Conceptos Básicos del Branding:

approach where the model is optimized as a space curve in scale-space rather than as a planar curve at each individual scale level will be discussed in chapter 10.

In this dissertation two main types of shape stacks are investigated: those obtained using active shape evolution, and those obtained via active shape focusing. Both concepts will be presented in the following.

7.2 Active Shape Evolution

Active shape evolution is similar to the classic multi-scale contour evolution and analysis (chap­ ter 3, section 3.4.1), as it starts from the “original” shape obtained via prior manual outlining by an expert or a suitable segmentation tool, or through some other form of higher level knowledge

(e.g. for analytical or artificial data). This shape is also called the true model or simply ground

truth. Instead of directly blurring the shape contour as in the classic multi-scale contour analy­ sis, the shape contour is embedded in its image context, and is taken as an initial active contour model. An image scale-space rather than a contour scale-space is constructed, along with an as­ sociated image feature scale-space. The multi-scale active contour model is then tracked through

this scale-space in a so-called fine-to-coarse fashion, for increasing levels of image scale and as­

sociated contour scale. In this way the model is attracted from the finest shape details to more and more global, higher scale image features. Active shape evolution in conjunction with any of the

presented scale-space dimensionalities can be formulated as a multi-scale, implicit segmentation

process o f 2D, 2^D , or 2 |D dimensionality. The resulting shape stack, consisting o f all interme­ diate scale results, is consequently of dimensionality 2^D , 3D, or 3 |D , respectively. Algorithm 7.1 illustrates the technique for active shape evolution of 2^D or 2 |D dimensionality (differing only in the dimensionality of the underlying image scale-space), which will in the following be explained in more detail.

7.3. Active Shape Focusing 148

L( x, y, Zk) , all ground truth models are optimized independently for increasing scales ai. As an optimization routine, any of the presented techniques of the previous chapter can be used.

For each scale level, the image energy terms of the energy functional are based on slice i of the

3D, S^ D, or 4D image scale-space. Consequently, each model Vz^(s’,ai) at scale level ai is of image-scale related contour scale ç^, enforced by an adaptive uniform sampling process (see

section 6.1.2.2). The fine-to-coarse tracking is performed by taking at each slice i of the image

scale-space the optimized contour models from the previous, next lower scale slice i — 1 as an initial estimate, until the highest or coarsest level of scale, n, is reached. The resulting set of

shapes {vz^(s; cr*)} represent the fine-to-coarse multi-scale shape stack, and can consequently

be structured in three different ways:

• as a single 2 \D shape stack, denoted by the set {v(s; ai)\i = 0, • • •, n — 1}, or

• as a set o f 2 ^ D shape stacks, organized as the set of all intermediate evolution results for

each image slice Zk, where each 2 ^ D stack is denoted by (s; ai)\i = 0, • • •, n — 1},

or

• as a single 3 \ D shape stack, which is given by the set of concatenated, volumetric shapes

for each ascending scale level ai, denoted by (s; ai)\zk = 1, • • • ,N} .

The scale samples are formulated in ascending order. Note that the latter two structures merely

refer to the re-organization of the resulting shapes, and not to a different active shape evolution method. However, in a following active shape description process, they give rise to different kinds of descriptors, which will be further discussed below.

7.3 Active Shape Focusing

Active shape focusing is the dual technique to active shape evolution, as it is performed in a coarse-to-fine fashion, similar to classic edge focusing (chapter 3, section 3.4.2). The true shape outline need not be known, as a very coarse initial estimate (e.g. a circle or an ellipse) is sufficient to capture the global shape outline at an adequately large scale. Taking such a coarse estimate as an initial model, this model is regularized or focused down for decreasing levels o f image scale. It is again important to note that the final result of this active shape focusing process is also an implicit rather than explicit multi-scale segmentation result o f 2Z>, 2^D , or 2 |D dimensionality,

as only the most prominent shape outline is followed in the coarse-to-fine tracking process. The

resulting multi-scale shape stack is consequently of the same dimensionality as when based on active shape evolution, yet it differs in the ordering of the scales. Algorithm 7.2 illustrates the

active shape focusing method of 2^ or 2 ^D dimensionality, which will be further explained in

7.3. Active Shape Focusing 149

H For all image slices

for Zk = I to N do

// Initialize with estimated model setv^,(s;an) =

// Optimize for decreasing scale levels

for i = n — 1 to 0 do

set (s; (Ti) = O p tim ize (s; o-j+i)) end for

end for