STA January 11, Object Data is Big Data of Complex Type. In this class, the focus is on the complexity of data,

Transcription

1 STA 6557 January 11, Object Data Analysis - an Introduction Object Data is Big Data of Complex Type. In this class, the focus is on the complexity of data, rather than on the size of such data. There are many disciplines in which data arises on object spaces which are manifolds. Among these are anthropology, astronomy, bioinformatics, computer vision, geology, image analysis, medical imaging, meteorology, and statistics. Over the next several chapters, this monograph will present the general theory and methodology underlying a nonparametric statistical analysis of data arising on manifolds. However, each application and type of data will have its own specific problems that need to be taken into consideration in the analysis, which will be addressed in later portions of this book. First, though, we wish to provide a number of examples of data lying on manifolds. In addition to showing applications in which such data arise, these example can help provide context in subsequent chapters focusing on theory.

2 2 Directional and Axial Data Statistical data analysis on spheres is a relatively old discipline. Watson (1983) [?] points out that one of the first statistical tests ever known is due to D. Bernoulli (1734) [?], who was asking whether the unit normals to orbital planes are uniformly distributed on the celestial sphere. Here, the angle of the orbital plane of a planet is in reference to the ecliptic, which is the apparent path of the Sun around the Earth. Let i be the inclination of the orbital plane of a planet to the ecliptic and Ω be the angle between a fixed line in the ecliptic (the line joining the Sun and the Earth at the time of the vernal equinox) and the line joining the ascending node of the planet (the point where the orbit of the planet rises to the positive side of the ecliptic). Then each orbit determines one directed unit vector n perpendicular to the orbital plane of the planet with the sense of direction given by the right hand rule, n = (sin(ω) sin(i), cos(ω) sin(i), cos(i)) The University of Uppsala data (Mardia and Jupp (2000), Table 10.2) [?] provides a set of measurements (i,ω) for the (at the time) nine planets in the solar system. From this data, Patrangenaru (1998) [?] derived the coordinates n x,n y,n z of the unit normals to orbital planes of the planets as of See Table 1 for these coordinates. Additional data on spheres or projective planes can be found in Fisher et al. (1987) [?]. One such example concerning wind directions at a given location on Earth can be found on p. 308 in that reference and is graphically displayed here in Figure 1. The data, itself, can be found in Table?? at the conclusion of this chapter. 2

3 Table 1: The normals to the orbital planes of the nine planets in the solar system Planet n x n y n z Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Similarity Shape Data and Size and Shape Data Images arise as data in a number of fields, including anthropology, biology, computer vision, and medicine. In a number of cases, the entire image may not be of interest to researchers. Instead, they may be interested only in describing certain geometric information, commonly called the shape, of key features of the image. Depending on the manner in which the images were obtained, this data may lend itself to similarity shape or size-and-shape analysis. We will now consider a few examples of such data. A useful data library, maintained by James Rohlf at the State University of New York, Stony Brook, can be found on his website [?]. This library includes classical data sets from Bookstein (1991) [?] in electronic format. Such data typically contains observations that consist of lists of coordinates for landmarks, which are points of key interest in an image. Two such data sets 3

4 Figure 1: Four views of a sample of wind directions on consecutive days of March 1981, near Singleton, United Kingdom, at a height of 300 meters. (Source: Bhattacharya et al.(2012), Figure 4. Reproduced by permission of John Wiley & Sons LTD). describe locations in the human skull. For instance, the University School Study data (Bookstein (1991), pp ) contains landmark coordinates from X-rays of children s midface bones. These observations can be found in Tables??,??,??, and?? at the conclusion of this chapter. The Apert data set (Bookstein (1991), pp ), as shown in Table??, consists of a number of landmarks describing children who have Apert syndrome. Apert syndrome is a genetic craniosynostosis, of a markedly deformed tower-shaped head resulting from the premature fusion of all cranial sutures. X-rays of a clinically normal and an Apert syndrome skull are displayed in Figure 2. In each of these examples, the data sets describe 2D, or planar, similarity shapes of the finite number of landmarks. For additional examples of 2D similarity shape data, we recommend the data sets in Dryden and Mardia (1998) [?]. While classical planar similarity shape analysis is concerned with the types of data shown in the preceding examples, in recent years, many researchers have begun to focus instead on analyzing the direct similarity shape of outlines or boundaries of objects, which are often referred to as contours, as infinite-dimensional objects rather than finite. For instance, consider 4

5 Figure 2: Lateral X-ray of a clinically normal skull (top, with landmarks) and an Apert syndrome skull (bottom). (Source: Bandulasiri et al.(2009), Figure 4. Reproduced by permission of Elsevier). 5

6 Figure 3: Sample of 4 contours of a hand gesture. (Source: Bhattacharya et al.(2012), Figure 4. Reproduced by permission of John Wiley & Sons LTD). the following sample of contours of hand gestures from Sharvit et. al. (1998) [?] shown in Figure 3. Here, the contours are represented by evaluating the underlying function at a sufficiently large number of sampling points, as suggested in Ellingson et al. (2013) [?]. A list of the 300 sampling points used for Figure 3 is given in Tables??,??,??,??,??,??,??, and??. Albert Einstein s brain was removed shortly after his death (most likely without prior family consent), weighed, dissected and photographed by a pathologist. Among other pictures, a digital scan of a picture of the General Relativity creator s half brain taken at the autopsy is displayed below; we extracted the contour of the corpus callosum(cc) from this Einstein s brain image, the shape of which would be set as a null hypothesis in our testing problem (see Figure 3). Fletcher (2013) [?] extracted contours of CC midsagittal sections from MRI images to study possible age related changes in this part of the human brain. His study points to certain age related shape changes in the CC. Given that Einstein passed away at 76, we consider a subsample of CC brain contours from Fletcher (2013) [?] in the age group to test how far the average 6

7 Figure 4: Right hemisphere of Einstein s brain including CC midsagittal section (left) and its contour (right). CC contour is from Einstein s. The data is displayed in Figure 3. Figure 5: Corpus callosum midsagittal sections shape data, in subjects ages - 65 to 83 With advancements in imaging technology, 3D-direct similarity shape and 3D size-andshape data are also produced in various fields, including the biological sciences and medicine. Our first example of such data arises from medical imaging. In the Louisiana State University Experimental Glaucoma Study (LEGS), the optic nerve head (ONH) region of both eyes of twelve mature Rhesus monkeys were imaged with a Heidelberg Retina Tomograph (HRT) device, also called Scanning Confocal Laser Tomograph (SCLT). The experimental glaucoma was induced in one eye of each animal and the second eye was kept as control. The images are arrays of elevation values which represent the depth of the ONH and are thus range images. Figure 6 shows contour plots for glaucomatous change (see Derado et al [?]). From clinical experience, it is known that the ONH area contains all the relevant information related to glaucoma onset. Figure 7 shows the relevant area with four landmarks. Namely, S for the superior aspect of the retina towards the top of the head, N for the nasal or nose side of the retina, T for temporal, the side of the retinal closest to the temple or temporal bone of the skull, 7

8 and V for the ONH deepest point. The first three are anatomical landmarks and the fourth one is a mathematical landmark. A fifth landmark, called I for inferior, was also recorded. The data set was obtained from a library of Heidelberg Retina Tomograph (HRT) images of the complicated ONH topography. Those images are so-called range images. A range image is, loosely speaking, like a digital camera image, except that each pixel stores a depth rather than a color level. It can also be seen as a set of points in 3D. The range data acquired by 3D digitizers, such as optical scanners, commonly consist of depths sampled on a regular grid. In the mathematical sense, a range image is a 2D array of real numbers which represent those depths. A combination of modules in C++ and SAS took the raw image output and processed it into arrays of height values Another byproduct was a file that we will refer to as the abxy file. This file contains the following information: subjects names (denoted by: 1c, 1d, 1e, 1f, 1g, 1i, 1j, 1k, 1l, 1n, 1o, 1p), observation points that distinguish the normal and treated eyes and the 10 or 15 degree fields of view for the imaging. Observation point 03 denotes a 10 degree view of the experimental glaucoma eye, 04 denotes 15 degree view of the experimental glaucoma eye, degree fellow normal eye, degree fellow normal eye. Recall that of the two eyes of one animal one was given experimental glaucoma, and the other was left untreated (normal) and imaged over time as a control. The coordinates of the ellipse that determines the border of the optic nerve head were determined by the software of the HRT as it interacts with the operator of the device. Two-dimensional coordinates of the center of these ellipses, as well as the sizes of the small and the large axes of the ellipses, are also stored in the abxy file. To find out more about the LSU study and the image acquisition, see Burgoyne et al. (2000) [?]. Files names (each 8

9 Figure 6: Range images as contour plots for HRT images of a Rhesus monkey ONH area before and after the induced glaucomatous shape change. (Source: Derado et al.(2004), Figure 3. Reproduced by permission of Taylor & Francis). file is one observation) were constructed from the information in these so-called abxy file. The list of all the observations was then used as an input for the program (created by G. Derado in C++), which determined the three dimensional coordinates of the landmarks for each observation considered in our analysis. The XY coordinates of the cardinal papilla landmarks, were recovered from abxy data file in the LEGS library, which further allowed a reading from each of the the Z coordinate from the corresponding array files. The original data was collected in experimental observations on Rhesus monkeys. Given that, after treatment, a healthy eye slowly returns to its original shape, for the purpose of IOP increment detection, only the first set of after-treatment observations of the treated eye were considered. Table?? contains the original sample coordinates, The filenames and their correspondingx r i coordinates, in microns, are given for 12 animals. A large source of 3D size-and-shape data is the RCSB Protein Data Bank (PDB) website at which provides a wealth of information about, including about their physical structures. As of March 1, 2015, there were 106, 858 structures 9

10 Figure 7: Magnified HRT image of the central region of the retina including ONH, and the four landmarks N, S, T, V. (Source: Derado et al.(2001), Figure 1. Reproduced by permission of Taylor & Francis). posted on the there. One of the many problems of interest in bioinformatics is the relationship of the physical structure and chemical sequence of a protein to its biological function and size-and-shape analysis provides one avenue for exploring this relationship. Here, we will consider the 3D atomic structures of two groups of binding sites, which are locations on the surface of a protein where binding activity occurs. In Figure 8, we show binding sites from three examples of hydrolase (serine proteinase) which were obtained by X- ray diffraction, as displayed using the software Rasmol. Their structure i.d.s on PDB are 1ela, 1eld and 1ele and their primary citation is Mattos et al. (1997) [?]. In Figure 8, the atoms are gray level coded where hydrogen is dark gray, oxygen is medium gray, and carbon is light gray. Similarly, Figure 9 displays binding sites for three examples of acid proteinase. The coordinates and chemical types of atoms from these acid proteinase binding sites are shown in Tables?? and??, where the units are measured inå. 10

11 Figure 8: Serine proteinase protein binding site structures.(source: Bhattacharya et al.(2012), Figure 4. Reproduced by permission of John Wiley & Sons LTD) Figure 9: Acid proteinase protein binding site structures. (Source: Bhattacharya et al.(2012), Figure 4. Reproduced by permission of John Wiley & Sons LTD). 11