3 IntroductionIn finding correspondences problem, several robust and optimal approaches have been developed by exploiting a prior geometric constraint.Typically this problem is formulated as an attributed graph matching problem where graph nodes represent the feature descriptors and edges represent the spatial relations between features.Enforcing consistency between the matches is formulate as a quadratic assignment problem.
4 IntroductionThis paper introduce a framework for feature matching among multiple sets of feature in one shot, where both the feature similarity in the descriptor space, as well as the local spatial geometry are enforced.Three contributions :Graph Matching through EmbeddingMatching Multiple sets in one shotScalability
6 Feature Embedding Framework Problem StatementWe are given K sets of feature points, in K images whereEach feature point is defined by its spatial location in its image plane and its feature descriptorFeature similarity and spatial structure must be preserved.
7 Feature Embedding Framework Problem StatementTo achieve a model that preserves these two constraints we introduce two data kernels based on the affinities in the spatial and descriptor domains separately.The spatial affinity in each image can be represented by a weight matrix whereThe feature affinity between image p and q can be represented by the weight matrix where.
8 Feature Embedding Framework Problem StatementLet denotes the embedding coordinate of pointfor each input feature setThe embedding should satisfy the following two constraints.The feature points from different point sets with high feature similarity should become close to each other.The spatial structure of each point set should be preserved in the embedding space.
9 Feature Embedding Framework Objective FunctionMinimize objective functionP. Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 2003.
10 Feature matching Matching Settings Pairwise Matching (PW) Given two sets of features, solve matching problems between two sets of embedding coordinates.Multiset Pairwise Matching (MP)Multiple sets of features and would like to find pairwisw matching between each pair of sets.Multiset Clustering (MC)multiple sets of feature points the unified embedding should bring correspondent features from different sets to be close to each other.
11 Feature MatchingHere used the Scott and Longuet-Higgins (SLH) algorithm as matching criterion in the embedding space.Define a matrix , represent the proximities between features in I and features in J.Orthogonal matrix P describe the one that correlate best with G.
12 Feature Matching Let F denote any orthogonal matrix. The indicate the matching pair between features and .
13 Feature and Spatial Affinities Spatial Structure WeightsIn this paper we used two different kinds of spatial weights.Euclidean-based weights : the weights are based on the Euclidean distances between features defined in each image coordinate system.Affine invariant-based weightSeveral kernels can be used to obtain the weights based on either the Euclidean or the affine including the Gaussian kernel and the Double exponential kernel
14 Feature and Spatial Affinities Feature WeightsThe feature weight matrix should reflect the feature to feature similarity in the descriptor space between the p-th and q-th sets.Solve for the feature weights in a way similar to SLH.Given the feature affinity G between features in sets p and q, we need to solve for a permutation matrix C that permutes the rows of G in order to maximize its trace.where
18 Experiments INRIA datasets Covers several effects such as viewpoint change, zooming, rotation, blurring and lighting change.
19 ConclusionThis paper shows that we can enforce spatial consistency for matching high-dimensional local appearance features in an efficient and scalable way.Since spacial structure is only measured within each set, there is no need to for quadratic edge consistency constraints. Therefore, the approach is linear and can scale to deal with large numbers of features.The approach can match multiple sets by solving a single eigenvalue problem.