State-of-the-Art Medical Image Registration Methodologies: A Survey

Abstract

Almost all computer vision applications, from remote sensing and cartography to medical imaging and biometrics, use image registration or alignment techniques that establish spatial correspondence (one-to-one mapping) between two or more images. These images depict either one planar (2-D) or volumetric (3-D) scene or several such scenes and can be taken at different times, from various viewpoints, and/or by multiple sensors. In medical image processing and analysis, the image registration is instrumental for clinical diagnosis and therapy planning, e.g., to follow disease progression and/or response to treatment, or integrate information from different sources/modalities to form more detailed descriptions of anatomical objects-of-interest. The unified registration goal – aligning a 2-D or 3-D target (sensed) image with a reference image – is reached by specifying a mathematical model of image transformations for and determining model parameters of the desired alignment. Frequently, the parameters provide an optimum of a goal function supported by the parameter space, so that the registration reduces to a certain optimization problem. This chapter overviews the 2-D and the 3-D medical image registration with special reference to the state-of-the-art robust techniques proposed for the last decade and discusses their advantages, drawbacks, and practical implementations.

Appendix A: List of Symbols

${I_{\rm{r}}}$ :

Reference image.

${I_{\rm{t}}}$ :

Target image.

${T_{\rm{g}}}( \cdot )$ :

Transformation function.

$\rho ( \cdot )$ :

Cost function.

${\mathbf{p}}$ :

Spatial coordinates vector.

${ \tilde{\mathbf{p}}}$ :

Homogeneous coordinates vector.

$\mu$ :

Contrast deviation factor.

$\Gamma ( \cdot )$ :

Random noise.

${\mathbf{W}}$ :

Rectangular window or neighborhood system (regular and irregular).

${\bar I_{\rm{r}}}$ :

Reference image mean value.

${\bar I_{\rm{t}}}$ :

Target image mean value.

$F(.,.)$ :

2-D Fourier transform.

${\hbox{CP}}{{\hbox{S}}_{{F_1},{F_2}}}$ :

Normalized cross-power spectrum.

${\mathbf{X}}$ and ${\mathbf{Y}}$ :

Finite signal sets.

$\phi$ and $\psi$ :

One-to-one mappings.

${p_i}$ and ${q_j}$ :

Marginal probability distributions.

${p_{ij}}$ :

Joint probability distribution of two random variables.

${p_{i|j}}$ :

Conditional probability distribution.

$H( \cdot )$ :

Shannon Entropy.

$H( \cdot | \cdot )$ :

Conditional Entropy.

$H\left( { \cdot, \cdot } \right)$ :

Joint Entropy.

${\mathbf{\Delta }} = ({\delta_x},{\delta_y},{\delta_z})$ :

Spatial offsets vector.

$E( \cdot )$ :

Gibbs energy.

${{\mathbf{V}}_{\mathbf{\Delta }}}$ :

Potential function.

${{\mathbf{F}}_{\mathbf{\Delta }}}$ :

Empirical probability of signal co-occurrences in the MGRF clique Family.

$\lambda$ :

Relative cardinality of the MGRF model.

$\tau$ :

Arbitrary scale factor.

$\theta$ :

Angel in radians.

${\mathbf{\alpha }} = ({\alpha_x},{\alpha_y},{\alpha_z})$ :

Scaling vector.

${\zeta_x}$ and ${\zeta_y}$ :

$x$and $y$-Shearing Factors.

$\begin{array}{llll} {\mathbf{a}} = ({a_{11}},{a_{12}},{a_{13}},{a_{14}},\ldots,\\ {a_{33}},{a_{34}}),\\ {\mathbf{b}} = ({b_{11}},{b_{12}},{b_{13}}, \ldots,{b_{23}}),\,{\hbox{and}}\,\\ {\mathbf{c}} = ({c_{11}},{c_{12}},{c_{21}},{c_{22}}) \end{array}$ :

Rigid transformation coefficients vectors.

$\begin{array}{llll}{\mathbf{A}} = ({a_{00}},{a_{10}},{a_{01}},\\ {a_{02}},{a_{20}}, {a_{03}},\\ {a_{000}},{a_{100}},{a_{010}},{a_{001}}),\\ {\mathbf{B}} = ({b_{00}},{b_{10}},{b_{01}},{b_{02}},{b_{20}},{b_{03}}) \end{array}$ :

Global polynomial and spline numerical coefficients.

${F_k}$ :

Spline distance weight coefficients.

${\eta^2}$ :

Spline stiffness coefficient.

$r$ :

Cartesian distance between two points.

${\mathbf{\Phi }}$ :

Control points lattice (mesh).

$\gamma$ :

Lattice (mesh) spacing.

$N$ :

Number of control points.

${\mathbf{\beta }} = \left( {{\beta_{ - 1}},{\beta_0},{\beta_1},{\beta_2}} \right)$ :

uniform cubic B-spline basis functions.

${R_k}(.,.)$ :

Radial basis function.

$\sigma { }$ :

Standard deviation.

${\mathbf{N}}$,${\mathbf{N^\prime}}$ :

Reference and target image planes (volumes).

${\beta^{\left[ n \right]}}\left( \cdot \right)$ :

$n$-order B-spline.