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Non-Iterative Hierarchical Registration for Medical Images

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Abstract

Digital images from diverse medical imaging modalities and from different imaging times are becoming an indispensable information resource for making clinical decisions. Image registration is an enabling technique for more fully utilizing the embedded heterogeneous image information. However, in addition to the complex differences and deformations inherent in the medical images, the increasing scope, resolution, and dimensionality of imaging pose significant challenges in this medical arena. Wavelets have shown great potential in multi-scale registration due to their superior capacity for representing image information at different resolutions and spatial frequencies. However, the application of wavelets in registration is hindered by their lack of rotation- and translation-invariance. To overcome this obstacle, this paper proposes a non-iterative hierarchical registration method based on points of interest which are extracted automatically from wavelet decompositions. The proposed algorithm for two-dimensional monomodal medical images has been validated by experiments on phantom data and clinical imaging data. This proposed non-iterative method provides a computationally efficient registration, as well as assists in avoiding the non-convergence problem.

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Acknowledgement

This work was supported by ARC and PolyU/UGC grants. The authors would like to express their appreciation to Dr. Ian Parkin for polishing the writing.

Author information

Authors and Affiliations

  1. Biomedical and Multimedia Information Technology (BMIT) Research Group, School of Information Technologies, University of Sydney, Sydney, Australia

    Xiuying Wang & David Dagan Feng

  2. Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong, China

    David Dagan Feng

Authors
  1. Xiuying Wang
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  2. David Dagan Feng
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Corresponding author

Correspondence to Xiuying Wang.

Appendices

Appendix 1: Wavelet Transform

The continuous wavelet transform of a function \(f\left( x \right) \in L^2 \left( \Re \right)\) can be defined as [20]:

$$\begin{array}{*{20}l}{\left( {W_\Psi f\left( x \right)} \right)\left( {a,b} \right) = \left\langle {f\left( x \right),\Psi _{ab} \left( x \right)} \right\rangle } \hfill \\{ = \frac{1}{{\sqrt {\left| a \right|} }}\int_\Re {f(x)\Psi \left( {\frac{{x - b}}{a}} \right)} dx} \hfill \\\end{array} $$
(1)

where the function ψ ab (x) defines the family of the wavelet functions with \(a,b \in \Re \); and \(a \ne 0\) is the dilation parameter; and b is the translation parameter. The definition also provides the similarity measurement of f(x) and wavelets ψab(x) obtained from mother wavelet ψ(x) by dilation and translation:

$$\Psi _{ab} \left( x \right) = \sqrt {2^{ - a} } \Psi \left( {2^{ - a} x - b} \right)$$

Similarly, the discrete wavelet of a one-dimensional discrete function f(n) is defined as:

$$\left( {W_\Psi f\left( n \right)} \right)\left( {j,k} \right) = \sum\limits_i {f\left( n \right)\Psi _{jk} \left( n \right)} $$
(2)

where \(\Psi _{jk} \left( n \right) = \sqrt {2^{ - j} } \Psi \left( {2^{ - j} n - k} \right)\); n = 2j + k for \(j = 0,1,...,\log _2 \left( N \right) - 1\), \(k = 0,1,...,2^j - 1\).

The mother wavelet ψ(x) can be constructed from a scaling function ϕ(x): \(\varphi \left( x \right) = \sqrt 2 \sum\limits_n {h_\varphi \left( n \right)} \varphi \left( {2x - n} \right)\) and \(\Psi \left( x \right) = \sqrt 2 \sum\limits_n {h_\Psi \left( n \right)} \varphi \left( {2x - n} \right)\) where \(h_\varphi \left( n \right)\) is the impulse response of a discrete filter, which needs to meet certain requirements to ensure the set of basis wavelet functions to be orthonormal and unique; while \(h_\psi \left( n \right)\) can be extracted from \(h_\varphi \left( n \right)\): \(h_\Psi \left( n \right) = \left( { - 1} \right)^n h_\varphi \left( {L - 1 - n} \right)\), where L is the length of \(h_\varphi \left( n \right) \cdot h_\varphi \left( n \right)\) and \(h_\psi \left( n \right)\) correspond to lowpass and highpass filters.

For fast discrete wavelet transform, the series of filtering and down-sampling operations used to compute \(W_\psi \left( {j,m,n} \right)\) and \(\left\{ {W_\Psi ^i \left( {\,j,m,n} \right)|i = D,V,H} \right\}\) can be mathematically expressed as:

$$\begin{array}{*{20}c} {W_{\Psi } {\left( {j,m,n} \right)} = h_{\varphi } {\left( m \right)}*{\left[ {h_{\varphi } {\left( n \right)}*W_{\Psi } {\left( {j - 1,m,n} \right)}|_{{n = 2k,k \geqslant 0}} } \right]}|_{{m = 2k,k \geqslant 0}} } \\ {W_{\Psi } ^{H} {\left( {j,m,n} \right)} = h_{\Psi } {\left( m \right)}*{\left[ {h_{\varphi } {\left( n \right)}*W_{\Psi } {\left( {j - 1,m,n} \right)}|_{{n = 2k,k \geqslant 0}} } \right]}|_{{m = 2k,k \geqslant 0}} } \\ {W_{\Psi } ^{V} {\left( {j,m,n} \right)} = h_{\varphi } {\left( m \right)}*{\left[ {h_{\Psi } {\left( n \right)}*W_{\Psi } {\left( {j - 1,m,n} \right)}|_{{n = 2k,k \geqslant 0}} } \right]}|_{{m = 2k,k \geqslant 0}} } \\ {W_{\Psi } ^{D} {\left( {j,m,n} \right)} = h_{\Psi } {\left( m \right)}*{\left[ {h_{\Psi } {\left( n \right)}*W_{\Psi } {\left( {j - 1,m,n} \right)}|_{{n = 2k,k \geqslant 0}} } \right]}|_{{m = 2k,k \geqslant 0}} } \\ \end{array} $$
(3)

where \(W_\psi \left( {j,m,n} \right)\) is approximation coefficients used to represent global (low frequency) information; \(W_\Psi ^H \left( {\,j,m,n} \right)\), \(W_\Psi ^D \left( {\,j,m,n} \right)\), and \(W_\Psi ^V \left( {\,j,m,n} \right)\) are horizontal coefficients, diagonal coefficients, and vertical coefficients respectively, which represent local (high frequency) information.

Appendix 2: Thin-plate Spline

The thin-plate spline interpolation was firstly introduced by Bookstein [32] into medical image registration area and since then, it has been one of commonly used elastic registration methods. Rohr gave a good study about elastic registration based on TPS [33].

The POI of reference image and study image are respectively \(P = \left\{ {\left. {p_i = \left( {x_i ,y_i } \right)} \right|i = 1,2,......n} \right\} \in I_R \) and \(Q = \left\{ {\left. {q_i = (x_i^\prime ,y_i^\prime )} \right|i = 1,2,......n} \right\} \in I_S \). To map the corresponding POI and to produce a smooth interpolation, a transformation function \(f:f\left( {\,p_i } \right) = q_i ,i = 1,2,...,n\) is to be determined to minimize the energy function E which reflects the amount of variation. The regulation parameter λ > 0 is used to control the fitness. The energy function can be expressed as:

$$\begin{array}{*{20}l}{E\left( f \right) = \sum\limits_{i = 1}^n {\left\| {q_i - \left. {f\left( {p_i } \right)} \right\|} \right.} ^2 } \hfill \\{ + \lambda \iint\limits_\infty {\left( {\left( {\frac{{\partial ^2 f}}{{\partial x^2 }}} \right)^2 + 2\left( {\frac{{\partial ^2 f}}{{\partial x\partial y}}} \right)^2 + \left( {\frac{{\partial ^2 f}}{{\partial y^2 }}} \right)^2 } \right))dxdy}} \hfill \\\end{array} $$
(4)

The thin-plate spline interpolation function can be written as:

$$f\left( {x,y} \right) = \sum\limits_{k = 1}^3 {a_k } \phi _k \left( t \right) + \sum\limits_{i = 1}^n {w_i U\left( {\left\| {\left( {x,y} \right) - p_i } \right\|} \right)} $$
(5)

where the coefficients a 1, a 2 and a 3 define the affine part of the transformation, and \(\left[ {\phi _1 \left( t \right),\phi _2 \left( t \right),\phi _3 \left( t \right)} \right] = \left[ {1,x,y} \right]\); whereas the coefficient w defines the elastic deformation, and \(U\left( {\left\| {\left( {x,y} \right) - p_i } \right\|} \right) = \left\| {\left( {x,y} \right) - p_i } \right\|^2 \log \left( {\left\| {\left( {x,y} \right) - p_i } \right\|} \right)\).

In order to keep f(x,y) having square integrable second derivatives, the following conditions must be satisfied: \(\sum\limits_{i = 1}^n {w_i = 0} \) and \(\sum\limits_{i = 1}^n {w_i x_i = } \sum\limits_{i = 1}^n {w_i y_i = 0} \).

The coefficient vector a = (a 1,a 2,a 3)T and w = (w 1,w 2,...w n )T can be computed through the following linear equations:

$$\left\{ {\begin{array}{*{20}c}{kw + Pa = v} \\{P^T w = 0} \\\end{array} } \right.$$
(6)

where v represents column vectors of landmarks; \(k_{ij} = U_i \left( {p_j } \right) = U\left( {\left\| {\left( {x_i ,y_i } \right) - \left( {x_j ,y_j } \right)} \right\|} \right)\); and (1,x,y) is the ith row in P. These two vector equations can be solved by:

$$\left\{ {\begin{array}{*{20}c}{w = K^{ - 1} (v - Pw){\text{ }}} \\{a = (P^T K^{ - 1} P)^{ - 1} P^T K^{ - 1} v} \\\end{array} } \right.$$
(7)

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Wang, X., Feng, D.D. Non-Iterative Hierarchical Registration for Medical Images. J Sign Process Syst Sign Image Video Technol 54, 65–77 (2009). https://doi.org/10.1007/s11265-008-0183-9

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