A Novel Similarity Learning Method via Relative Comparison for Content-Based Medical Image Retrieval

Abstract

Nowadays, the huge volume of medical images represents an enormous challenge towards health-care organizations, as it is often hard for clinicians and researchers to manage, access, and share the image database easily. Content-based medical image retrieval (CBMIR) techniques are employed to facilitate the above process. It is known that a few concrete factors, including visual attributes extracted from images, measures encoding the similarity between images, user interaction, etc. play important roles in determining the retrieval performance. This paper concentrates on the similarity learning problem of CBMIR. A novel similarity learning paradigm is proposed via relative comparison, and a large database composed of 5,000 images is utilized to evaluate the retrieval performance. Extensive experimental results and comprehensive statistical analysis demonstrate the superiority of adopting the newly introduced learning paradigm, compared with several conventional supervised and semi-supervised similarity learning methods, in the presented CBMIR application.

Appendix

In Eq. 6, the second term $\frac {\text {exp}\big (2(\ell _{(q,y)} - \ell _{(q,x)})\big )-1}{\text {exp}\big (2(\ell _{(q,y)} - \ell _{(q,x)})\big )+1}$ of the right-hand side (RHS) of SKT can be viewed as a coefficient in the derivation, since it is not related to parameters a to learn in this study. Thus, the following derivation only focuses on the first term $\frac {\text {exp}\big (2(s_{(q,x)} - s_{(q,y)})\big )-1}{\text {exp}\big (2(s_{(q,x)} - s_{(q,y)})\big )+1}$ of RHS in SKT. For the ease of writing, let us denote the second term $\frac {\text {exp}\big (2(\ell _{(q,y)} - \ell _{(q,x)})\big )-1}{\text {exp}\big (2(\ell _{(q,y)} - \ell _{(q,x)})\big )+1}$ as term coeff. After differentiation, Eq. 6 becomes:

$$\begin{array}{rll}\lefteqn{\bigtriangledown \textrm{SKT}(a) =\frac{1}{N_{n}}\cdot\left(\sum\limits_{x,y \in D, x \neq y}\right.\frac{2\cdot\text{exp}\big(2(s_{(q,x)} - s_{(q,y)})\big)\big(s_{(q,x)} - s_{(q,y)}\big)^{\prime}\cdot\Big(\text{exp}\big(2(s_{(q,x)} - s_{(q,y)})\big)+1\Big)}{\Big(\text{exp}\big(2(s_{(q,x)} - s_{(q,y)})\big)+1\Big)^{2}}}\\&&{\kern83pt}\left.\frac{-2\cdot\text{exp}\big(2(s_{(q,x)} - s_{(q,y)})\big)\big(s_{(q,x)} - s_{(q,y)}\big)^{\prime}\cdot\Big(\text{exp}\big(2(s_{(q,x)} - s_{(q,y)})\big)-1\Big)}{\Big(\text{exp}\big(2(s_{(q,x)} - s_{(q,y)})\big)+1\Big)^{2}}\cdot\textrm{\textit{coeff}}\right)\\&&{\kern30.5pt}= \frac{1}{N_{n}}\cdot\left(\sum\limits_{x,y \in D, x \neq y}\frac{4\cdot\text{exp}\big(2(s_{(q,x)} - s_{(q,y)})\big)\big(s_{(q,x)} - s_{(q,y)}\big)^{\prime}}{\text{exp}^{2}\big(2(s_{(q,x)} - s_{(q,y)})\big)+2\text{exp}\big(2(s_{(q,x)} - s_{(q,y)})\big)+1}\cdot \textrm{\textit{coeff}}\right)\\&&{\kern30.5pt}=\frac{1}{N_{n}}\cdot\left(\sum\limits_{x,y \in D, x \neq y}\frac{4\cdot\big(s_{(q,x)} - s_{(q,y)}\big)^{\prime}}{\text{exp}\big(2(s_{(q,x)} - s_{(q,y)})\big)+\text{exp}\big(2(s_{(q,y)} - s_{(q,x)})\big)+2}\cdot \textrm{\textit{coeff}}\right)\end{array} $$

(11)

Then, after replacing the term coeff with its original mathematical form, the gradient can be rewritten as:

$$ \bigtriangledown \textrm{SKT}(a) =\frac{1}{N_{n}}\cdot\left(\sum\limits_{x,y \in D, x \neq y}\frac{4\cdot\left(\frac{\partial s_{(q,x)}}{\partial a} - \frac{\partial s_{(q,y)}}{\partial a}\right)\cdot\frac{\text{exp}\big(2(\ell_{(q,y)} - \ell_{(q,x)})\big)-1}{\text{exp}\big(2(\ell_{(q,y)} - \ell_{(q,x)})\big)+1}}{\text{exp}\big(2(s_{(q,x)} - s_{(q,y)})\big)+ \text{exp}\big(2(s_{(q,y)} - s_{(q,x)})\big)+2}\right) $$

(12)

which is the same as Eq. 7.