{3}} into four corner vertices of the rectangle. The ratio of width and height is the conformal module. Four vertices {v_{0}, v_{1}, v_{2}, v_{3}} divide the boundary of the topological quadrilateral into four segments, denoted as Γ_{0}, Γ_{1}, Γ_{2}, Γ_{3} and ∂Γ_{k} = v_{k + 1} − v_{k}. Solving two Dirichlet equations, there are two harmonic functions f_{1}, f_{2} : G → R which satisfy the Dirichlet boundary and the Neumann boundary:
where n is the normal vector. The gradient fields ∇f_{1} and ∇f_{2} are mutually vertical lines but are not conjugate to each other. A constant λ is necessary for the conformal mapping ϕ with \( \phi =\lambda {f}_2+\sqrt{1}{f}_1 \). So λ is the width of the canonical rectangle. Due to the harmonic function with conformal invariant property, there is a mapping energy
and
4 Experimental results and discussions
All experiments are conducted on a Windows 7 platform, with a single 2.3GHz Intel CPU, 16GB RAM memory. All algorithms have been developed in generic C++, compiled using Visual Studio 2010. The sparse linear systems are solved using the numerical library Eigen [31].
The experimental work is to apply the combinatorial Hodge theory to compute the harmonic functions on the graph setting and use these harmonic functions to get the combinatorial conformal module of a given graph. The combinatorial conformal module can be viewed as the fingerprint of a graph. For each image, we make a feature graph for it. Based on the combinatorial conformal modules of graphs, matching work can be done by finding correspondences between two sets of features extracted from images. We test the proposed method in the following applications.
4.1 Viewpoint detection
Figure 3 illustrates the experiment for viewpoint detection by the combinatorial conformal modules of graphs. We use the house images with features from CMU house datasets. Although all the images are the same building with the same feature, the combinatorial structure of the Delaunay triangulation varies according to the viewpoint change. If the viewpoints are close to each other, the corresponding combinatorial structure of the feature graph is similar. We expect that the combinatorial conformal modules of the feature graphs with close viewpoints are close as well. We select four corners which are consistent among all the house images and separate the exterior face boundary into four segments, denoting as left, bottom, right, and top respectively. Applying our proposed method into these graphs, we can compute the conformal module of each image. Each image is associated with a conformal module, then the data is cast on the plane as shown in Fig. 4. It can be easily seen that there are three clusters of the conformal modules, which correspond to three major view positions. This means that the combinatorial conformal modules of the feature graphs with close viewpoints are close as well.
4.2 Posture detection
Another experiment is that we use CMU Graphics Lab Motion Capture Database for the application of the posture detection. The human actor wears 41 markers and makes a motion sequence. There are devices which capture the 3D position of the markers. In our experiment, we fix a virtual camera and project the markers onto the 2D image plane to analyze the result. Firstly, we use Delaunay triangulation to construct the graph. Then we compute the combinatorial conformal module of the graph. Four canonical postures are selected: standing, jumping, leaning back and walking as shown in Fig. 5. We project the posture onto the XOY,YOZ and ZOX plane respectively and calculate the corresponding Delaunay triangulation which is showed in the 3D posture scatter figures. We project the 3D posture onto the plane to find the most left vertex, the most bottom vertex, the most right vertex and the most top vertex as the corners. The proposed method is applied to compute the combinatorial conformal module of a posture graph. The combinatorial conformal module can be used to distinguish these postures. We use the proposed method to process the whole video sequence with 2751 frames and use Knearest neighbor method to classify the postures. The correct rate of posture classification result is nearly 90%.
5 Conclusion
Graph matching has been intensively applied in computer vision for finding correspondences between two sets of features extracted from images. Most of the existing graph matching methods require high time complexity and are not usable to large graphs. Furthermore, these methods can hardly handle the circumstances where two graphs to be matched have large nonrigid transformations. In this work, we propose a novel graph matching and classification method by the combinatorial conformal module.
We generalize the classical Hodge theory to the combinatorial graph setting. The process is that we apply combinatorial Hodge theory to compute the combinatorial harmonic functions on graphs. By defining combinatorial Laplace matrices on graphs, the solution of harmonic functions is converted into solving linear Laplace matrices. The combinatorial harmonic 1forms can give the combinatorial conformal invariant of a given graph, called as conformal module. According to the mathematical result, the conformal module can be used as the fingerprint of a given graph. Thus we can use the conformal module to do some matching work. Comparing to other graph matching methods, the proposed method can discover the combinatorial invariant of a given graph. Furthermore, the proposed method can tolerate the nonrigid transformation on a given graph. Due to considering the proposed method from the algebraic topology viewpoint, the computation of matching problem can be done in linear time. More importantly, our proposed method is adaptable to not only planar graphs but also graphs embedded in high genus surfaces.
In the future work, we purpose to explore the inexact graph matching method based on the conformal module and we hope the proposed method can be used to more applications in computer vision.
Abbreviations
 GM:

Graph matching
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Acknowledgements
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Jialing Zhang was born in Yunnan, P.R. China, in 1981. She received her Master of Science degree from Yunnan Normal University in 2006. She has been a faculty in Kunming University of Science and Technology since 2006. Her main research interests focus on Computational GeometryTheory and Applications.
Kun Qian was born in Yunnan, P.R. China, in 1982. He received his M.S. degree in Mechanical and Electronic Engineering from Xidian University in China in 2009. He was a visiting scholar at the State University of New York at Stony Brook and University of Louisiana at Lafayette, US from January 2015 to January 2016. He is currently pursuing his Ph.D. degree in Kunming University of Science and Technology in China. His research interests are in the area of computational geometry, computer graphics and computer vision.
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Zhang, J., Qian, K. Graph matching using conformal module. J Image Video Proc. 2019, 26 (2019). https://doi.org/10.1186/s136400190407x
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DOI: https://doi.org/10.1186/s136400190407x