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... subjects1.1
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.
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... MRI1.2
See list of acronyms and abbreviation in the Appendix cha:Appendix-EList-of.
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... object1.3
The word ``object'' will from here onwards refer to a structure of interest in $n$-dimensional space.
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... aperture1.4
In the case of medical imaging, there are even more factors to be considered, as opposed to a camera's aperture.
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... rigid1.5
``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.
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... Arguably1.6
Ideas such as this are overly optimistic perhaps.
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... scientific1.7
This text is goal-oriented and it embraces the technical, not much of the inter-personal and curricular.
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... resources2.1
All Web resources are listed at the end of this report.
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... data2.2
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.
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... posed2.3
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.
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... blow-up2.4
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.
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... contents2.5
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.
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... cases2.6
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.
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... range2.7
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.
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... grey-level2.8
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.
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... models2.9
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.
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... superset2.10
They can be thought of as a superset, simply being shape models which hold some additional information and the correlations between all encapsulated data.
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... variation2.11
Offsets of standard deviation units from the mean of each mode then illustrate the effect each variation mode has.
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... above2.12
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.
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... Component2.13
Plainly speaking, PCA only picks up $n$ Eigen-values whose Eigen-values are the greatest.
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... space2.14
Normalisation step as such is similar to the mapping onto a sphere, for instance.
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... shape2.15
Oftentimes, the choice of the mean shape proves to be the least damaging choice.
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... patch2.16
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.
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... spot2.17
Analogically, in the case of shape, sharp-bended descriptors result from the low number of sample points.
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... appearances2.18
One of the main aims and great power of appearance models is full synthesised portrayal, so photo-realism is at a premium.
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... greatest2.19
If one thinks of the cloud in $n$ dimensional space as a placement of characteristics $(c_{1},c_{2}...c_{n})$, 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'.
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... for2.20
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.
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...2.21
As an ad-hoc example, intensity frequently takes values in the range $0..255$ whereas normalised shape coordinates lies between 0 and 1 so fractions such as $\frac{1}{255}$ can be used as coefficients. The two should then scale almost indifferently.
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... pixel2.22
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.
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... image2.23
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.
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... set2.24
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.
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... model2.25
There are some more complex considerations as the model needs to be aligned properly as well as change in its form.
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... intensity2.26
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.
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... negligible2.27
This is reminiscent of the need for a median measure, where average is sensitive to erratic values or salt-and-pepper noise.
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... effect2.28
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 $Transformation\Longleftrightarrow Error$.
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... not2.29
The vector's distribution of values, i.e. positions with high absolute values, can answer this question quite grossly.
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... matrix2.30
The matrix $\mathbf{A}$ can be obtained using linear regression.
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... problem2.31
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.
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... target2.32
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.
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... lie2.33
Advanced knowledge about the problem is highly conductive at this stage, otherwise some bottom-up image analysis is a must.
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... off-line2.34
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.
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... compact2.35
A sparse collection of pixels (or voxels) can be encoded using a lossy function with an even smaller number of parameters.
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... down2.36
When models break down, fitting defaults to a local (and hence false) minimum.
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... synthesising2.37
This can be considered as being a reversal of interpretation, in fact. This binds with the notable computer vision/graphics differentiation.
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... lower2.38
Although some results support this claim, it is quite likely that better implementations and further improvements will prove otherwise.
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... ASM2.39
This is not necessarily so in the like-for-like comparison.
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...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.
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...3.2
More strictly, the inclusion of scaling makes this a similarity transformation rather than rigid.
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...3.3
Such a process is a very fundamental one in computer graphics modelling and various books cover shape-normalisation techniques and algorithms.
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... values3.4
1 value for scaling, 3 for x, y and z coordinates and 2 for rotation, e.g. the xy and yz angles $\theta_{1}$ and $\theta_{2}$.
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...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.
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... structure3.6
A random uncontrollable transformation will dispart basic structures in the image and can make valid interpretation impossible.
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... issue3.7
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.
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... image3.8
More generally, arbitrary data of any complexity should be applicable.
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... transformations3.9
This so-called mapping or transformation can be thought of as being a standard function, for example $f(x,y)=(x',y')$ in 2-D and it is applied to all the pixels within a predefined range.
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... image3.10
More generally, the functions are mappings defined over a matrix or a vector which is analogous to an image.
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... effect3.11
A pixel of course can be mapped onto the exact same original position, but the idea is that a continuous flow must prevail.
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... shapes3.12
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.
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... reference3.13
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.
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... points3.14
A continuous function is independent of the number of points. Therefore, the complexity can be increased progressively to obtain finer, more accurate results.
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... discussion3.15
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 [*].
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... space3.16
One can think of images as a vector of pixel values that define a position in a high-dimensional space
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... way3.17
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.
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... literature3.18
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.
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... sum3.19
One could suggest an extension to such a method and assign weights to differentiate regions of varying significance.
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...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.
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... correlation4.2
To make this appear more practical, one can think of a large ($>100000$ pixels) image where patterns are present.
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... these4.3
This can be portrayed as a uniform plain-white image.
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... sequence4.4
General problem reducibility to a sequence is axiomatic as Turing Machines suggest.
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... landmarks5.1
Often the choice is random so that no assumption are made about the problem.
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... message5.2
An alternative method involving B-fitting was proposed by Thacker et al. [].
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... similarity5.3
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.
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...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 $\mathbf{I}_{m}'$ and the reference $\mathbf{I}_{r}$ in this case.
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... high5.5
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.
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... exist5.6
To name several more methods: dynamic programming, genetic algorithms, Powell's, simulated annealing and steepest descent.
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... axes5.7
Optimisation is a multi-dimensional problem that searches along hyper-spaces, some of which are orthogonal to the many existing axes.
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... search5.8
Exhaustivity is impossible for continuous functions, but digital images are luckily discrete.
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... example6.1
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).
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... data6.2
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.
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... search6.3
This similarity computation is incorporated in the objective function and it usually comprises a collection of pair-wise similarity measures.
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... involved6.4
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).
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... well-integrated6.5
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.
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... investigated7.1
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.
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... online7.2
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.
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... operates7.3
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.
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... laid7.4
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).
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... enclosed7.5
It seems likely that a CD-ROM will accompany later surveys (a la thesis). AART can generate several movie types with little user involvement.
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... bump7.6
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).
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... identify7.7
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.
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... single-point7.8
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.
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... model7.9
The fact that shape component was chosen to be the reparameterisation curve has not been enlightened yet.
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... data7.10
The data was generated by extending the 1-D bump data generator. It is not a Gaussian.
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... implementations7.11
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.
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... direction7.12
This idea is borrowed from technical analysis in finance. It can be useful in science as well.
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... terminology7.13
In AART, this definition of iteration is repeatedly referred to as warping step/s.
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... instances7.14
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.
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... linear7.15
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.
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... Eigen-analysis7.16
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.
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... distances7.17
This measure is better immune to lage local misalignment. This is similar to arguments presented in Equation eq: delta start.
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... resolution7.18
The number of pixels that data comprises is an argument that may be changed.
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... domain7.19
This domain is the selection of landmarks in shapes for the construction of statistical models of shape. Modification of this will be described later.
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... handled7.20
Examples of human hands were later tested as well . They were an easier case that is quicker to reach convergence.
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... dynamic7.21
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.
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... modern9.1
Usually Pentium 4 processor with 256 Megabyte of random access memory (RAM). These were not computational servers in essence.
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... Report10.1
The literature report is located at:
http://www.danielsorogon.com/Webmaster/Research/Literature_Report
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... non-specific10.2
These requirements are intentionally very general. They can be applicable to most computer-scientific research.
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... database10.3
Actually, these are hierarchical nested indices of experiments, sorted chronologically.
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... document10.4
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.
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... times10.5
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.
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... selection10.6
Most recently it turned out to have been complicated. Solution are agreed upon in present days and will soon be implemented and devised.
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...10.7
This has been work in progress since early June 2004.
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... follows10.8
Note that where work has been done already, suitable explanations and examples were provided a previous chapter (Chapter [*]) on experiments and results.
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... descent10.9
This excludes the start when subsets require time to stabilise by preliminary warps.
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... recognitionA.1
Much of the popularity of this method has been imputed to face recognition tasks.
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... slowA.2
The algorithms currently used for demonstration purposes take 3 days to run, but substantial speed-up is expected soon.
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... diffeomorphicB.1
While the bump may have its form tweaked and manipulated, its highest peak should be preserved although it may move leftward or rightward.
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... matrixB.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.
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... modelB.3
It also used proper MDL terms rather than an approximation as Smith did.
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... endeavoursB.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.
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... interactiveB.5
A reasonable response time depends on the purpose of the system, the level of detail, etc.
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... compositeB.6
It will be prematurely assumed that the new synthetic data type possesses several distinct morphological attributes.
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... choicesB.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.
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... issueB.8
Warps placement truly seemed tactless and poor at the time, but this needed to be confirmed by actual evidence.
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... SNRB.9
The signal-to-noise ratio in medical images can be lower by orders of magnitude in comparison with visual images.
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... aimsB.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.
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... productivelyC.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.
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