For the second part of the experiments, real MRI data were used. In order to get quantitative results and to estimate ground truth deformation fields, the registration was independently performed between corresponding pairs of images that were used to compare with those estimated using the prediction system. The overall robustness in terms of the accuracy with respect to the ground truth deformation fields is summarised in the 2nd row of Table 6.2 whereas the 1st row characterises the ground
truth data. Based on these results, it can be concluded that the proposed method is able to estimate the dense deformation field with error less than 2.00.62mm. Additionally, the computation speed is significantly reduced when compared with the classical registration. The average time of prediction when the motion model is already trained is about 1s while the registration process takes about 180s.
Figure 6.4 presents the variation of the first three modes of the implicit shape representation. The first mode can be seen as the mode linked to the changes of the bladder size. Also some changes of the rectum shape are noticeable. The second mode seems to be related mostly to the rectum and the bladder shape changes
Figure 6.3: Example of synthetic deformation field used in the experiments (top left), Estimation error examples: using PDM with Gaussian noise µ 0.0, σ 1.0 (top right), PDM with gross error ς 5%, τ40
20 (bottom left) and using the implicit
representation (bottom right)
.
Figure 6.4: The variability of the major modes of the implicit shape representation of the prostate (blue), bladder (red) and rectum (green).
that are likely to model the patient variability. The most remarkable impact of the third mode is linked to changes of the prostate size. Although the implicit shape representation does not preserve entirely the neighbourhood topologies, the shapes in the presented visualisation do not overlap between each other.
The practical advantage of the deformation fields parameterisation using the sta- tionary velocity fields is shown in Table 6.3. The deformation fields estimated using the major modes of the motion model that is built using the diffeomorphic defor- mation fields can be non-diffeomorphic, whereas the deformation fields estimated
Deformation field representation
-3.0λr -1.5λr 0.0λr 1.5λr 3.0λr
Minimum determinant of Jacobian
r 1 -0.306 0.080 0.109 0.113 0.017
r 2 -0.062 0.094 0.109 0.125 0.136
r 3 0.086 0.104 0.109 0.114 -0.213
Velocity field representation
-3.0λr -1.5λr 0.0λr 1.5λr 3.0λr
Minimum determinant of Jacobian
r 1 0.048 0.097 0.113 0.119 0.123
r 2 0.064 0.096 0.113 0.118 0.121
r 3 0.100 0.106 0.113 0.115 0.109
Table 6.3: Minimum of the determinant of the Jacobian matrix for the variability of the trained motion model using the deformation fields and the stationary velocity fields.
utilising the log-domain paramererised deformation fields to build the motion model are always diffeomorphic. Thus, the parameterisation of the transformations via stationary velocity fields is indicated as the robust methodology due to preserving the one-to-one properties of the estimated deformation fields.
6.7
Summary
The chapter describes a novel technique for model-based dense deformation field estimation with an implicit surface representation used as an effective and robust deformation descriptor. The proposed framework uses motion model estimated from a training data set of shapes and corresponding displacement fields parameterised via stationary velocity fields estimated using a fast and efficient diffeomorphic regis- tration scheme, formulated in the log-Euclidean framework. It has been also demon- strated that with the help of the proposed method, it is possible to predict dense displacement fields solely from the measured deformations of the implicit surface. The experiments conducted with the real data show that it is possible to predict deformation field thereby position of the prostate from shape deformations of the bladder/rectum. As it is relatively easier to segment bladder (rectum) in the CBCT data when compared to prostate segmentation, it can be concluded that the proposed methodology can be potentially useful for adaptive radiation therapy of prostate. Thus, further investigation is suggested to combine the motion model built from MRI data with shape descriptors extracted from radiotherapy imaging.
Facial expression recognition using
log-Euclidean statistical shape
models
This chapter presents a new method for facial expression modelling and recognition based on the diffeomorphic image registration parameterised via stationary velocity fields in the log-Euclidean framework, that was described in the previous chapters. First, Section 7.1 briefly presents an overview of the current facial expression rep- resentations and introduces the concept of using the motion fields as a feature for face recognition and facial expression recognition systems. In Section 7.2, the pro- cess of common face space generation is utilised based in the implicit group-wise registration algorithm adapted to the facial expression modelling. Then, Section 7.3 introduces the velocity field based representation of facial expressions (described in Section 7.3.1), and the Point Distribution Model (presented in Section 7.3.2). The robustness of the proposed facial expression representation is demonstrated based on the experimental results of a qualitative and quantitative evaluation (shown in Section 7.4). The obtained results show that the facial expression representation based on stationary velocity fields can be successfully utilised in facial expression recognition, and this parameterisation produces a slightly higher recognition rate than the facial expression representation based on deformation fields. Finally, the concluding remarks are given in Section 7.5.
7.1
Introduction
Face is an important medium used not only by humans to communicate, but also reflecting a person’s emotional and awareness states, cognitive activity, personality or well-being. Over the last ten years automatic facial expression representation
and recognition have become an area of significant research interest by the com- puter vision community, with applications in human-computer interaction (HCI) systems, medical/psychological sciences, and visual communications to name a few. Although, significant efforts have been undertaken to improve the facial features ex- traction process and the recognition performance, automatic facial expression recog- nition is still a challenging task due to an inherent subjective nature of the facial expressions and their variation over different gender, age, and ethnicity groups. De- tailed overviews of existing methodologies, recent advances and challenges can be found in the literature surveys [37, 95] and standard textbooks [86, 139].
The facial expression representation can be seen as a process of extracting fea- tures, that can be generic such as local binary patterns [120] or Gabor coefficients [13] or more specific such as landmarks of characteristic points located in areas of major facial changes due to articulation [69], or a topographic context (TC) that treats the intensity levels of an image as a 3-D terrain surface [149]. Recently, in [103, 105] authors postulated that the shape space vectors (SSV) of the statistical shape model (SSM) can constitute a significant feature space for the recognition of facial expressions. The SSM can be constructed in many different ways, and it was developed based on the point distribution model originally proposed by [33]. In [104], the SSM is built based on the control points of the B-Spline surface of the training data set, and in [106] an improved version with multi-resolution correspon- dence search and multi-level model deformation was proposed. In this chapter, the SSM is generated using the stationary velocity fields obtained from the diffeomorphic face registration.
The idea of using the motion fields as features in computer vision and pattern recognition was used previously for face recognition where the optical flow was com- puted to robustly recognise face with different expressions based on a single sample per class in the training set [60].
The previous chapters introduced and then utilised the parameterisation of the diffeomorphic transformations via the principal logarithm of non-linear geometrical deformations in medical applications. As the facial shapes (mouth, eyes, eye brows) can be assumed to have constant intra- and inter- subject topology, it is interesting to check the adequacy of the facial expressions represented using stationary velocity fields as a result of performing the diffeomorphic image registration that preserves the spatial topology of objects by maintaining diffeomorphism (detailed description in Section 3.2). Additionally, using this framework (described in details in Chapter 3.2), the log-Euclidean vectorial statistics can be performed on the diffeomorphic vector fields via their logarithm, which always preserve the invertibility constraint contrary to the Euclidean statistics on the deformation fields. In other words, this
provides a mathematically consistent framework to generate the SSM based on the deformation fields parameterised via the stationary velocity fields. Finally, facial expression representation based on the deformation field is compared with facial ex- pression representation based on the stationary velocity field in terms of separability in feature space and recognition performance is evaluated.