Variational Image Registration for Inhomogeneous-Resolution Pairs

Abstract

We propose a variational image registration method for a pair of images with different resolutions. Traditional image registration methods match images assuming that the resolutions of the reference and target images are homogeneous. For the registration of inhomogeneous-resolution image pairs, we first introduce a resolution-conversion method to harmonise the resolution of a pair of images using the rational-order pyramid transform.Then, we develop a variational method for image registration using this resolution-conversion method.

Appendix: Proofs of Theorems 1, 2 and 3

Using the matrix expression of downsampling for vectors $\varvec{S}_q=\varvec{I}_n\otimes \varvec{e}_1^q$, where $\varvec{e}_1^q=(1,0,\cdots ,0)^\top \in \mathbf{R}^q$, the two-dimensional downsampling is expressed as $\varvec{G}=\varvec{S}_{q}\varvec{F}\varvec{S}_{q}^\top$. This expression derives the following lemma.

Lemma 1

Assuming that the domain of images is $\mathcal {L}\{\varvec{\varphi }_i\varvec{\varphi }_i^\top \}_{i,j=0}^{n-1}$, the range of images downsampled by factor q is $\mathcal {L}\{\varvec{\varphi }_i\varvec{\varphi }_j^\top \}_{i,j=0}^{\frac{1}{q}n-1}$.

The $2p+1$-dimensional diagonal matrix $\varvec{N}_p=\left( \left( n_{|i-j|} \right) \right)$, where $n_k=\frac{p-k}{p}$ for $0\le p \le k$. is expressed as $\varvec{N}_p=\sum _{k=0}^p a_k\varvec{D}_{n}^k,$ where $\varvec{D}_{n}^0=\varvec{I}_n$, for an appropriate collection of coefficients $\{a_k\}_{k=0}^p$. The linear interpolation for the two-dimensional image $\varvec{F}$ is $\varvec{N}_p\varvec{S}_p\varvec{F}(\varvec{N}_p\varvec{S}_p)^\top =\varvec{N}_p\varvec{S}_p\varvec{F}\varvec{S}_p^\top \varvec{N}_p^\top$. This expression implies the following lemma.

Lemma 2

Assuming that the domain of images is $\mathcal {L}\{\varvec{\varphi }_i\varvec{\varphi }_j^\top \}_{i,j=0}^{n-1}$ the range of images interpolated by order p is $\mathcal {L}\{\varvec{\varphi }_i\varvec{\varphi }_j^\top \}_{i,j=0}^{pn-1}$.

Furthermore, the pyramid transform of factor q is $\frac{1}{q^2}\varvec{S}_q\varvec{N}_q\varvec{F}\varvec{N}_q^\top \varvec{S}_q^\top$, since the pyramid transform is achieved by downsampling after shift-invariant smoothing. This expression of the pyramid transform implies the following lemma.

Lemma 3

With the Neumann boundary condition, the pyramid transform of order q is a linear transform from $\mathcal {L}\{\varvec{\varphi }_i\varvec{\varphi }_j^\top \}_{i,j=0}^{n-1}$ to $\mathcal {L}\{\varvec{\varphi }_i\varvec{\varphi }_i^\top \}_{i,j=0}^{\frac{1}{q}n-1}$, assuming $n=kq$.

Moreover, the matrix form of the $q\mathrm{{/}}p$-pyramid transform for the two-dimensional image $\varvec{F}$ is

$$\begin{aligned} \varvec{R}_{q/p}\varvec{F}\varvec{R}_{q/p}^\top&=\frac{1}{q^2}\varvec{S}_q\varvec{N}_{q/p}\varvec{S}_p \varvec{F} (\varvec{S}_q\varvec{N}_{q/p}\varvec{S}_p)^\top ,&\varvec{N}_{q/p}&=\varvec{N}_q\varvec{N}_p. \end{aligned}$$

(45)

This expression implies the following lemma.

Lemma 4

With the Neumann boundary condition, the $q\mathrm{{/}}p$ pyramid transform is a linear transform from $\mathcal {L}\{\varvec{\varphi }_i\varvec{\varphi }_j^\top \}_{i,j=0}^{n-1}$ to $\mathcal {L}\{\varvec{\varphi }_i\varvec{\varphi }_j^\top \}_{i,j=0}^{\frac{p}{q} n -1}$.