Fusion and comparison of images is an important task in image processing. Examples include motion correction in Dynamic Contrast Enhanced Magnetic Resonance Imaging (DCE-MRI) or reconstruction in Histological Serial Sectioning (HSS). These tasks require a suitable image similarity measure which takes application dependent image features into account. For example, in DCE-MRI or HSS the measure has to be invariant under intensity variations due to contrast uptake or staining artefacts, respectively.
For pairs of two images, a variety of options has been proposed and is well-understood. Among the various choices are
based measures, normalized gradient fields (NGF), mutual information or Kullback-Leibler divergence; see, e.g.[5, 6] and references therein. For scenarios where more than two images , , are to be aligned, the standard approach is to adapt the pairwise procedure in a sequential fashion. More precisely, the -th image is aligned with respect to the neighbours and for all . Since the correct aligned neighbours are yet unknown, the procedure has to be repeated until convergence.
This paper proposes an alternative so-called approach, which is based on the rank of image gradients of the whole sequence. A key feature of this new approach is that similarity information is made globally. The approach is inspired by work of Möllenhoff et. al.  and Haber et. al. . In , Schatten--Norms are used for color image denoising and in  local normalized gradient fields are introduced as for pairwise image registration. In our paper, is used as a data fitting term and globally normalized image gradients of the whole sequence with arbitrary many images are used as a starting point.
We derive and motivate the similarity measure. We also compare the performance to state-of-the-art measures on real life data. Our computations for a serial sectioning of a mouse brain indicate that results are qualitatively comparable to a sequential based registration but can be obtained about six times faster; see also .
2 The novel similarity measure SqN
The basic idea of our novel measure derives from color image denoising . There, the linear dependency of the gradients of the three color channels is used for regularization. This dependency is quantified by a Schatten--Norm . We generalize this idea and use it in the context of image registration. In particular, we extend the idea to an arbitrary number of images. Moreover, we also use normalized gradient fields rather than image gradients. Finally, we use the functional as a datafit rather than a regularizer.
The Schatten--Norm of a matrix is essentially the
-norm of the vector of its singular values. More precisely, for any
there exists a singular value decomposition (SVD), where denotes the -by-identity matrix, is a diagonal matrix with diagonal entries and . The Schatten--(quasi)-norm of is then defined as
We assume that dimensional images are given, where and the spatial dimension is denoted by . Following , regularized normalized gradients are given by
Here, the parameter discrimiantes signal from noise; see  for a discussion of choices of and discretization issues. Setting we define the new image similarity measure:
The registration model is to minimize , with and ; cf. .
3 Numerical Results and Discussion
We present results for a histological serial sectioning of a sectioned mouse brain, data courtesy of Oliver Schmitt. For results on DCE-MRI data we refer to . Fig. 1 shows results for a sequential linear pre-registration, the new based registration, and a standard sequential based registration as a reference (robust and fast to compute).
Although only one iteration was performed for the sequential registration, the computing time is about six times as for the approach. The result also shows a much stronger spatial correlation, indicating that the sequential approach has not yet converged.
Our future work addresses the optimal choice of (currently as in ) and the extension to 3D.
The authors acknowledge the financial support by the Federal Ministry of Education and Research of Germany in the framework of MED4D (project number 05M16FLA)
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