- ... subjects
^{1.1} - From this point onwards, there will be a clear focus on 2-D imaging
of the anatomical. This narrower view can be considered a case study
for image registration and it allows descriptions to be easier to
follow.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... MRI
^{1.2} - See list of acronyms and abbreviation in the Appendix cha:Appendix-EList-of.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... object
^{1.3} - The word ``object'' will from here onwards refer to a structure
of interest in -dimensional space.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... aperture
^{1.4} - In the case of medical imaging, there are even more factors to be
considered, as opposed to a camera's aperture.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... rigid
^{1.5} - ``Rigid'' refers to constrained variability and low model generalisability
as explained later. It is significantly different from the term ``rigid''
in the actual context of registration.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Arguably
^{1.6} - Ideas such as this are overly optimistic perhaps.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... scientific
^{1.7} - This text is goal-oriented and it embraces the technical, not much
of the inter-personal and curricular.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
resources
^{2.1} - All Web resources are listed at the end of this report.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... data
^{2.2} - The chapter considers images to be the default case. These methods
are usually operable over an arbitrary number of dimensions, but 2-D
proves to be easier for a reader (and the paper) to visualise.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
posed
^{2.3} - A bottom-up approach will look at low-level data and build up towards
knowledge of higher complexity which has a meaning. Top-down is an
opposite approach which 'knows' what it tries to find so it searches
for a best lower-level match.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... blow-up
^{2.4} - Current model-based methods typically deal with only the order of
tens of thousands of pixels. High-resolution medical images can contain
millions of pixels.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... contents
^{2.5} - In most cases, edge detection is sufficient to capture regions or
points of greater significance in the image. Edges and corners usually
hold more information of use for subsequent analysis and aid segmentation.
They lead to better identification of the different objects residing
in the image.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... cases
^{2.6} - Contrarily, analysis of mammograms needs to account for texture as
well as shape. The boundaries of a breast are not sufficient to characterise
the distinguishable data that is of most interest.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... range
^{2.7} - The legal range can be thought of as the values a parameters may take.
In reality, a Gaussian distribution usually fits the observed range
rather well.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... grey-level
^{2.8} - Colour can be simply thought of as an extension of the single grey-scale
band being divided up into red, green and blue components. There are
different possible colour schemes [] which have no affect
on the actual principle of intensity sampling.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... models
^{2.9} - In principle, they (appearance models) can be made just as powerful,
but in practice they suffer from requirements for high speed. As this
text shall later explain, they can sometimes lead algorithms to getting
trapped in local minima.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... superset
^{2.10} - They can be thought of as a superset, simply being shape models which
hold some additional information and the correlations between all
encapsulated data.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... variation
^{2.11} - Offsets of standard deviation units from the mean of each mode then
illustrate the effect each variation mode has.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... above
^{2.12} - The process which was proposed by Manfred Eigen allows the calculation
of Eigen-vectors and Eigen-values. For a given matrix, Eigen-vectors
describe directions in space that are derived from the matrix and
the corresponding Eigen-values describe their magnitude.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Component
^{2.13} - Plainly speaking, PCA only picks up Eigen-values whose Eigen-values
are the greatest.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... space
^{2.14} - Normalisation step as such is similar to the mapping onto a sphere,
for instance.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... shape
^{2.15} - Oftentimes, the choice of the mean shape proves to be the least damaging
choice.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... patch
^{2.16} - It is helpful to think of two different triangles and the relationship
between points within these triangles. Centre of gravity (COG) is
used here to assign approximate correspondence.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... spot
^{2.17} - Analogically, in the case of shape, sharp-bended descriptors result
from the low number of sample points.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... appearances
^{2.18} - One of the main aims and great power of appearance models is full
synthesised portrayal, so photo-realism is at a premium.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... greatest
^{2.19} - If one thinks of the cloud in dimensional space as a placement
of characteristics
, the principal component
is one characteristic which best separates instances of the data.
It takes the largest range of variation. To simplify the concept,
it can be assistive to think of a standard keyboard. The number of
key will poorly distinguish one keyboard from another, but since the
names and labels of manufacturer are diverse, this may as well be
the one 'principal component'.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... for
^{2.20} - A common choice is 98% of the observed variation which means 2%
of the variation is not accounted for. This 2% of variation is usually
the least informative though - being exactly what PCA is intended
to accomplish.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{2.21} - As an ad-hoc example, intensity frequently takes values in the range
whereas normalised shape coordinates lies between 0 and
1 so fractions such as can be used as coefficients.
The two should then scale almost indifferently.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
pixel
^{2.22} - For colour it is common to use 24 bits and for grey-level just 8 bits.
For more compact statistical appearance models, less than 8 bits (256
shades of grey) might suffice to achieve good results and in medical
imaging 12 bits are nearly a standard in acquisition.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... image
^{2.23} - The distinction here is hard because the model can describe more than
one valid independent object and usually represents only a partial
section of the entire image. In a medical context, the term
*atlas*fits somewhat more nicely and it usually describes a single organ or anatomical structure.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... set
^{2.24} - There is a subtlety which makes this phrasing a bit deceiving and
inaccurate. The word ``range'' is a gross terminologically equivalent
to the area that stretches in between the space of training set instances.
It can be conceived as the space defined by a Gaussian distribution
cloud that is deduced from the training set.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... model
^{2.25} - There are some more complex considerations as the model needs to be
aligned properly as well as change in its form.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... intensity
^{2.26} - A simple raster scan that account for all pixels should clearly be
fast under most contemporary computer architectures. This is indeed
the case if simple mundane operations like subtractions are pipelined
on the ALU.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... negligible
^{2.27} - This is reminiscent of the need for a median measure, where average
is sensitive to erratic values or salt-and-pepper noise.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... effect
^{2.28} - It is possible to learn the properties of rotation, as an exemplar,
by applying a rotation and looking at the difference between the resulting
image and the original image. That is the main concept that this step
is based upon, namely inferring
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... not
^{2.29} - The vector's distribution of values, i.e. positions with high absolute
values, can answer this question quite grossly.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... matrix
^{2.30} - The matrix can be obtained using linear regression.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... problem
^{2.31} - The problem is nearly identical owing to one basic assumption - the
assumption that similar objects are examined with some known pattern
of model placement in the target image. The location of mismatches
(indicated by high difference values, i.e. white) tend to show where
supplemental model deformation is yet necessary.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... target
^{2.32} - This process of fitting strives to converge to the global minimum
(of difference measure). Realistically speaking, the model and the
target never reach complete equivalence, namely the difference value
of absolute 0. Even if the target was used to train the model, PCA
would corrupt the obscure the connection between the two.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... lie
^{2.33} - Advanced knowledge about the problem is highly conductive at this
stage, otherwise some bottom-up image analysis is a must.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... off-line
^{2.34} - If these are not available, some guessing would be an alternative.
It is important, however, to learn from the experience gained during
this independent run of the program or else the optimisation would
behave senselessly and lead to improvements being identified very
slowly. General optimisers ought to make a good judgement as such.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... compact
^{2.35} - A sparse collection of pixels (or voxels) can be encoded using a lossy
function with an even smaller number of parameters.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
down
^{2.36} - When models break down, fitting defaults to a local (and hence false)
minimum.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
synthesising
^{2.37} - This can be considered as being a reversal of interpretation, in fact.
This binds with the notable computer vision/graphics differentiation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... lower
^{2.38} - Although some results support this claim, it is quite likely that
better implementations and further improvements will prove otherwise.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... ASM
^{2.39} - This is
*not*necessarily so in the like-for-like comparison.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{3.1} - The names sometimes change in the literature despite standardisation.
What is important is the description of transformations and not the
names or mnemonics that wound up describing them.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{3.2} - More strictly, the inclusion of scaling makes this a similarity transformation
rather than rigid.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{3.3} - Such a process is a very fundamental one in computer graphics modelling
and various books cover shape-normalisation techniques and algorithms.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... values
^{3.4} - 1 value for scaling, 3 for
*x, y*and*z*coordinates and 2 for rotation, e.g. the*xy*and*yz*angles and .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{3.5} - Popular transformations such as skew, shear and taper, on the contrary,
are not parallelism-preserving. The importance of this rigorous constraint
is that the distance between any two points remains proportional to
the transformation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... structure
^{3.6} - A random uncontrollable transformation will dispart basic structures
in the image and can make valid interpretation impossible.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... issue
^{3.7} - This includes the Structure and Function Grand Challenge. The Grand
Challenge aims to unify the different stages of analysis. It will
be referred to yet again in Chapter which
deals with recent and on-going work, including that on non-rigid registration.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... image
^{3.8} - More generally, arbitrary data of any complexity should be applicable.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... transformations
^{3.9} - This so-called mapping or transformation can be thought of as being
a standard function, for example
in 2-D and it is
applied to all the pixels within a predefined range.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... image
^{3.10} - More generally, the functions are mappings defined over a matrix or
a vector which is analogous to an image.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... effect
^{3.11} - A pixel of course can be mapped onto the exact same original position,
but the idea is that a continuous flow must prevail.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... shapes
^{3.12} - The principles are better described by borrowing some concepts from
work on landmark selection in shapes (to be further seen in Section
). Similar methods as applied to images have
not thoroughly been investigated yet.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... reference
^{3.13} - This can be thought of as a space which defines a common (non-linear)
grid. In this grid, mappings between corresponding points become clearer,
visually.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
points
^{3.14} - A continuous function is independent of the number of points. Therefore,
the complexity can be increased progressively to obtain finer, more
accurate results.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... discussion
^{3.15} - In Manchester University, Cootes and others are in favour of many
small warps, but some are in favour of few rather more complex warps
that are controlled by a larger number of parameters. More details
on such issues appear in later discussions on current work on page
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... space
^{3.16} - One can think of images as a vector of pixel values that define a
position in a high-dimensional space
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... way
^{3.17} - The discovery of this mutual information is actually attributed to
Maes as well. The thesis worked on by Viola in the mid-nineties received
great recognition though and MI is ascribed to him.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... literature
^{3.18} - There is an additional distinction between symmetric and asymmetric
normalised mutual information, but rationalè for this requires the
full technical recipe. The dissertation at http://www.lans.ece.utexas.edu/ strehl/diss/node107.html
summarises the way in which NMI evaluated.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... sum
^{3.19} - One could suggest an extension to such a method and assign weights
to differentiate regions of varying significance.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{4.1} - Binary representation is quite complete in the sense that any data,
e.g. programs and text, can be coded in a binary form. However, this
representation might be very greedy of space and the issue of representation
compactness then arises.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... correlation
^{4.2} - To make this appear more practical, one can think of a large (
pixels) image where patterns are present.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... these
^{4.3} - This can be portrayed as a uniform plain-white image.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... sequence
^{4.4} - General problem reducibility to a sequence is axiomatic as Turing
Machines suggest.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... landmarks
^{5.1} - Often the choice is random so that no assumption are made about the
problem.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... message
^{5.2} - An alternative method involving B-fitting was proposed by Thacker
*et al.*[]*.*. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... similarity
^{5.3} - It will temporarily be assumed that for an objective function that
needs to be minimised, the similarity measure will return small values
for good similarity and vice versa.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{5.4} - Being slightly more specific, this function is said to minimise the
sum of the difference between two images and another less significant
term. The two images compared are the transformed image
and the reference
in this case.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... high
^{5.5} - The behaviour of such a problem is not linear and it may cross over
to the realms of quadratic programming (QP) where various parameters
simultaneously control a function and minimisation is therefore by
no means trivial.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... exist
^{5.6} - To name several more methods: dynamic programming, genetic algorithms,
Powell's, simulated annealing and steepest descent.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... axes
^{5.7} - Optimisation is a multi-dimensional problem that searches along hyper-spaces,
some of which are orthogonal to the many existing axes.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... search
^{5.8} - Exhaustivity is impossible for continuous functions, but digital images
are luckily discrete.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... example
^{6.1} - There was also a further investigation into the optimisation scheme.
All shapes can be optimised over simultaneously (also known as joint
optimisation) or one can be optimised at any single iteration (known
as sequential optimisation).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
data
^{6.2} - The data type is irrelevant. It makes no difference whether it is
an image of full appearance or just a 'brick-and-bump' as was repeatedly
the case.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... search
^{6.3} - This similarity computation is incorporated in the objective function
and it usually comprises a collection of pair-wise similarity measures.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... involved
^{6.4} - Although knowledge of the problem is an integral part of most program
optimisation steps, the more formal methods can be used to identify
dependencies. A dependency graph can reliably indicate where re-evaluation
is indeed necessary (e.g. Figure cap:AART-dependencies_struct).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... well-integrated
^{6.5} - The parallel development in both fields, especially the need to identify
homologous structures, is what makes this GC suitably arranged and
increases its potential of resulting in success.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
investigated
^{7.1} - Issues which yet cannot be ignored are related to efficiency. By scaling
down the problem though, proof of method appropriateness is both possible
and traceable.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
online
^{7.2} - The author is aware that Web references and expansions through remote
text are frowned upon. However, these are collectively an 'open door'
to the large majority of work (some 2,000+ files); much of it is completely
omitted from this report.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... operates
^{7.3} - On a separate note on knowledge- and code-sharing, GUI components
and demos were made available at MATLAB Central. They received nearly
4,000 downloads, accredited to the author and his affiliation with
ISBE and Manchester University. These contributions ranked him amongst
the world's top 5 for popularity in July 2004. Confer [WWW-17]
for more information.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... laid
^{7.4} - The user interface is courtesy of Java and it runs over a Java virtual
machine (this can be seen as either having pros or cons).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... enclosed
^{7.5} - It seems likely that a CD-ROM will accompany later surveys (a la thesis).
AART can generate several movie types with little user involvement.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... bump
^{7.6} - This is rather a different type of data than the one mentioned in
past work (Appendix where
bumps are less composite) and that which has been tested in MDL shape
optimisation (Chapter where brick topped by
a bump is looked at).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... identify
^{7.7} - In real-life circumstances, there will rarely be a correct solution
for inter-subject registration. There may, however, be one for intra-subject
registration, e.g. in the case of correction for movement.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... single-point
^{7.8} - This refers to the number of knot-points that are involved in the
calculation of the transformation. A single point fully describes
the Green's function which CPS builds upon.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... model
^{7.9} - The fact that shape component was chosen to be the reparameterisation
curve has not been enlightened yet.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... data
^{7.10} - The data was generated by extending the 1-D bump data generator. It
is
*not*a Gaussian.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... implementations
^{7.11} - More cunning implementations would have involved better 'dialogue'
between the similarity measures and the warps chosen, for example.

Rather than that, each of the two components was treated as a black box, fully independent from the other.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... direction
^{7.12} - This idea is borrowed from technical analysis in finance. It can be
useful in science as well.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... terminology
^{7.13} - In AART, this definition of iteration is repeatedly referred to as
*warping step/s*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
instances
^{7.14} - As sub:The-MDL-based-Objective explains, an extra term, epsilon,
is used to refrain from multiplication by 0. Due to the finite precision
of digital systems, Eigen-values may be assigned this zero measure.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
linear
^{7.15} - The curves were all going from the bottom left to the top right corner,
meaning that each point mapped onto itself and no changes were made.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Eigen-analysis
^{7.16} - There is a serious flaw present because this analysis is cubic in
its complexity. Profiling is yet to be considered an option for improvements
discovery.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... distances
^{7.17} - This measure is better immune to lage local misalignment. This is
similar to arguments presented in Equation eq: delta start.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... resolution
^{7.18} - The number of pixels that data comprises is an argument that may be
changed.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... domain
^{7.19} - This domain is the selection of landmarks in shapes for the construction
of statistical models of shape. Modification of this will be described
later.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... handled
^{7.20} - Examples of human hands were later tested as well . They were an easier
case that is quicker to reach convergence.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... dynamic
^{7.21} - As an example, the famous travelling salesman problem speaks of a
pre-set value for each edge in a graph. This means that the problem
never changes. What if the values of edges changed for each choice
of a path? This renowned problem would then become less workable then
it has become. Each choice then introduces a new, yet unknown, optimisation
problem.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... modern
^{9.1} - Usually Pentium 4 processor with 256 Megabyte of random access memory
(RAM). These were not computational servers in essence.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
Report
^{10.1} - The literature report is located at:

http://www.danielsorogon.com/Webmaster/Research/Literature_Report. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... non-specific
^{10.2} - These requirements are intentionally very general. They can be applicable
to most computer-scientific research.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... database
^{10.3} - Actually, these are hierarchical nested indices of experiments, sorted
chronologically.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... document
^{10.4} - These have faithfully and quickly served the need to produce figures
for this report, as well as some of the results which were earlier
outlined. In fact, the experiments pages can serve as a document of
progress in their raw state. They might require some additional annotation
to be legible.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... times
^{10.5} - It was discovered, for instance, that values returned by the optimiser
can be lower than the preliminary input values. That suggested that
evaluations can be exacerbated as registration was performed and additional
code was composed to resolve this.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... selection
^{10.6} - Most recently it turned out to have been complicated. Solution are
agreed upon in present days and will soon be implemented and devised.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{10.7} - This has been work in progress since early June 2004.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... follows
^{10.8} - Note that where work has been done already, suitable explanations
and examples were provided a previous chapter (Chapter )
on experiments and results.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... descent
^{10.9} - This excludes the start when subsets require time to stabilise by
preliminary warps.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
recognition
^{A.1} - Much of the popularity of this method has been imputed to face recognition
tasks.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... slow
^{A.2} - The algorithms currently used for demonstration purposes take 3 days
to run, but substantial speed-up is expected soon.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... diffeomorphic
^{B.1} - While the bump may have its form tweaked and manipulated, its highest
peak should be preserved although it may move leftward or rightward.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... matrix
^{B.2} - This will indicate the
*volume*of the model's scatter in space. The more compact a model appears, the lower this volume. More importantly, it is an approximation to the MDL objective function.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... model
^{B.3} - It also used proper MDL terms rather than an approximation as Smith
did.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... endeavours
^{B.4} - Although none of the points is far-fetched, not a single one of them
proposes an unfamiliar approach; and yet, an open mind is the key
to advancements.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... interactive
^{B.5} - A reasonable response time depends on the purpose of the system, the
level of detail, etc.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... composite
^{B.6} - It will be prematurely assumed that the new
*synthetic*data type possesses several distinct morphological attributes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... choices
^{B.7} - Some discussions also suggested that data should be similar to that
used in Davies' thesis, i.e. a bump on top of a rectangular brick.
Analysis based on current understanding then becomes viable too.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... issue
^{B.8} - Warps placement truly seemed tactless and poor at the time, but this
needed to be confirmed by actual evidence.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... SNR
^{B.9} - The signal-to-noise ratio in medical images can be lower by orders
of magnitude in comparison with visual images.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... aims
^{B.10} - Chapter was bound to take a pessimistic point-of-view
to describe worst-case scenarios. A more optimistic contemplation
would have discussed the obtainable goals and the factors that make
these goals arduous if not impossible to reach.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
productively
^{C.1} - Frequently it appears to be the case that in order to get reasonable
results, high computational power is mandatory. In the absence of
this power, experiments might fail or become impractical.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .