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Assessing the Accuracy of Non-Rigid
Registration With and Without Ground Truth

R. S. Schestowitz^{1}, W. R. Crum^{2}, V. S. Petrovic^{1},
C. J. Twining^{1}, T. F. Cootes^{1} and C. J. Taylor^{1}


^{1}Imaging Science and Biomedical Engineering, University
of Manchester
Stopford Building, Oxford Road, Manchester M13 9PT,
United Kingdom


^{2}Centre for Medical Image Computing, Department of
Computer Science, University
College London, Gower Street, London WC1E 6BT, United Kingdom





We present two methods for assessing the performance of
non-rigid registration algorithms. One of them requires
ground-truth solutions, whereas the other does not need
any form of ground truth. The former method is based on
label overlap, which can be computed using Tanimoto's
formulation. The method which requires no ground truth
exploits the fact that, given a set of non-rigidly
registered images, a generative statistical appearance
model can be constructed. The quality of the model
depends on the quality of the registration, and can be
evaluated by comparing images sampled from it with the
original image set. We derive indices of model
specificity and generalisation, and show that they
demonstrate the loss of registration as a set of
correctly registered images is progressively perturbed.
We finally compare the two methods of assessment and
show that the latter method, which requires no ground
truth, is in fact more sensitive than the one that does.

Over the past few years, non-rigid registration (NRR)
has been used increasingly as a basis for medical image
analysis. Applications include structural analysis,
atlas matching and change analysis. Many different
approaches to NRR have been proposed, for registering
both pairs and groups of images. These differ in terms
of the objective function used to assess the degree of
mis-registration, the representation of spatial
deformation fields, and the approach to minimizing the
mis-registration with respect to the deformations. The
problem is highly under-constrained and, given a set of
images to be registered, each approach will, in
general, give a different result. This leads to a
requirement for methods of assessing the quality of registration.

Hereby we outline two methods for assessment, one of
which requires ground-truth solutions to be provided a
priori while the other does not. We shall present
results which confirm that both methods are valid and
proceed to calculating their sensitivities. We find
that the method which requires ground-truth solutions
is not as sensitive as the method which requires
nothing but the raw images and the corresponding
deformation fields, i.e. the registration.

The first among the methods to be described relies on the
existence of ground-truth data such as boundaries of
image structures, produced by manual markup of
distinguishable points. Having registered an image set,
the method can measure overlap between structures that
have been annotated, thereby implying how good a
registration was.

Our latter method is able to assess registration
without ground truth of any form. The approach involves
automatic construction of appearance models from the
registered data, subsequently evaluating, using model
syntheses, the quality of that model. Quality of the
registration is tightly-related to the quality of its
resulting model and the two tasks, namely model
construction and image registration, are innately the
same one. Both involve the identification of corresponding
points, also known as landmarks in the context of
model-building. Expressed differently, a registration
produces a dense set of correspondences and models
of appearance require the images and these
correspondences in order to be built.

To put the validity of both methods to the test, we
assembled a set of 2-D 38 MR images of the brain. Each
of these images was carefully annotated to identify
different compartments within the brain. These
anatomical compartments can be perceived as simplified
labels that faithfully define the structure of the brain. Our
first method of assessment uses the Tanimoto overlap
measure to calculate the degree to which labels across
the image set concur. In that respect, it exploits
ground truth, which has been identified by an expert,
to reason about registration quality.

The second method takes an entirely different approach.
It feeds on the results of a registration algorithm,
where correspondences have been highlighted, and builds
an appearance model given the images and their
correspondences. From that model, many synthetic brain
images are derived. Vectorisation of these images
allows us to embed them in a
high-dimensional space. We can then compare the spatial
cloud that these synthetic images form with the cloud
that gets composed from the original image set -- the set
from which the model has been build. Computing the
overlap between these clouds gives insight into the
quality of the registration. Simply put, it is a model
fit evaluation paradigm. The better the registration,
the greater the overlap between those clouds will be.

To compute overlap between two clouds of data, we have
devised measures that we refer to as Specificity and
Generalisablity. The former tells how well the model
fits its seminal data, whereas the latter tells how
well the data fits its derived model. It is a
reciprocal relationship that 'locks' data to its
model and vice versa. We calculate Specificity and
Generalisablity by measuring distances in space. As we
seek a distance measure that is tolerant to slight anatomical differences,
we use the shuffle distance, not neglecting to compare
it against Euclidean distance. The shuffle distance compares each point in one image with a larger corresponding region in another image. It adheres to the best fit, i.e. matches the two points whose distance is minimal.

Our assessment framework, by which we test both
methods, uses non-rigid registration, whereby many
degrees of freedom are involved in image
transformations. To systematically generate data over
which our hypotheses can be tested, we perturb the
brain data using clamped-plate splines, which are diffeomorphic. In the brain
data which we use, correspondences among images are said to be
perfect so they can only ever be degraded. We wish
to show that as the degree of perturbation increases,
so do the measures of our registration assessment methods.

In an extensive batch of experiments we perturbed the
datasets at progressively increasing levels, which led
to well-understood mis-registration of the data. We
repeated these experiments 10 times to demonstrate that
both approaches to assessment are consistent and
results are unbiased. Having investigated and plotted the
measures of overlap for each perturbation extent, we
see a rather linear decrease in the amount of overlap
(Figure X). This means that, when ground-truth-based
registration is eroded, the overlap-based measure is
able to detect that and the response is very
well-behaved, thus meaningful and reliable.

<Graphics file: ./Graphics/1.eps>
    <Graphics file: ./Graphics/2.eps>


          Figures X&Y. The measured quality of registration as perceived
          by the overlap-based evaluation (left) and the model-based
          evaluation (right).

We then undertake another assessment task, this time
exploiting the method which does not make use of ground truth.
We notice a very similar behaviour (Figure Y), which is
evidence that the latter is a powerful and reliable
method of assessing the degree of mis-registration -- or
conversely -- the quality of registration.

As a last step, we embark on the task of comparing the
two algorithms, identifying sensitivity as the factor
which is most important. Sensitivity reflects on our
ability to confidently tell apart a good registration
from a worse one. The slighter the difference which can
be detected reliably, the more sensitive the method.
To calculate sensitivity, we compute the amount of
change in terms of mean pixel displacement --
deviation from the correct solution, that is. We then
look at differences in our assessor's value, be it
overlap, or Specificity, or Generalisation. We also must
stress the need to take account of the errors bars as
there is both an inter-instantiation error and a
measure-specific error; the two must be composed
carefully. The derivation of sensitivity can be
expressed as follows:

placeholder

where X is... (TODO)

<Graphics file: ./Graphics/3.eps>


          Figure Z. The sensitivity of registration assessment methods.
          note to self: exclude Gen.? Combined? Plots?
          -.

Figure Z suggests that, for roughly any selection of
shuffle distance neighbourhood, the method which does
not require ground truth is more sensitive than the
method which depends on it. When the trends of these
curves are inspected closely, it can be observed that
they are approximately parallel, which implies that the two
methods are very closely correlated.

In summary, we have shown two valid methods for
assessing non-rigid registration. The methods are
correlated in practice, but the principles they build
upon are quite separable. Their pre-requisites -- if
any -- likewise. Registration can be evaluated with or
without ground-truth annotation and the behaviour our measures is consistent across distinct datasets, is
well-behaved, and is sensitive. Both methods have been
successfully applied to assessment of non-rigid
registration algorithms and both led to the
expected conclusions. That aspect of the work,
nonetheless, is beyond the scope of this paper.
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