Multi-vector feature space based on pseudo-Euclidean space and oblique basis

Abstract

The Earth Mover’s Distance (EMD) and the quadratic-form distance (QFD) are representative distances used in similarity searches of images. Although the QFD greatly outperforms the EMD in speed, the EMD outperform the QFD in performance. The EMD, however, has almost no theoretical justification and requires high computation costs. We propose a feature space model we call a “multi-vector feature space based on pseudo-Euclidean space and an oblique basis (MVPO).” In MVPO, an object such as an image is represented by a vector set (roughly speaking, a solid) and the EMD is reinterpreted as the distance between vector sets while the QFD is reinterpreted as the distance between the centroids of vector sets. Therefore MVPO gives a common geometrical view to these distances. We hypothesized that in MVPO the entity of an image is represented by a vector set (solid) and geometrical reasoning is applicable to MVPO. Our hypothesis explains well that the EMD outperforms the QFD in performance because the centroid of a solid is the simplest approximation of it. Our hypothesis implies that the performance of the QFD should be good when solids are far apart but bad when they are close together. We conjectured that discriminability would decline—that is, dissimilar images would be judged to be similar—when the centroids of solids are very close. Our experiment supported this conjecture. And from our hypothesis we conjectured that by making an original solid simpler, we can make an approximation method that has better performance than the QFD and faster than the EMD. The results of our experiment with this method supported our conjecture and consequently our hypothesis.

Appendix

1.1 Determining an Oblique Basis

Instead of using the eigenvalues of the similarity matrix, we show a method based on simultaneous equations taking non-regular matrices into account.

• (1) Method for Regular Similarity Matrix.

  Let $T = {\left( {e_{1} ,e_{2} , \ldots ,e_{n} } \right)} = {\left( {\begin{array}{*{20}c} {{e_{{11}} }} & {{e_{{12}} }} & { \cdots } & {{e_{{1n}} }} \\ {0} & {{e_{{22}} }} & { \cdots } & {{e_{{2n}} }} \\ { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ {0} & {0} & { \cdots } & {{e_{{nn}} }} \\ \end{array} } \right)}$

  Then Conditions (C1) and (C3) are regarded as being simultaneous equations with the components of T, e ₁₁, e ₁₂···, e _nn . Given that e _ii  ≠ 0, these equations can be solved by determining e ₁₁, e ₁₂,···, e _1n , e ₂₂,···, e _2n ,···, e _nn in that order. The value of e _ij (i < j) and e _ii are as follows:

  $$e_{{ij}} = {{\left( {r_{{ij}} - {\sum\limits_{k = 1}^{i - 1} {e_{{ki}} e_{{kj}} } }} \right)}} \mathord{\left/ {\vphantom {{{\left( {r_{{ij}} - {\sum\limits_{k = 1}^{i - 1} {e_{{ki}} e_{{kj}} } }} \right)}} {e_{{ii}} }}} \right. \kern-\nulldelimiterspace} {e_{{ii}} }{\text{ and }}e_{{ii}} = {\sqrt {1 - {\sum\limits_{k = 1}^{i - 1} {e_{{ki}} ^{2} } }} }.$$

• (2) Oblique Basis for Any Similarity Matrix

  We assumed a similarity matrix is regular, but using a method similar to the one above, we can get a linearly independent oblique basis for any similarity matrix in 2n-dimensional pseudo-Euclidean space. In fact, the similarity matrix in the circular case is not regular.

  Let ${\mathbf{T}} = {\left( {e_{1} ,e_{2} , \ldots ,e_{n} } \right)} = {\left( {\begin{array}{*{20}c} {1} & {{e_{{12}} }} & { \cdots } & {{e_{{1n}} }} \\ {0} & {1} & { \cdots } & {{e_{{2n}} }} \\ { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ {0} & {0} & { \cdots } & {1} \\ {0} & {0} & { \cdots } & {0} \\ {0} & {{e_{{n + 2,2}} i}} & { \cdots } & {0} \\ { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ {0} & {0} & {0} & {{e_{{2n,n}} i}} \\ \end{array} } \right)}.$

  This matrix consists of two square sub-matrices. The upper one is an upper triangular matrix, and its diagonal components are 1. Therefore, e ₁, e ₂,..., e _n are obviously linearly independent. And the lower sub-matrix is a diagonal matrix whose diagonal components are imaginary numbers or 0. And using a method similar to the one above, we can determine an oblique basis. Here all the e _ij ’s obtained are real numbers.