A principal component approach to space-based gravitational wave astronomy

The current approach to data analysis for the Laser Interferometry Space Antenna (LISA) depends on the time delay interferometry observables (TDI) which have to be generated before any weak signal detection can be performed. These are linear combinations of the raw data with appropriate time shifts that lead to the cancellation of the laser frequency noises. This is possible because of the multiple occurrences of the same noises in the different raw data. Originally, these observables were manually generated starting with LISA as a simple stationary array and then adjusted to incorporate the antenna's motions. However, none of the observables survived the flexing of the arms in that they did not lead to cancellation with the same structure. The principal component approach is another way of handling these noises that was presented by Romano and Woan which simplified the data analysis by removing the need to create them before the analysis. This method also depends on the multiple occurrences of the same noises but, instead of using them for cancellation, it takes advantage of the correlations that they produce between the different readings. These correlations can be expressed in a noise (data) covariance matrix which occurs in the Bayesian likelihood function when the noises are assumed be Gaussian. Romano and Woan showed that performing an eigendecomposition of this matrix produced two distinct sets of eigenvalues that can be distinguished by the absence of laser frequency noise from one set. The transformation of the raw data using the corresponding eigenvectors also produced data that was free from the laser frequency noises. This result led to the idea that the principal components may actually be time delay interferometry observables since they produced the same outcome, that is, data that are free from laser frequency noise. The aims here were (i) to investigate the connection between the principal components and these observables, (ii) to prove that the data analysis using them is equivalent to that using the traditional observables and (ii) to determine how this method adapts to real LISA especially the flexing of the antenna. For testing the connection between the principal components and the TDI observables a 10x 10 covariance matrix containing integer values was used in order to obtain an algebraic solution for the eigendecomposition. The matrix was generated using fixed unequal arm lengths and stationary noises with equal variances for each noise type. Results confirm that all four Sagnac observables can be generated from the eigenvectors of the principal components. The observables obtained from this method however, are tied to the length of the data and are not general expressions like the traditional observables, for example, the Sagnac observables for two different time stamps were generated from different sets of eigenvectors. It was also possible to generate the frequency domain optimal AET observables from the principal components obtained from the power spectral density matrix. These results indicate that this method is another way of producing the observables therefore analysis using principal components should give the same results as that using the traditional observables. This was proven by fact that the same relative likelihoods (within 0.3%) were obtained from the Bayesian estimates of the signal amplitude of a simple sinusoidal gravitational wave using the principal components and the optimal AET observables. This method fails if the eigenvalues that are free from laser frequency noises are not generated. These are obtained from the covariance matrix and the properties of LISA that are required for its computation are the phase-locking, arm lengths and noise variances. Preliminary results of the effects of these properties on the principal components indicate that only the absence of phase-locking prevented their production. The flexing of the antenna results in time varying arm lengths which will appear in the covariance matrix and, from our toy model investigations, this did not prevent the occurrence of the principal components. The difficulty with flexing, and also non-stationary noises, is that the Toeplitz structure of the matrix will be destroyed which will affect any computation methods that take advantage of this structure. In terms of separating the two sets of data for the analysis, this was not necessary because the laser frequency noises are very large compared to the photodetector noises which resulted in a significant reduction in the data containing them after the matrix inversion. In the frequency domain the power spectral density matrices were block diagonals which simplified the computation of the eigenvalues by allowing them to be done separately for each block. The results in general showed a lack of principal components in the absence of phase-locking except for the zero bin. The major difference with the power spectral density matrix is that the time varying arm lengths and non-stationarity do not show up because of the summation in the Fourier transform.

Three dimensional study to quantify the relationship between facial hard and soft tissue movement as a result of orthognathic surgery

Introduction Prediction of soft tissue changes following orthognathic surgery has been frequently attempted in the past decades. It has gradually progressed from the classic “cut and paste” of photographs to the computer assisted 2D surgical prediction planning; and finally, comprehensive 3D surgical planning was introduced to help surgeons and patients to decide on the magnitude and direction of surgical movements as well as the type of surgery to be considered for the correction of facial dysmorphology. A wealth of experience was gained and numerous published literature is available which has augmented the knowledge of facial soft tissue behaviour and helped to improve the ability to closely simulate facial changes following orthognathic surgery. This was particularly noticed following the introduction of the three dimensional imaging into the medical research and clinical applications. Several approaches have been considered to mathematically predict soft tissue changes in three dimensions, following orthognathic surgery. The most common are the Finite element model and Mass tensor Model. These were developed into software packages which are currently used in clinical practice. In general, these methods produce an acceptable level of prediction accuracy of soft tissue changes following orthognathic surgery. Studies, however, have shown a limited prediction accuracy at specific regions of the face, in particular the areas around the lips. Aims The aim of this project is to conduct a comprehensive assessment of hard and soft tissue changes following orthognathic surgery and introduce a new method for prediction of facial soft tissue changes.   Methodology The study was carried out on the pre- and post-operative CBCT images of 100 patients who received their orthognathic surgery treatment at Glasgow dental hospital and school, Glasgow, UK. Three groups of patients were included in the analysis; patients who underwent Le Fort I maxillary advancement surgery; bilateral sagittal split mandibular advancement surgery or bimaxillary advancement surgery. A generic facial mesh was used to standardise the information obtained from individual patient’s facial image and Principal component analysis (PCA) was applied to interpolate the correlations between the skeletal surgical displacement and the resultant soft tissue changes. The identified relationship between hard tissue and soft tissue was then applied on a new set of preoperative 3D facial images and the predicted results were compared to the actual surgical changes measured from their post-operative 3D facial images. A set of validation studies was conducted. To include: • Comparison between voxel based registration and surface registration to analyse changes following orthognathic surgery. The results showed there was no statistically significant difference between the two methods. Voxel based registration, however, showed more reliability as it preserved the link between the soft tissue and skeletal structures of the face during the image registration process. Accordingly, voxel based registration was the method of choice for superimposition of the pre- and post-operative images. The result of this study was published in a refereed journal. • Direct DICOM slice landmarking; a novel technique to quantify the direction and magnitude of skeletal surgical movements. This method represents a new approach to quantify maxillary and mandibular surgical displacement in three dimensions. The technique includes measuring the distance of corresponding landmarks digitized directly on DICOM image slices in relation to three dimensional reference planes. The accuracy of the measurements was assessed against a set of “gold standard” measurements extracted from simulated model surgery. The results confirmed the accuracy of the method within 0.34mm. Therefore, the method was applied in this study. The results of this validation were published in a peer refereed journal. • The use of a generic mesh to assess soft tissue changes using stereophotogrammetry. The generic facial mesh played a major role in the soft tissue dense correspondence analysis. The conformed generic mesh represented the geometrical information of the individual’s facial mesh on which it was conformed (elastically deformed). Therefore, the accuracy of generic mesh conformation is essential to guarantee an accurate replica of the individual facial characteristics. The results showed an acceptable overall mean error of the conformation of generic mesh 1 mm. The results of this study were accepted for publication in peer refereed scientific journal. Skeletal tissue analysis was performed using the validated “Direct DICOM slices landmarking method” while soft tissue analysis was performed using Dense correspondence analysis. The analysis of soft tissue was novel and produced a comprehensive description of facial changes in response to orthognathic surgery. The results were accepted for publication in a refereed scientific Journal. The main soft tissue changes associated with Le Fort I were advancement at the midface region combined with widening of the paranasal, upper lip and nostrils. Minor changes were noticed at the tip of the nose and oral commissures. The main soft tissue changes associated with mandibular advancement surgery were advancement and downward displacement of the chin and lower lip regions, limited widening of the lower lip and slight reversion of the lower lip vermilion combined with minimal backward displacement of the upper lip were recorded. Minimal changes were observed on the oral commissures. The main soft tissue changes associated with bimaxillary advancement surgery were generalized advancement of the middle and lower thirds of the face combined with widening of the paranasal, upper lip and nostrils regions. In Le Fort I cases, the correlation between the changes of the facial soft tissue and the skeletal surgical movements was assessed using PCA. A statistical method known as ’Leave one out cross validation’ was applied on the 30 cases which had Le Fort I osteotomy surgical procedure to effectively utilize the data for the prediction algorithm. The prediction accuracy of soft tissue changes showed a mean error ranging between (0.0006mm±0.582) at the nose region to (-0.0316mm±2.1996) at the various facial regions.