doi:10.1016/j.imavis.2006.12.009 
Copyright © 2007 Elsevier B.V. All rights reserved.
A minimum description length objective function for groupwise non-rigid image registration
aInstitute of Information Sciences, Massey University, Private Bag 11222, Palmerston North, New Zealand
bImaging Science and Biomedical Engineering (ISBE), Stopford Building, University of Manchester, Manchester M13 9PL, UK
Received 18 February 2005;
revised 26 April 2006;
accepted 8 December 2006.
Available online 19 December 2006.
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Abstract
Non-rigid registration finds a dense correspondence between a pair of images, so that analogous structures in the two images are aligned. While this is sufficient for atlas comparisons, in order for registration to be an aid to diagnosis, registrations need to be performed on a set of images. In this paper, we describe an objective function that can be used for this groupwise registration. We view the problem of image registration as one of learning correspondences from a set of exemplar images (the registration set), and derive a minimum description length (MDL) objective function.
We give a brief description of the MDL approach as applied to transmitting both single images and sets of images, and show that the concept of a reference image (which is central to defining a consistent correspondence across a set of images) appears naturally as a valid model choice in the MDL approach.
In this paper, we demonstrate both rigid and non-rigid groupwise registration using our MDL objective function on two-dimensional T1 MR images of the human brain, and show that we obtain a sensible alignment. The extension to the multi-modal case is also discussed. We conclude with a discussion as to how the MDL principle can be extended to include other encoding models than those we present here.
Keywords: Image registration; Non-rigid registration; Groupwise registration; Minimum description length (MDL)
Fig. 1. (Left) A consistent correspondence (solid arrows) between a set of images for a single point. (Right) A consistent correspondence (dashed arrows) generated via correspondence to a reference (solid arrows).
Fig. 2. From the left: original shape, rotated shape with correct correspondence (as indicated by point size and colour), rotated shape with incorrect correspondence.
Fig. 3. Graphs showing description length as a function of δσ for three datasets with different variances, with N = 50. Crosses: the exact description length (Eq. (10)), with the minimum circled; solid line: the continuum approximation (Eq. (19)), with the position of the minimum shown by the dashed line.
Fig. 4. (Top row) The description lengths per pixel for a set of images, encoded using the two different models, optimised Gaussian: grey crosses; empirical distribution: black circles. (Middle row) Thumbnails of the images with image dimensions in pixels. (Bottom row) The centred image histograms, all to the same scale.
Fig. 5. The original image with various amounts of Gaussian white noise.
Fig. 6. The ratio of description lengths for transmitting an image set with and without using an 8-bit reference image, as a function of the number of images in the set, for varying values of the noise variance. Values of the ratio which are less than one show that using a reference produces a smaller description length than not using a reference image.
Fig. 7. The ratio of description lengths for transmitting a set of images with and without using a reference image, as a function of the number of grey levels in the reference, for varying numbers of images in the set (ns). The minimum point of each graph is circled.
Fig. 8. The set of transformations between reference and image frames.
Fig. 9. Rigid groupwise registration. (Top row) The group of five images to be aligned (translated versions of the same image). (Second row) The description length in nats divided by the total number of pixels in the group of images as a function of iteration number. (Bottom two rows) The mean/reference image at each iteration.
Fig. 10. Non-rigid registration. (Top row) The group of five images to be aligned, with the reference image knotpoints positions superimposed. (Second row) The description length in nats divided by the total number of pixels in the group of images as a function of iteration number. (Bottom row) The mean/reference image at the start, and at the 2nd, 4th, 6th, 8th, and 10th iterations.
Fig. 11. (Top row) The set of training images. (Other rows) The reference image as the registration progresses, with the value of the objective function (the total description length for the set in nats).
Fig. 12. The mean and median of the aligned training set from Fig. 11 compared to the seed (original) image. The value of the total description length for the two choices of reference is given below the image.
Fig. 13. Non-rigid registration. (Top row) The set of training images. (Other rows) The reference image as the registration progresses, with the value of the objective function (the total description length for the set in nats divided by the total number of pixels in all the images). The graph shows the objective function decreasing as the registration progresses.
Fig. 14. (Top row) The two seed images, and the absolute difference between them. (Bottom row) The reference images for the two subsets, and the combined set, with the total description length in nats.
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