Every unit matrix is a LULU

AbstractThe four matrices L0U0L1U1 at the end of the title are triangular with ones on their main diagonals. Their product has determinant one. Following a question and theorem of Toffoli, we show that any matrix with determinant one can be factored in this way. A transformation of the plane becomes a sequence of one-dimensional shears, with n2 — 1 free parameters

High Quality Alias Free Image Rotation

This paper presents new algorithms for the rotation of images. The primary design criteria for these algorithms is very high quality. Common methods for image rotation, including convolutional and separable approaches, are examined and shown to exhibit significant high frequency aliasing problems. A new resampling filter design methodology is presented which minimizes the problem for conventional convolution-based image rotation. The paper also presents a new separable image rotation algorithm which exhibits improved performance in term of reduction in artifacts and an efficient $O(N^{2} log N)$ running time

Sayısal görüntülerin boyutlarının ara değerleme yöntemi ile büyütülmesi

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Ara değerleme, görüntü işlemenin temel uygulamalarından biridir ve görüntüyeniden boyutlandırılmasında sıkça kullanılmaktadır. İdeal ara değerleme fonksiyonusonsuz impuls cevaplı olduğundan sonlu impuls cevaplı ara değerleme fonksiyonlarıgeliştirilmiştir. Sonlu impuls cevaplı yöntemlerin iki çeşidi vardır: Uyarlamalımetotlar ve uyarlamalı olmayan metotlar. Ara değerleme görüntü kalitesi seçilenmetot ile yakından ilgilidir. Tezde bu metotların incelenmesi ve kıyaslanmasıamaçlamıştır. İncelenen metotlardan uyarlamalı olmayanları; 1) sınırlandırılmış vepencerelendirilmiş sinc, 2) en yakın komşuluk, 3) doğrusal, 4) karesel, 5) kübik, 6)B-spline ve 7) Lagrange yöntemleridir, uyarlamalı yöntemler ise 1) kenar duyarlı ve2) yerel eğim özelliklerini temel alan yöntemlerdir. Yöntemler, zaman ve Fourieruzayı analizi, görüntü kalitesi, nicel ölçümler ve hesap yükü gibi ölçütler hesabakatılarak kıyaslanmışlardır. Yapılan kıyaslamaların sonucunda şu sonuçlaraulaşılmıştır: En iyi görüntü kalitesini veren yöntem yerel eğim özelliklerini temelalan uyarlamalı kübik ara değerlemedir. Uyarlamalı olmayan metotlar arasında en iyigörüntü kalitesi sağlayan kübik ara değerleme, en hızlı metotlar ise en yakınkomşuluk ve doğrusal ara değerlemedir. Hız ve kalite bakımından optimum yöntemise karesel ara değerlemedir. Her ne kadar tezde en iyi yöntemler belirlenmeyeçalışılmışsa da görüntünün özelliklerine ve uygulamaya göre en doğru yöntemdeğişebilir. Bu nedenle her bir uygulama ve görüntü için yöntemlerin veparametrelerinin yeniden gözden geçirilmesi önerilir.Image interpolation is a key aspect of digital image processing and frequently usedfor resampling of images. Since the ideal interpolation function is spatially unlimited,several interpolation kernels of finite size have been introduced. There are twodifferent kinds of finite interpolation methods: adaptive and non-adaptiveinterpolation techniques. The interpolated image quality is closely related to chooseninterpolation technique. This thesis is aimed to investigate and compare differentinterpolation techniques. The non-adaptive methods discussed include 1) truncatedand windowed sinc, 2) nearest neighbor, 3) linear, 4) quadratic, 5) cubic, 6) B-splineand 7) Lagrange methodes. The adaptive ones examined are 1) edge sensitive and 2)adaptive image interpolation based on local gradient features. The comparison isdone by means of spatial and Fourier analysis, interpolated image quality,quantitative analysis and computational cost. According to the comparisons, the bestmethod for interpolated image quality appears to be adaptive image interpolationbased on local gradient features. Best method among the non-adaptive methods iscubic interpolation. The fastest algorithms seem to be nearest neighbour and linearinterpolation, and the optimum method that satisfies both speed and qualityrequirements is quadratic interpolation. Although the thesis tries to identify the bestinterpolation method, the most suitable method for an image or application differs.Hence, comparing the selected methods for a given application is stronglyrecommended

An Integrated Multi-modal Registration Technique for Medical Imaging

Registration of medical imaging is essential for aligning in time and space different modalities and hence consolidating their strengths for enhanced diagnosis and for the effective planning of treatment or therapeutic interventions. The primary objective of this study is to develop an integrated registration method that is effective for registering both brain and whole-body images. We seek in the proposed method to combine in one setting the excellent registration results that FMRIB Software Library (FSL) produces with brain images and the excellent results of Statistical Parametric Mapping (SPM) when registering whole-body images. To assess attainment of these objectives, the following registration tasks were performed: (1) FDG_CT with FLT_CT images, (2) pre-operation MRI with intra-operation CT images, (3) brain only MRI with corresponding PET images, and (4) MRI T1 with T2, T1 with FLAIR, and T1 with GE images. Then, the results of the proposed method will be compared to those obtained using existing state-of-the-art registration methods such as SPM and FSL. Initially, three slices were chosen from the reference image, and the normalized mutual information (NMI) was calculated between each of them for every slice in the moving image. The three pairs with the highest NMI values were chosen. The wavelet decomposition method is applied to minimize the computational requirements. An initial search applying a genetic algorithm is conducted on the three pairs to obtain three sets of registration parameters. The Powell method is applied to reference and moving images to validate the three sets of registration parameters. A linear interpolation method is then used to obtain the registration parameters for all remaining slices. Finally, the aligned registered image with the reference image were displayed to show the different performances of the 3 methods, namely the proposed method, SPM and FSL by gauging the average NMI values obtained in the registration results. Visual observations are also provided in support of these NMI values. For comparative purposes, tests using different multi-modal imaging platforms are performed

Phase-shifting Haar Wavelets For Image-based Rendering Applications

In this thesis, we establish the underlying research background necessary for tackling the problem of phase-shifting in the wavelet transform domain. Solving this problem is the key to reducing the redundancy and huge storage requirement in Image-Based Rendering (IBR) applications, which utilize wavelets. Image-based methods for rendering of dynamic glossy objects do not truly scale to all possible frequencies and high sampling rates without trading storage, glossiness, or computational time, while varying both lighting and viewpoint. This is due to the fact that current approaches are limited to precomputed radiance transfer (PRT), which is prohibitively expensive in terms of memory requirements when both lighting and viewpoint variation are required together with high sampling rates for high frequency lighting of glossy material. At the root of the above problem is the lack of a closed-form run-time solution to the nontrivial problem of rotating wavelets, which we solve in this thesis. We specifically target Haar wavelets, which provide the most efficient solution to solving the tripleproduct integral, which in turn is fundamental to solving the environment lighting problem. The problem is divided into three main steps, each of which provides several key theoretical contributions. First, we derive closed-form expressions for linear phase-shifting in the Haar domain for one-dimensional signals, which can be generalized to N-dimensional signals due to separability. Second, we derive closed-form expressions for linear phase-shifting for two-dimensional signals that are projected using the non-separable Haar transform. For both cases, we show that the coefficients of the shifted data can be computed solely by using the coefficients of the original data. We also derive closed-form expressions for non-integer shifts, which has not been reported before. As an application example of these results, we apply the new formulae to image shifting, rotation and interpolation, and demonstrate the superiority of the proposed solutions to existing methods. In the third step, we establish a solution for non-linear phase-shifting of two-dimensional non-separable Haar-transformed signals, which is directly applicable to the original problem of image-based rendering. Our solution is the first attempt to provide an analytic solution to the difficult problem of rotating wavelets in the transform domain

Estimation of translation and rotation by Fourier transform

In the research area of vision-aided motion sensors, the rotation parameters can be computed from the motion in the picture . Th e properties of translation and rotation in the frequency domain of the Fourier transform are used here . This study is restricted to rigid-body transformations, but other application domains, such as matching of rigidly misaligned images, also exist .Dans l'idée de compléter les capteurs mécaniques de mouvement par des techniques à base de vision, nous analysons le déplacement d'une image pour en déduire les paramètres de rotation de la caméra. L'approche choisie est celle de la transformation de Fourier dont on utilise les propriétés d'invariance par rotation et de déphasage par translation. L'application, réduite pour cette étude aux rotations de caméra, peut s'étendre à tous les domaines liés au recalage d'images

Key duplicating software using digital image processing

In order to cut a key duplicate, it is necessary to measure the original key very precisely. There is a great variety in the key measuring devices currently in use; however, they follow the common basic principle of using a known light source of some kind and measuring the features of the light reflected off the key. In some variations of these systems, only a small part of the key can be measured at one time, and therefore some part of the system needs to be physically mobile to measure the entire key; be it the key itself, the light source or the camera measuring the reflected light. Most of the systems also require the key to be positioned exactly straight in the device in order to be measured, and even a slight deviation means that the blank key will be cut wrong and will be unusable

Image enhancement in digital X-ray angiography

Anyone who does not look back to the beginning throughout a course of action, does not look forward to the end. Hence it necessarily follows that an intention which looks ahead, depends on a recollection which looks back. | Aurelius Augustinus, De civitate Dei, VII.7 (417 A.D.) Chapter 1 Introduction and Summary D espite the development of imaging techniques based on alternative physical phenomena, such as nuclear magnetic resonance, emission of single photons ( -radiation) by radio-pharmaceuticals and photon pairs by electron-positron annihilations, re ection of ultrasonic waves, and the Doppler eect, X-ray based im- age acquisition is still daily practice in medicine. Perhaps this can be attributed to the fact that, contrary to many other phenomena, X-rays lend themselves naturally for registration by means of materials and methods widely available at the time of their discovery | a fact that gave X-ray based medical imaging an at least 50-year head start over possible alternatives. Immediately after the preliminary communica- tion on the discovery of the \new light" by R¨ ontgen [317], late December 1895, the possible applications of X-rays were investigated intensively. In 1896 alone, almost one 1,000 articles about the new phenomenon appeared in print (Glasser [119] lists all of them). Although most of the basics of the diagnostic as well as the therapeutic uses of X-rays had been worked out by the end of that year [289], research on im- proved acquisition and reduction of potential risks for humans continued steadily in the century to follow. The development of improved X-ray tubes, rapid lm changers, image intensiers, the introduction of television cameras into uoroscopy, and com- puters in digital radiography and computerized tomography, formed a succession of achievements which increased the diagnostic potential of X-ray based imaging. One of the areas in medical imaging where X-rays have always played an im- portant role is angiography,y which concerns the visualization of blood vessels in the human body. As already suggested, research on the possibility of visualization of the human vasculature was initiated shortly after the discovery of X-rays. A photograph of a rst \angiogram" | obtained by injection of a mixture of chalk, red mercury, and petroleum into an amputated hand, followed by almost an hour of exposure to X-rays | was published as early as January 1896, by Hascheck & Lindenthal [139]. Although studies on cadavers led to greatly improved knowledge of the anatomy of the human vascular system, angiography in living man for the purpose of diagnosis and intervention became feasible only after substantial progress in the development yA term originating from the Greek words o (aggeion), meaning \vessel" or \bucket", and -' (graphein), meaning \to write" or \to record". 2 1 Introduction and Summary of relatively safe contrast media and methods of administration, as well as advance- ments in radiological equipment. Of special interest in the context of this thesis is the improvement brought by photographic subtraction, a technique known since the early 1900s and since then used successfully in e.g. astronomy, but rst introduced in X-ray angiography in 1934, by Ziedses des Plantes [425, 426]. This technique al- lowed for a considerable enhancement of vessel visibility by cancellation of unwanted background structures. In the 1960s, the time consuming lm subtraction process was replaced by analog video subtraction techniques [156, 275] which, with the in- troduction of digital computers, gave rise to the development of digital subtraction angiography [194] | a technique still considered by many the \gold standard" for de- tection and quantication of vascular anomalies. Today, research on improved X-ray based imaging techniques for angiography continues, witness the recent developments in three-dimensional rotational angiography [88, 185, 186, 341,373]. The subject of this thesis is enhancement of digital X-ray angiography images. In contrast with the previously mentioned developments, the emphasis is not on the further improvement of image acquisition techniques, but rather on the development and evaluation of digital image processing techniques for retrospective enhancement of images acquired with existing techniques. In the context of this thesis, the term \enhancement" must be regarded in a rather broad sense. It does not only refer to improvement of image quality by reduction of disturbing artifacts and noise, but also to minimization of possible image quality degradation and loss of quantitative information, inevitably introduced by required image processing operations. These two aspects of image enhancement will be claried further in a brief summary of each of the chapters of this thesis. The rst three chapters deal with the problem of patient motion artifacts in digital subtraction angiography (DSA). In DSA imaging, a sequence of 2D digital X-ray projection images is acquired, at a rate of e.g. two per second, following the injection of contrast material into one of the arteries or veins feeding the part of the vasculature to be diagnosed. Acquisition usually starts about one or two seconds prior to arrival of the contrast bolus in the vessels of interest, so that the rst few images included in the sequence do not show opacied vessels. In a subsequent post-processing step, one of these \pre-bolus" images is then subtracted automatically from each of the contrast images so as to mask out background structures such as bone and soft- tissue shadows. However, it is clear that in the resulting digital subtraction images, the unwanted background structures will have been removed completely only when the patient lied perfectly still during acquisition of the original images. Since most patients show at least some physical reaction to the passage of a contrast medium, this proviso is generally not met. As a result, DSA images frequently show patient-motion induced artifacts (see e.g. the bottom-left image in Fig. 1.1), which may in uence the subsequent analysis and diagnosis carried out by radiologists. Since the introduction of DSA, in the early 1980s, many solutions to the problem of patient motion artifacts have been put forward. Chapter 2 presents an overview of the possible types of motion artifacts reported in the literature and the techniques that have been proposed to avoid them. The main purpose of that chapter is to review and discuss the techniques proposed over the past two decades to correct for 1 Introduction and Summary 3 Figure 1.1. Example of creation and reduction of patient motion artifacts in cerebral DSA imaging. Top left: a \pre-bolus" or mask image acquired just prior to the arrival of the contrast medium. Top right: one of the contrast or live images showing opacied vessels. Bottom left: DSA image obtained after subtraction of the mask from the contrast image, followed by contrast enhancement. Due to patient motion, the background structures in the mask and contrast image were not perfectly aligned, as a result of which the DSA image does not only show blood vessels, but also additional undesired structures (in this example primarily in the bottom-left part of the image). Bottom right: DSA image resulting from subtraction of the mask and contast image after application of the automatic registration algorithm described in Chapter 3. 4 1 Introduction and Summary patient motion artifacts retrospectively, by means of digital image processing. The chapter addresses fundamental problems, such as whether it is possible to construct a 2D geometrical transformation that exactly describes the projective eects of an originally 3D transformation, as well as practical problems, such as how to retrieve the correspondence between mask and contrast images by using only the grey-level information contained in the images, and how to align the images according to that correspondence in a computationally ecient manner. The review in Chapter 2 reveals that there exists quite some literature on the topic of (semi-)automatic image alignment, or image registration, for the purpose of motion artifact reduction in DSA images. However, to the best of our knowledge, research in this area has never led to algorithms which are suciently fast and robust to be acceptable for routine use in clinical practice. By drawing upon the suggestions put forward in Chapter 2, a new approach to automatic registration of digital X-ray angiography images is presented in Chapter 3. Apart from describing the functionality of the components of the algorithm, special attention is paid to their computationally optimal implementation. The results of preliminary experiments described in that chapter indicate that the algorithm is eective, very fast, and outperforms alterna- tive approaches, in terms of both image quality and required computation time. It is concluded that the algorithm is most eective in cerebral and peripheral DSA imag- ing. An example of the image quality enhancement obtained after application of the algorithm in the case of a cerebral DSA image is provided in Fig 1.1. Chapter 4 reports on a clinical evaluation of the automatic registration technique. The evaluation involved 104 cerebral DSA images, which were corrected for patient motion artifacts by the automatic technique, as well as by pixel shifting | a manual correction technique currently used in clinical practice. The quality of the DSA images resulting from the two techniques was assessed by four observers, who compared the images both mutually and to the corresponding original images. The results of the evaluation presented in Chapter 4 indicate that the dierence in performance between the two correction techniques is statistically signicant. From the results of the mutual comparisons it is concluded that, on average, the automatic registration technique performs either comparably, better than, or even much better than manual pixel shifting in 95% of all cases. In the other 5% of the cases, the remaining artifacts are located near the borders of the image, which are generally diagnostically non-relevant. In addition, the results show that the automatic technique implies a considerable reduction of post-processing time compared to manual pixel shifting (on average, one second versus 12 seconds per DSA image). The last two chapters deal with somewhat dierent topics. Chapter 5 is concerned with visualization and quantication of vascular anomalies in three-dimensional rota- tional angiography (3DRA). Similar to DSA imaging, 3DRA involves the acquisition of a sequence of 2D digital X-ray projection images, following a single injection of contrast material. Contrary to DSA, however, this sequence is acquired during a 180 rotation of the C-arch on which the X-ray source and detector are mounted antipo- dally, with the object of interest positioned in its iso-center. The rotation is completed in about eight seconds and the resulting image sequence typically contains 100 images, which form the input to a ltered back-projection algorithm for 3D reconstruction. In contrast with most other 3D medical imaging techniques, 3DRA is capable of provid- 1 Introduction and Summary 5 Figure 1.2. Visualizations of a clinical 3DRA dataset, illustrating the qualitative improvement obtained after noise reduction ltering. Left: volume rendering of the original, raw image. Right: volume rendering of the image after application of edge-enhancing anisotropic diusion ltering (see Chapter 5 for a description of this technique). The visualizations were obtained by using the exact same settings for the parameters of the volume rendering algorithm. ing high-resolution isotropic datasets. However, due to the relatively high noise level and the presence of other unwanted background variations caused by surrounding tissue, the use of noise reduction techniques is inevitable in order to obtain smooth visualizations of these datasets (see Fig. 1.2). Chapter 5 presents an inquiry into the eects of several linear and nonlinear noise reduction techniques on the visualization and subsequent quantication of vascular anomalies in 3DRA images. The evalua- tion is focussed on frequently occurring anomalies such as a narrowing (or stenosis) of the internal carotid artery or a circumscribed dilation (or aneurysm) of intracra- nial arteries. Experiments on anthropomorphic vascular phantoms indicate that, of the techniques considered, edge-enhancing anisotropic diusion ltering is most suit- able, although the practical use of this technique may currently be limited due to its memory and computation-time requirements. Finally, Chapter 6 addresses the problem of interpolation of sampled data, which occurs e.g. when applying geometrical transformations to digital medical images for the purpose of registration or visualization. In most practical situations, interpola- tion of a sampled image followed by resampling of the resulting continuous image on a geometrically transformed grid, inevitably implies loss of grey-level information, and hence image degradation, the amount of which is dependent on image content, but also on the employed interpolation scheme (see Fig. 1.3). It follows that the choice for a particular interpolation scheme is important, since it in uences the re- sults of registrations and visualizations, and the outcome of subsequent quantitative analyses which rely on grey-level information contained in transformed images. Al- though many interpolation techniques have been developed over the past decades, 6 1 Introduction and Summary Figure 1.3. Illustration of the fact that the loss of information due to interpola- tion and resampling operations is dependent on the employed interpolation scheme. Left: slice of a 3DRA image after rotation over 5:0, by using linear interpolation. Middle: the same slice, after rotation by using cubic spline interpolation. Right: the dierence between the two rotated images. Although it is not possible with such a comparison to come to conclusions as to which of the two methods yields the smallest loss of grey-level information, this example clearly illustrates the point that dierent interpolation methods usually yield dierent results. thorough quantitative evaluations and comparisons of these techniques for medical image transformation problems are still lacking. Chapter 6 presents such a compar- ative evaluation. The study is limited to convolution-based interpolation techniques, as these are most frequently used for registration and visualization of medical image data. Because of the ubiquitousness of interpolation in medical image processing and analysis, the study is not restricted to XRA and 3DRA images, but also includes datasets from many other modalities. It is concluded that for all modalities, spline interpolation constitutes the best trade-o between accuracy and computational cost, and therefore is to be preferred over all other methods. In summary, this thesis is concerned with the improvement of image quality and the reduction of image quality degradation and loss of quantitative information. The subsequent chapters describe techniques for reduction of patient motion artifacts in DSA images, noise reduction techniques for improved visualization and quantication of vascular anomalies in 3DRA images, and interpolation techniques for the purpose of accurate geometrical transformation of medical image data. The results and con- clusions of the evaluations described in this thesis provide general guidelines for the applicability and practical use of these techniques