Index of /rozprawy2/11456

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AGH University of Science and Technology

Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering

Ph.D. dissertation

Mobile optical tracking system

in computer-assisted surgery

mgr in˙z. Adrian Goral

Ph.D. advisor: prof. dr hab. Józef Kozak

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Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

Wydział Elektrotechniki, Automatyki, Informatyki i In˙zynierii Biomedycznej

Rozprawa doktorska

Mobilny lokalizator wizyjny

w komputerowym wspomaganiu chirurgii

mgr in˙z. Adrian Goral Promotor: prof. dr hab. Józef Kozak

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Abstract

This dissertation presents a concept of a new method for tracking of surgical instruments based on the data captured with a standard video camera and a time of flight (ToF) camera, working in a calibrated setup. The detection of the instruments is based on combined photometric and geometric data, and their orientation is estimated using a model-based 3D matching algorithm. The method was implemented on a mobile device, turning it into a prototype of a mobile optical tracking system. The prototype was tested in laboratory experiments, in cadaver study, and in phantom study in order to test performance and compatibility with various intraoperative settings.

Most of the existing optical systems for intraoperative instrument tracking rely on light-reflective or light-emitting markers. The markers are organized into groups of unique spatial configurations and mounted on the instruments in form of arrays. The tracking system recognizes the instrument by the shape of the array and determines its spatial orientation. The task of attaching and detaching the arrays diverts the attention of the surgeon from the procedure itself and increases intraoperative time. Further limitations of marker-based approach become evident in orthopedic surgery, where the instruments are the subject of vigorous motion and large impact forces. As a result, the arrays may dislocate or detach making the guidance infeasible without additional steps. These limitations prevent the widespread adoption of computer guidance in some areas of orthopedic surgery, despite the fact that the instrument orientation is of key importance in most of the orthopedic procedures.

This work focuses on applying the proposed method to navigated implantation of acetabular com-ponent in computer-assisted total hip replacement surgery. In clinical practice, the target orientation of the component is described using two angles between the geometrical axis of the component and main anatomical planes. In this work, two sets of requirements were formulated based on the literature data: one that demands the tested system to be as accurate as the existing guidance systems that use marker-based tracking, and one that demands it to be visibly more accurate than non-guided (freehand) implantation. These requirements were used as baselines in the experimental part of this work, where the proposed prototype was tested in three different settings.

The tested system met the more rigorous set of requirements only in laboratory conditions, in absence of occlusions and with both localizer and target instruments stabilized, although only at closer distances. As new adverse conditions emerged, the accuracy declined, rendering the system only marginally more accurate than freehand positioning in phantom and in cadaver study. The exper-imental data were used to determine the factors that mostly affected the accuracy and recognition rate of the instruments. It was found that both aspects of performance were affected mostly by the size of the dataset used in estimating the orientation of the instrument. Thus, the most important limitation was related with the sensor. The current performance of the system is not completely satisfactory, but it can be improved by using a range sensor of greater resolution. Considering fast progress in the field of depth sensing, introducing the system into clinical settings may be possible in the near future.

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Streszczenie

Niniejsza rozprawa doktorska przedstawia now ˛a metod˛e pomiaru orientacji narz˛edzi chirurgicznych na podstawie danych zarejestrowanych przy pomomcy skalibrowanego zestawu kamer: klasycznej kamery wideo oraz kamery gł˛ebi bazuj ˛acej na czasie propagacji ´swiatła (time of flight, ToF). De-tekcja narz˛edzia opiera si˛e na danych geometrycznych i fotometrycznych pozyskanych z obydwu kamer. Okre´slenie jego orientacji polega na dopasowaniu trójwymiarowych chmur punktów, z któ-rych pierwsza reprezentuje model narz˛edzia a druga otrzymywana jest w wyniku akwizycji danych. Prezentowana metoda jest podstaw ˛a działania prototypu mobilnego lokalizatora wizyjnego, skonstru-owanego na potrzeby pracy i przetestskonstru-owanego w serii eksperymentów pomiarowych w warunkach laboratoryjnych i w warunkach zbli˙zonych do klinicznych.

Istniej ˛ace systemy ´sródoperacyjnego ´sledzenia narz˛edzi wykorzystuj ˛a najcz˛e´sciej techniki wi-zyjne, wymagaj ˛ace zamocowania dodatkowych markerów emituj ˛acych lub odbijaj ˛acych ´swiatło. Mar-kery te umo˙zliwiaj ˛a identyfikacj˛e narz˛edzi i okre´slenie ich poło˙zenia, komplikuj ˛a jednak zabieg chi-rurgiczny i wydłu˙zaj ˛a czas jego trwania. Dodatkowe ograniczenia tego rozwi ˛azania ujawniaj ˛a si˛e zabiegach ortopedycznych. Narz˛edzia podlegaj ˛a tu znacznym siłom i przyspieszeniom, mog ˛acym na-ruszy´c stabilno´s´c mocowania markerów. Ograniczenia te przyczyniaj ˛a si˛e do niskiej akceptacji metod wspomagania komputerowego w´sród chirurgów, pomimo tego, ˙ze orientacja narz˛edzi ma decyduj ˛ace znaczenie dla powodzenia wi˛ekszo´sci zabiegów ortopedycznych.

W niniejszej pracy analizowano mo˙zliwo´sć zastosowania zaproponowanej metody do nawigo-wanej implantacji komponentu panewkowego w zabiegu alloplastyki stawu biodrowego. Docelowa orientacja panewki opisywana jest standardowo za pomoc ˛a dwóch k ˛atów, które o´s panewki tworzy z płaszczyznami anatomicznymi. Na podstawie danych literaturowych w pracy sformułowano dwa zestawy kryteriów dotycz ˛acych dokładno´sci pomiaru k ˛atów implantacji: pierwszy z nich zakłada, ˙ze badany system powinien być tak samo dokładny jak istniej ˛ace systemy nawigacji, za´s drugi – ˙ze jego dokładno´sć powinna być wyra´znie lepsza ni˙z szacowana dokładno´sć implantacji bez u˙zycia narz˛edzi pomiarowych. W cz˛e´sci eksperymentalnej pracy przeprowadzono trzy rodzaje testów dokładno´sci: w warunkach laboratoryjnych, w badaniach na zwłokach, oraz w badaniu na fantomie.

Testowany system spełniał rygorystyczne wymagania jedynie w warunkach laboratoryjnych, w któ-rych narz˛edzia znajdowały si˛e w bliskiej odległo´sci od kamer i były przesłoni˛ete przez inne obiekty w nieznacznym stopniu. W trakcie badania na zwłokach oraz testów na fantomie zaobserwowano wi˛eksze bł˛edy pomiarowe: testowany system spełniał tu jedynie wymagania sformułowane w odnie-sieniu do zabiegów nie wykorzystuj ˛acych nawigacji. W celu wyja´snienia otrzymanych wyników dane zebrane w trakcie eksperymentów wykorzystano do przeanalizowania czynników wpływaj ˛acych na dokładno´sć systemu i skuteczno´sć detekcji narz˛edzi. Stwierdzono, ˙ze najistotniejszym jest rozmiar zbioru danych wykorzystywanych do dopasowania. Najistotniejszym ograniczeniem dla dokładno´sci systemu była wi˛ec rozdzielczo´sć przestrzenna kamery gł˛ebi. Szybki rozwój technologii w zakresie kamer ToF stwarza szans˛e na zastosowanie przedstawionej metody ju˙z w najbli˙zszej przyszło´sci.

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Autor pracy był stypendyst ˛a w ramach projektu Doctus – Małopolski fundusz stypendialny dla doktorantów,

współfinansowanego ze ´srodkow Unii Europejskiej w ramach Europejskiegu Funduszu Społecznego.

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Acknowledgements

I thank my PhD advisor, Prof. Józef Kozak, for motivating me to work, for many thoughtful discus-sions, and for all his help and care during my stays in Germany.

I thank my PhD co-advisor, Dr. Mirosław Socha, for his help with preparing and conducting the measurements and for his comments on the contents of this dissertation.

I thank Dr. Andrzej Kochman, for sharing his clinical expertise, for his comments on the medical aspects of the dissertation, and for his help with final measurment experiments.

I thank Prof. Janusz Gajda, for his care and support during my PhD program.

I thank my colleagues from Department of Measurement and Electronics at AGH: Katarzyna Heryan, Dr. Andrzej Skalski, and Marek Wodzi´nski, for careful reading of the dissertation and for their con-structive comments.

I thank my colleagues at B. Braun Aesculap: Andreas Alk, Mateusz Danioł, and Tobias Martin, for helping me with measurement experiments. I also thank Mateusz Danioł for proofreading of the dissertation.

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Notation

X: matrix

(·)⊤: transposition

x: column vector, x = [x1 x2 x3]⊤

˜x : homogeneous representation of vector x, ˜x = [x1 x2 x3 1]⊤

x : scalar

tr(A) : trace of a square matrix A, tr(A) =∑N_i=1ai,i

I_n×n: n× n identity matrix

diag(a1, . . . , an) : n× n diagonal matrix

   a1 . .. an   

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Introduction

Recent technologies allow surgeons to reach virtually any location in the human body, sometimes without even seeing it in the operative field. Instead, the surgeons can see through the tissues by looking on a computer screen, where the current position of surgical instruments is superimposed on a high quality image obtained pre- or intraoperatively from computed tomography or other imaging modality. There, they can see the anatomical target, the path to access it, and the critical structures, like blood vessels or nerves, that must remain intact. With the help of computer technologies, sensors, and imaging techniques they navigate through the tissues as if they explored an unknown area with a GPS device (although using the word “unknown” would be doing an injustice to the surgeons, given their vast knowledge of human anatomy). The tools and methods that make use of computer techology for planning, monitoring, guidance, and partial automation of surgical procedures are known under a common name of computer-assisted surgery (CAS).

Aim and scope

One branch of surgery that has particularly benefited from introduction of computer guidance is or-thopedics, where resection angles, implant orientations, and drilling trajectories are of key importance in restoring patient’s health and mobility. Also, in contrast to conventional orthopedic surgery, which is typically very traumatic, computer-assisted approach allows to perform the same type of surgical procedure using a minimally invasive technique. Finally, from technical standpoint, the bones are rigid structures, much easier to model mathematically than soft tissues, which allows the orthopedic guidance systems to work efficiently and reliably.

Computer-assisted surgical procedures often rely on measurement devices called localizers that track the location of the instruments in the operative field. Most of the existing tracking systems require additional objects that must be attached to the instruments and to patient’s anatomical structures, so that the former can be tracked relative to the latter. In optical localizers, the structures have a form of arrays of light-reflective or light-emitting markers that can be recognized by the system. The task of attaching and detaching the arrays diverts the attention of the surgeon from the procedure itself. Further limitations of marker-based approach become evident in computer-assisted orthopedic surgery, where the instruments are often subject to vigorous motion and large impact forces. In such conditions, the arrays may dislocate or detach, making the navigation inaccurate or infeasible without repeating certain steps. In addition, attaching the arrays to the bones requires using screws or pins and may itself cause trauma and intraoperative complications.

This work attempts to address the limitations of marker-based optical tracking in orthopedic surgery, particularly in implantation of the acetabular component of total hip endoprosthesis. In total hip replacement, the negative consequences of these limitations are undisputable: up to a quarter of patients who undergo the procedure need to have their implant replaced due to mechanical

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complica-tions that often occur as a result of misaligned implant components. Yet, the orthopedic community approaches computer-guided total hip replacement with moderate enthusiasm. The main barriers to the widespread adoption of computer guidance in total hip replacement surgery are extended intraopera-tive time, related with complicated handling of the CAS system, and spatial limitations of the operating room [1]. As a result, the majority of hip replacements are done without any assistive techniques.

This work aims at making computer-assisted total hip replacement less complicated, and more space-efficient, by abandoning the marker-based approach in favor of a technique that requires no instrument-mounted marker arrays. The main concept that is proposed to accomplish this task is a portable optical localizer that combines a video camera and a time of flight camera, a new type of sensor capable of acquiring 3-dimensional geometry of the scene in a single image frame. The two sensors make the system capable of tracking the instrument’s orientation based only on its shape and appearance. Additionally, the portability of the system allows to minimize the obstruction of the instruments by other structures in the operative field and to save space in the operating room. Thus, it eliminates two common limitations of stationary optical localizers: the lack of a clear line of sight, and the occlusions of the tracked object in the operative field.

Structure of the dissertation

This dissertation is composed of two main parts. The first part including chapters 1 to 3 provides the background information from both technical and medical perspective. This background is necessary to understand what are the requirements for a guidance system for total hip replacement surgery and what are the available techniques to satisfy these requirements.

Chapter 1 presents the concept of computer-assisted surgery. It describes CAS systems in terms of their taxonomy, architecture, and methodology. As the main components of the system relevant to this work, it identifies intraoperative measurement tools and the methods for maintaining correspondence between 3-dimensional datasets. Chapter 2 focuses on total hip replacement: the procedure that was chosen as the intended application of the proposed method and system. After briefly discussing the medical background, it presents more technical aspects of hip replacement: the typical surgical proce-dure, the components of the implant, and the instruments whose alignment determines its orientation. Finally, it shows the importance of implant alignment in context of postoperative complications. The concepts discussed in chapters 1 and 2 intertwine in chapter 3, which shows how the CAS techniques are implemented in the existing guidance systems. Chapter 3 presents the typical workflow of a computer-assisted hip replacement procedure. It describes the most common approaches to accom-plish each stage of this workfow that can be found in existing guidance systems. The remaining part of chapter 3 presents some limitations of these approaches and shows that, despite these limitations, the skepticism about the efficiency of computer guidance in total hip replacement is largely unfounded. Chapter 4 provides a transition to the second part of this dissertation. It describes the general methodology of testing the CAS systems, including assessment procedures and various measures of performance. The description is focused mainly on accuracy, and it is followed by a set of requirements that should be satisfied in order for the system to be clinically useful.

The second part (chapters 5, 6 and 7) presents the original contribution of this work: the mo-bile optical localizer and its application in various measurement experiments. Chapter 5 describes the hardware components of the prototype and the algorithms that were adapted or implemented to automatically detect the instruments within the operative field and measure their spatial orientation. Chapters 6 and 7 present the methodology, and the results of experimental evaluation. The experiments

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3

described in chapter 6 were performed in laboratory settings, in reproducible conditions, while those presented in Chapter 7 aimed at recreating the intended working configuration of the system.

The dissertation is concluded by a summary, which recapitulates the contributions of this work, points out its main limitations, and outlines the challenges for the future research.

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Chapter 1 Computer-assisted surgery

Computer-assisted surgery (CAS) is a set of tools and methods that make use of computer technology for planning, monitoring and partial automation of surgical procedures [2]. The subject of planning may be the surgical approach, the range of the procedure, or other aspects that serve the purpose of restoring either the function or the structure of tissues and organs affected by a trauma or a disease. The key mechanism for intraoperative implementation of the plan is monitoring the course of the procedure. Computer-assisted surgery provides the operator with methods that facilitate accurate assessment of the operative field. As a result, the actual operative field can be easily related with the preoperative plan, even in poor visibility or in case of complicated morphology. CAS also allows for partial automation of the surgery, e.g. by means of a robotic arm which carries out the tasks that demand high repeatability or precision, under supervision of the surgeon.

1.1 Evolution and classification of CAS systems

The concept of CAS evolved in the late 1980s from mechanical guidance systems which were then used in neurosurgery, ENT (Ear, Nose, Throat) surgery and otholaryngology [3]. These techniques, called stereotactic surgery, required fitting patient’s head with a rigid frame. The frame allowed to introduce the surgical tool along a predefined trajectory, into a destination located within the patient’s cranium [4]. The target location was specified in frame’s own coordinate system, based on two preoperative X-ray images taken in perpendicular projections. The stereotactic techniques were used for ablation, biopsies, injections, electrical stimulation, and in experimental research [5]. Surgical navigation

The advances in computer technologies, sensors, and medical imaging techniques have enabled the introduction of frameless stereotactic surgery based solely on 3-dimensional medical images. In frame-less stereotaxy, the frame is replaced with a tracking device called a localizer, which gives the operator more freedom in manipulating with surgical instruments. In the early applications, the system super-imposed the position of the instrument on a preoperatively acquired computed tomography (CT) image and displayed it in an operating panel. Due to its resemblance to navigating with a map in an unknown environment, the method was called neuronavigation, or, more general, surgical navigation.

The emergence of surgical navigation has inspired many attempts to apply minimally invasive operating techniques in the procedures, where it was not possible before. For example, in orthopedics navigation was initially used for screw guidance in transforaminal lumbar interbody fusion (TLIF), and for recoustruction of anterior cruciate ligaments [6, 7]. In both procedures, it allowed to plan an appropriate trajectory of bone drilling. As the trajectory could be seen on the screen, it was no longer needed to expose the operated site to the extent that was required for the conventional, non-navigated

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approach. As a result, the procedures became less invasive and the recovery time shrank substantially. Not long after navigation proved to be useful for spine and knee surgery, it was applied in guided implantation of prosthetic joints: in total knee and hip replacement. In the former case the aim of navigation was to maintain the mechanical axis of the lower limb [7] while in the latter the main objective was to preserve the limb length and the location of the joint’s rotation center [8].

Image-free navigation

Although initially the navigated orthopedic procedures required a CT scan of the operated structures, the first prototypes have shown that the anatomical model required to perform the procedure was much more simple than a full 3D reconstruction of patient’s bones. For both hip and knee replacement a simplified model was proposed, that included the basic geometric relationships between the joints and the mechanical axes of the lower limb. In image-free navigation the relationships are acquired in-traoperatively with a hand-held pointer placed by the operator in the anatomical or artificial landmarks of the bones. The system tracks the position of the pointer relative to the respective bones and builds the simplified kinematic model of the limb [7].

Active and semi-active systems

The main motivation for CAS in orthopedics is to improve accuracy and repeatability of conventional surgery. In navigation though, the role of the system is confined to indicating the correct drilling trajectory or cutting plane, and the outcome still depends on operator’s skills. The partial automation of the surgical procedures, although not possible in early CAS systems, was considered as an one of the directions for evolution of CAS from the very first concepts. In order to distinguish the partially autonomous systems from the surgical navigation, the robotic systems were called active systems. The idea of building the first active system, ROBODOC (Fig. 1.1B), emerged in 1980s at the University of California-Davis, USA [9]. Soon after, a prototype was presented, and in 1992 the robot was tested in a clinical trial. Since 1995 ROBODOC was routinely used in hip replacement surgery, for milling bone cavities for a prosthetic joint [10]. The cavity was shaped individually for each patient, according to a plan built upon a preoperative CT image. The coherence between the preoperative plan and the action of the robot was maintained using bone screws. Based on the location of the screws in CT image and their location measured intraoperatively using a pointing device, the system could calculate the desired intraoperative position of the mill. Although the active systems were used under constant surveillance of the surgeon, who could immediately abort the procedure in a hazardous situation, they still raised a question of exposing patients to an unnecessary risk. The use of robots was initially rationalized by favourable clinical outcomes. However, the long-term studies have shown an increased risk of complications in patients who underwent the robot-assisted hip replacement. As a result, the use of ROBODOC was partially abandoned [1].1

Semi-active systems are an attempt to combine the repeatability of the robotic CAS with the full human control over the surgery typical for navigation. As in active robotic systems, in semi-active CAS the tools, although installed on the arm of a robot, are driven by hand by the surgeon. The robotic arm is activated only at an attempt to move the tool outside the range specified by the intraoperative plan.

1_{The decisive moment for withdrawing the most of ROBODOC units was a suit filed in 2005 by a group of German} patients against Integrated Surgical Systems (who were developing the system at that time), over the allegations that the robot was “defective and dangerous”.

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1.1. Evolution and classification of CAS systems 7

This interaction mode, called the haptic feedback, is used in bone resection procedures, where the aim is to give the bone a specific shape [10]. Outside orthopedics, the semi-active systems are useful in laparoscopic surgery, for positioning the trocar – a pipe-like device that serves as a portal for inserting miniature surgical instruments into the abdomen. Here, the arm of the robot allows the operator to manipulate the trocar’s angle while maintaining a constant entry point [3]

One example of semi-active system that is currently used in clinical practice is RIO (Stryker, USA, Fig. 1.1C). RIO is built upon two projects, formerly developed by MAKO Surgical Corp. (USA) and Acrobot Co. (UK) [11]. It is used in orthopedics, in unicompartmental knee replacement and total hip replacement, to shape the periarticular parts of the bones, so that the implant can be inserted in a correct orientation ensuring maximum contact surface with the bone.

The concept of semi-active CAS is close to that of remote surgery (or telesurgery), where the surgeon is not present at the operating table but instead operates at a console with joysticks. The console may be located in the same operating theatre or at some remote site. The movements of the surgeon’s hands are transferred to the operative field by a robotic arm and the system provides the surgeon with visual and haptic feedback from the operative field. In such model, the methods of semi-active CAS are built into the system, between the console and the robotic unit, to restrict the robot’s operating range to a safe space. The indirect control also allows to scale down the movement of the surgeon’s hands and to filter out the hand tremor. As a result, much smaller structures can be operated with this technique than with conventional surgery. The best known and the most widely used example of remote surgical system is DaVinci (Intuitive Surgical Inc., USA), with 4200 active units around the world in 2017.2

Patient-specific templates

In computer-assisted orthopedic surgery, the desired tool trajectories or implant orientations can be secured also by mechanically restricting the tool’s path. Here, the role of computer technology is confined to the preoperative phase. Before the surgery, a patient-specific template (PST) is configured based on a CT scan. The template is essentially a customized cutting block, fixed on the bone surface immediately before resection. The shape of the block’s base is complementary to the bone, so that the block fits the bone in exactly one orientation and therefore uniquely defines a cutting plane or a drilling trajectory. Patient-specific templates were first described in the literature in 1998 [12]. Originally, they were realized as adjustable guidance instruments, meant for multiple use after re-calibration, but due to complex maintenance they did not gain much popularity. In recent years, the development of rapid manufacturing techiques, such as 3D printing, has revived the interest in mechanical guidance. Unlike the adjustable instruments, the 3D printed PSTs are single-use, and come pre-sterilized from the manufacturer, which makes them easy to integrate into the surgical workflow. One recent example is Visionaire, a PST system from Smith & Nephew, shown in Fig. 1.1D.

Apart from the personalized templates, the computer technologies and 3D medical imaging found their way to the traditional adjustable mechanical systems, resulting in a new generation of devices. An example of such device is HipXpert, an adjustable guide from Surgical Planning Associates (Fig. 1.1E). The spatial configuration of the device, including the bone support points and implantation angles, is established in a preoperative simulation on a CT model and then recreated intraoperatively [13].

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Figure 1.1:Examples of various CAS systems. A – OrthoPilot (Aesculap, Germany) image-free navigation for hip and knee surgery. B – ROBODOC (Curexo, USA) active robotic system. C – RIO (Stryker, USA) semi-active robotic system. D – Visionaire (Smith& Nephew, UK) patient-specific cutting blocks for femoral and tibial resection in TKR. E – HipXpert (Surgical Planning Associates, USA), adjustable instrument for mechanical guidance in THR.

Classification

The most commonly used system for classification of CAS techniques is based on the extent and on the nature of their interaction with the operator. The three main categories are passive, active and semi-active (or active-constraint) techniques (Fig. 1.2). The distinction between active and semi-active systems is natural: in active systems the robotic arm takes over during some repetitive parts of the surgery, while in semi-active systems the role of the robot is limited to preventing the surgeon from moving the tool outside a predefined area or trajectory. Together, the active and the semi-active systems are known by the common term of robotic surgery. More subtle is the distinction between semi-active and passive systems. The passive systems include surgical navigation and mechanical guides, either adjustable or patient-specific. The passive character of the navigation is clear, since the interaction between the surgeon and the system takes place only at the level of visualization, to show the surgeon the preoperative plan in the intraoperative context. The mechanical guides serve the same purpose as a semi-active robotic arm, but they need no power source, so they are also considered as passive CAS devices.

For navigation, a further distinction is introduced between image-based and image-free techniques, depending on whether planning or carrying out the procedure requires some form of preoperative or intraoperative imaging. In the literature, the distinction is sometimes extended beyond the category of passive systems, and all the image-based CAS techniques are called Image-Guided Surgery (IGS).

1.2 Methodology of CAS

The general methodology of CAS is shown in Fig. 1.3. Typically, the computer-assisted procedure includes three main stages: perception, decision, and action. In many systems, the two former stages are split between the preoperative and the intraoperative phase. In the preoperative phase, the computer

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1.2. Methodology of CAS 9 Computer-Assisted Surgery Image-Guided Surgery Image-free Robotic surgery Image-based (pre- and/or intra-operative imaging) Active Semi-active (Haptic) Patient-specific templates (PST) Mechanical guides (frame-based) Passive Navigation (frameless stereotaxy)

Figure 1.2:Taxonomy of CAS methods.

assistance begins with acquiring the data that describe the anatomy of the patient and the relevant physiological parameters (see Fig. 1.3, step 1). These data, together with general medical knowledge, are integrated into a model, using a set of alignment methods that transform the input datasets into a common reference frame. In order to perform the alignment, it is necessary to find some structural analogies within the data obtained from different sources. Therefore, if the data are images, it may be helpful to perform segmentation of structures and organs that are important during the surgery (2). The model resulting from the alignment allows to build a preoperative plan of the surgery (3). The plan may include, for example, the location of the target anatomical structure and the preferred surgical approach.

In the intraoperative phase, the acquisition of the patient-related data is repeated (4) to establish the correspondence between the preoperative model and the intraoperative setting (5). Based on the updated model, a plan for further steps can be defined (6). The planning is typically interactive and involves various forms of visualization which provide the operator with all necessary information. The plan is implemented through an interaction between the system and the operator (7). Depending on the system, the interaction may assume various forms, from simple passive guidance, through haptic feedback in semi-active systems to active techniques, where the role of the operator is limited to supervision.

Once completed, each step of the procedure is a subject to verification, done with the intraoper-ative diagnostic devices, such as fluoroscopes, CT or magnetic resonance imaging (MRI) scanners, endoscopes and electrophysiological recorders (8). Simultaneously, some auxiliary devices are used to monitor the state of the system itself, e.g. the position of the tools and reference objects that enable tracking the tools in an appropriate coordinate frames. The key requirement for the auxiliary devices is their non-invasiveness to the patient. Also, from the surgeon’s perspective, they should absorb as little time and attention as possible. These techniques, as the central element of this thesis, are discussed in detail in Section 1.4. Although the measurements they provide are not directly related with the patient, they allow for precise positioning of the tools and guarantee the correct alignment of the new diagnostic data. In active systems, the auxiliary measurements are also used as an input for control

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Perception Data acquisition and modelling Anatomical models + physiological knowledge Imaging (CT, MRI, PET) Preoperative plan Intraoperative plan Control Decision

Definition of the operative strategy Action Performing a surgical intervention Vision, telemetry, tactile sensors (for matching & control)

Intraoperative

Intraop. imaging (e.g. fluoroscopy, iCT, USG),

physiological signals Preoperative Passive guidance (visual/mechanical) Semi-active guidance (haptic constraints) Active control (autonomous sub-tasks) Segmentation/ Registration Segmentation/ Registration 1 2 3 4 6 5 7 8 9

Figure 1.3:Methodology of computer-assisted surgery.

algorithms (9). System components

Implementing the methodology presented in the previous section requires a number of key components that build up a CAS system. These components are:

• a virtual model representing the operated tissues or anatomical structures,

• a measurement technique that is used to track the intraoperative position of surgical instruments and other objects, relative to patient’s anatomical structures,

• a mapping method, to align the model to the intraoperative situation, and

• a method to perform the intervention according to the plan, including the way of interaction between the operator and the system.

Two of the above components, i.e. the mapping methods and the intraoperative measurement tech-niques seem particularly relevant to this work and will be discussed in greater detail in the rest of this chapter, in Sections 1.3 and 1.4, respectively. Before moving on, however, all four components mentioned will be briefly characterized.

The model of the operative field is a formal description of the operated structure and the spa-tial relationships between this structure and its surroundings. A simple example of such model for neurosurgical procedures is a preoperative MRI scan of the brain. The model can be extended by some a priori knowledge which is common to all interventions of a given type, e.g. the course of the main blood vessels or the nerves that should remain intact during the operation. The model also contains the data that are unique for a given case related to the individual anatomical variation or the

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1.3. Registration 11

existing pathological changes. Some advanced models that are designed to describe soft tissues can also include physiological data, that help to predict the deformation patterns, such as those related to cardiac or respiratory motion [14].

For the model to remain useful throughout the procedure, a continuous update is required in order for the model to remain useful, according to the progress in the operative field. One way to keep the model up-to-date is to use an intraoperative imaging technique. However, introducing the necessary equipment into the operating room is usually difficult due to the lack of space and sterility requirement. Also, for some imaging techniques, the patient and the medical personnel may be exposed to an excessive dose of ionizing radiation. Therefore, unless the imaging is absolutely necessary, more basic tools called localizers or trackers are preferred for intraoperative measurements in CAS.

Presenting the intraoperative data, such as the current location of the tool in the context of the preoperative model is only possible with the spatial mapping between the model and the intraoperative setting. There are three common cases, when the mapping is needed:

1. The data are acquired at different stages of the procedure, e.g. before and after resection of a tumor.

2. The data are acquired simultaneously, but with two different modalities, e.g. with an image-based and a non-image modality (such as a localizer).

3. The data describe analoguous, but different objects or structures, e.g. they describe the same organ, but they were obtained from different patients.3

The last component of the CAS is the model of interaction between the operator and the system, that was already mentioned in the methodology description. The model defines what range and type of data is available to the surgeon, how it is made available, and what actions are expected of the surgeon to execute the plan of the intervention.

1.3 Registration

The term of registration originally belonged in the domain of medical image processing, where it de-noted a procedure in which two different images of analoguous anatomical structures are transformed into one coordinate system [15]. In CAS, the aim of registration is similar, with two main differences:

1. The target coordinate frame is usually the intraoperative one.

2. Not all of the data that are subject of registration are images; the data can also be collected through mechanical contact or contactless by using optical methods.

Despite the differences, most of the registration techniques devised for images can be adopted for non-image or hybrid datasets, typical of CAS applications.

3_{In medical image processing, this type of data is called an atlas. Atlasses are useful for image analysis tasks, such as} the segmentation of non-distinctive anatomical structures, by correlating their position with other structures, better visible in the image. An example from computer-assisted intervention is a neurophysiological atlas used in deep brain stimulation, where an electrode is inserted into the brain. The atlas allows to precisely determine the target stimulation region, despite the homogeneous structure of the brain, based on the data previously collected for other patients.

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A detailed taxonomy of registration techniques was proposed by Maintz and Viergever in their survey, originally published in 1998 and revisited in 2016 [16, 17]. The three following sections discuss the subject of registration using three of the criteria proposed in these works that seem the most important in context of CAS: the type of the data (see section 1.3.1), the transformation models (section 1.3.2) and the optimization methods (section 1.3.3). The discussion is also limited to the data types, methods, and models that are relevant for this work.

1.3.1 Types of data and similarity measures

In order to perform a correct registration, some analogies must exist between the source and the target datasets. The analogies may relate to voxel intensity distribution or to some specific geometric features. The latter can be the surfaces, planes and points, obtained either from image segmentation or from direct pre- and intraoperative measurements. Some of them, such as the surfaces of internal organs, represent the actual anatomical structures. Other, e.g. midsaggital plane of the brain, bifurca-tions of blood vessels or mechanical axes of bones, are some abstracbifurca-tions used only to describe the morphology. In a broader sense, registration may be applied to non-anatomical data, e.g. for tracking the surgical tools.

The similarity measure for a given registration technique must be defined according to the type of the data. For intensity-based registration, the similarity measure is computed based on the predicted relationship between the voxel values of the corresponding regions of images. Similarity measures for geometry-based registration are defined in terms of distance between the corresponding geometric entities that can be found in both datasets. The correspondence is established by matching local descriptors which characterize the most distinctive properties of the entities. The intensity- and the geometry-based registration can also be combined to improve the efficiency of registration [15]. Geometry-based registration

Geometry-based methods were the first registration techniques used in CAS. Initially, they relied entirely on fiducials – miniature implants placed preoperatively in patient’s body. Fiducials could be reliably indentified in both source and target dataset and thus provide the correspondence between the preoperative and the intraoperative situation. For instance, in neuronavogation systems which were mentioned at the beginning of this chapter, the fiducials were fixed on patient’s skull. The target dataset included the positions of the fiducials in the preoperative CT and the source dataset was acquired intraoperatively, by measuring the positions of the fiducials by an optically tracked pointer.

The fiducials have proved to be effective in multimodal registration. This is because their shape and physical properties can be designed so that their locations are easy to determine in both source and target datasets. Additionally, they work equally good for image and non-image modalities.4However, the need for implantation of additional objects complicates the surgical workflow and increases the invasiveness of the procedure. Therefore, much research effort has been dedicated to abandon the use of artificial implants and to replace them with characteristic anatomical structures that are naturally present in the patient’s body. The recent developments in medical image processing allow in some cases to use either characteristic points (so called landmarks) or more complex entities obtained as a result of image segmentation, such as outlines or surfaces.

4_{The registration techniques that rely on fiducials are also the basis of the optical tracking systems, the most widely} used approach for tracking the intraoperative position of the tools, which is discussed in more detail in section 1.4

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Despite the progress that was made in geometry-based approaches, the most common intraopera-tive registration techniques are still based on sets of individual points [17]. The main reason for this is that point-based registration is reliable and fast, and many implementations readily available. What is also important for CAS, point-based algorithms are entirely compatible with the data obtained from intraoperative measurements. Most of the localizers in CAS capture the data as a set of points, which can be directly matched to their counterparts, identified in the image.

Control points

Many of the registration procedures make use of so called control points to describe the transformation between the source and the target dataset [18]. In geometry-based techniques the control points can be considered as a simplified representation of the entities upon which the registration is based. If the input datasets comprise of individual points, the control points are simply situated at these points’ locations. Otherwise, they can be obtained by sampling the surfaces or outlines. The registration requires an initial step of establishing the correspondence between the control points of the two sets. Once the correspondence is known, an iterative algorithm or a closed-form method is applied to estimate the final transformation.

1.3.2 Transformation models

Registration, as a task of finding a function that maps one observation of an object to another, is an inverse problem that has no unique solution. However, the set of possible solutions may be narrowed down to a certain class, by choosing an appropriate transformation model. The model describes the transformation by a function, that assigns each point in the source set, xs, with its counterpart in the

target set, xs:

xt = f (xs;θ), xs∈ S,xt∈ T, (1.1)

The vectorθ in Equation 1.1 is a set of parameters that uniquely define the transformation f .

According to the general classification, the transformation models can be divided into linear (global) and deformable transformations. The linear models preserve co-linearity and co-planarity of the datasets that undergo the transformation, so they are not suited to model local deformations of the datasets. Estimating the local deformations requires a deformable transformation model.

Linear models

The most basic example of a linear transformation is the isometry, a transformation that preserves the distances between any two points of the transformed set. A special case of isometry is a Euclidean transform, composed of translation and rotation. For a three-dimensional case, this operation can be written as the following matrix equation:

˜xt = [ R3×3 t3×1 0₁_×3 1 ] | {z } T_rig ˜xs, (1.2)

where R denotes the rotation matrix and t – the translation vector. If the source and the target sets have different scales, a more general model is needed. After including the scale, the Equation 1.2 takes the

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following form: ˜xt =     s1 0 0 0 0 s2 0 0 0 0 s3 0 0 0 0 1     | {z } Ts Trig˜xs. (1.3)

where the parameters s1, s2, and s3define the scale along each of the three dimensions.

Apart from the scale the linear transformations may include the global distortions which are intro-duced during the acquisition process. The global distortions are characteristic of image-based datasets, acquired with a projective techniques. The two most common examples are the skew distortion and the perspective projection distortion. The former results from non-perpendicular alignment of the beam with respect to the detector plane. The model that can reflect the skew distortion is obtained by extending the transformation from Equation 1.3 by three parameters, h12, h13i h23, which describe

the degree of distortion for respective planes of the global coordinate system:

˜xt=     1 h12 h13 0 0 1 h23 0 0 0 1 0 0 0 0 1     | {z } Th TsTrig˜xs. (1.4)

The latter type of global distortions results from finite size of radiation source, such as an X-ray tube in standard radiography or fluoroscopy. The imaging beam reaches each part of the detector at a different angle of incidence. Consequently, the trasformation model for registration between three-and two-dimensional images needs to include the parameters of the projection distortion that is present in the two-dimensional image. For example, the transformation between a preoperative CT and an intraoperative fluorogram can be found by generating a virtual fluorogram based on the CT. This is done by projecting the CT data onto a virtual plane. In to ensure that the corresponding anatomical points have the same coordinates, ut, in both datasets, the registration procedure has to determine

the orientation of the projection plane and to find the values of the projection parameters of the fluoroscope. The transformation that describes the projection is obtained by further extension of the model given by Equation 1.4:

˜ut = K3×4Trig˜xs, (1.5)

where the transformation Trig describes the relative alignment of the virtual projection plane and the

matrix K includes the parameters of the projection itself.

1.3.3 Optimization methods

Estimating the parameters of a transformation model, xt= f (xs;θ), can be formulated as an

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The objective function as be formulated as a sum of squares and can be written in a matrix form:

g(θ) = z⊤z, z =      xt,1− f (xs,1;θ) x_t,2− f (x_s,2;θ) .. . x_t,N− f (x_s,N;θ)      D×1 (1.6)

where (xt,i, xs,i), i = 1, . . . , N are the pairs of the observations that are the basis of the registration, and

θ = [θ1,θ2, . . . ,θM]⊤ is the estimated vector of parameters. In particular, the function 1.6 takes the

form of the sum of squared distances between the elements of the target set and their corresponding elements in the source set after applying the estimated transformation, that is

g(θ) =

N

∑

i=1

||xt,i− f (xs,i)||2. (1.7)

Depending on the assumed model of transformation, the parameters θ can be computed as a closed-form solution or estimated iteratively. The closed form-solution is available to linear trans-formations, such as the Euclidean transform, and to simple deformable models. In the remaining cases the parameters of the model cannot be computed directly, and an iterative solution is required [16]. Closed-form solution

An example of a closed-form solution of the registration task is the best-fit algorithm [19, 20]. The procedure is defined for two sets of corresponding points, interrelated with a Euclidean transform, as in Equation 1.2. For practical reasons, the algorithm optimizes the rotation given in a form of a unit quaterion, written as a 4-element vector, qr= [q0 q1 q2 q3]⊤, where q20+ q21+ q22+ q23= 1. As a result,

the vector of the optimized parameters is built by concatenating two vectors that denote the rotation and the translation:

θ = [ q_r t ] . (1.8)

As the objective function, the best-fit algorithm uses the formula described by Equation 1.7, that takes the matrix form

g(θ) = 1 N N

∑

i=1 ||xt,i− R(qr)xs,i− t||2 (1.9)

where N is number of points in each of the sets, and R(qr) is the rotation matrix corresponding to the

unit quaternion, qr.5The Euclidean transform between the source and the target sets is found in two

steps: First, the optimal rotation is computed using the covariance matrix of the two input sets, and then the translation is determined. Prior to computing the covariance matrix, the input datasets, xsand

x_t, are translated to their respective centroids,

mt = 1 N N

∑

i=1 xt oraz ms= 1 N N

∑

i=1 xs. (1.10)

5_{For the method to calculate the rotation matrix for a given unit quaternion, the reader is referred to the article by} Horn [20]

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Then, the covariance matrix is computed as S_t,s= 1 N N

∑

i=1 (x_t,i− mt)(xs,i− ms)⊤. (1.11)

The optimal rotation in terms of the objective function 1.9 is given by a unit quaternion qr, that can be

determined by normalization of a vector that corresponds to the largest eigenvalue of the matrix Q(S_t,s) = [ tr(St,s) ∆⊤ ∆ S_t,s+ S⊤_t,s− tr(St,s)I3×3, ] (1.12) where∆ is a vector composed of the elements of a matrix A = St,s−S⊤t,s, such that∆ = [A23 A31 A12]⊤.

In the second, final step of the best-fit procedure, the quaternion qris used to compute the translation:

t = mt− R(qr)ms. (1.13)

Iterative solution

In case of more complex transformation models the estimation of the transformation parameters is then performed with general non-linear optimization methods. In such methods, the objective function is minimized by iteratively updating the parameters, according to the scheme given by the general expression:

θ[k+1]=θ[k]+λδ[k], (1.14)

where the functionδ returns a vector of updates for the parameters, θ, based on the previous iteration, and λ defines the amount by which the vector is updated. The detailed form of the update term, λδ, depends on the particular optimization method. The most basic example is the steepest descent method, in which the direction of the update is determined by the gradient of the objective function at θ, andλ is a small constant value. The algorithm is proven to converge to the local minimum of the objective function, but with the basic update scheme the process may be slow. In order to speed up the convergence, some modifications of the steepest gradient method are used, such as the conjugate gradient. In conjugate gradient method, the vectorθ is updated recursively, based on the combination of the gradient of the objective function computed for the current set of parameters,θ[k], and on the update from previous iteration,δ[k]=−∇g(θ[k]) +βkδ[k−1].

Another method to improve the convergence of iterative optimization algorithms is to use the second derivative of the objective function. High value of the second derivative suggests a rapid change in gradient, i.e. the closeness of a local extremum of the objective function. An example of a method which involves the second derivative is Newton’s algorithm, that updates the parameters of the model according to the formula:

θ[k+1]=θ[k]−λ(∇2g(θ[k]))−1∇g(θ[k]) (1.15)

The inversely proportional relationship between the amount of the update and the second derivative in the Equation 1.15 allows for more precise updates in the proximity of the extremum, that lower the risk of omitting it. The more computationally simple variant of the Newton’s method is a Gauss-Newton algorithm, that does not require determining the second derivative of the objective function. Instead, the values of the second derivative for the current set of parameters,θ[k], are approximated

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with a Jacobian J∈ RD×M. The individual elements of the matrix J are computed as jdm=∂zd/∂θm,

where zd denotes the elements of the vector of differences, used to compute the value of the objective

function according to the Equation 1.6. The second derivative of the objective function is approximated according to the expression:

∇2_g(

θ)≈ J⊤J, (1.16)

and the update scheme for the transformation parametersθ from the Equation 1.15 takes the form:

θ[k+1]=θ[k]−λ(J⊤J)−1∇g(θ[k]). (1.17)

A combination of the steepest descent and the Gauss-Newton methods is the Levenberg-Marquardt algorithm. Here, the update term is a linear combination of the terms that are used in both these methods:

θ[k+1]=θ[k]− (J⊤J +ζI)−1∇g(θ[k]). (1.18)

based on the registration error the algorithm adaptively adjusts the effect of each term on the update vector, by changing the weighting coefficient,ζ.

1.3.4 Model-based recognition and matching

Registration can be the component of a more general task of determining whether an object of interest, defined in the source set, is present in the target set. In the literature, the task is known as recognition and matching. In CAS, the problem of recognition and matching is often circumvented by design of the acquisition procedure. For instance, the image-free navigation systems for computer assisted orthopedic surgery (CAOS) prompt the operator to acquire the anatomical bony landmarks with a pointer tool in a predefined order. Consequently, the recognition and matching is performed manually by the operator. The fact that all the required points can be found and measured implies the presence of the structure in the target dataset. The sequence of acquisition solves the problem of correspondence between the points of the two datasets.

As either the complexity of the data or the desired rate of recognition and matching increases, the manual approach to the problem becomes inefficient or even infeasible. For example, manual recognition and matching is not suited for estimating the positions of surgical tools, that needs to be done continuously and in short time intervals. In such cases, the automatic recognition and tracking is more suitable. The most frequently used approach is optical tracking. The technique is is based on artificial objects with distinctive optical properties, called fiducials or markers. The location of each individual fiducial can be measured using triangulation, by capturing an image of the fiducials with a calibrated camera setup. The fiducials are fixed on each target object in a unique rigid configuration that allows for unambiguous identification. The aim is then to find a rigid transformation given by Equation 1.2, where the source dataset is the set of markers’ coordinates measured during the tool calibration, and the target dataset are the corresponding coordinates measured with the localizer during the actual tracking. This is an example of model-based recognition and matching, since the source dataset is assumed to be the true description of the object of interest, i.e. the model. The target dataset is usually obtained intraoperatively and is sometimes referred to as the input dataset.

In model-based recognition and matching, the input data are commonly a set of 3-dimensional points, obtained either directly from the operative field or from an already pre-processed datasets. The former include those based on palpation of anatomical structures or artificial landmarks, as well as the contactless optical methods. The indirect measurements depend largely on the data obtained with

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medical imaging modalities, such as CT, MRI or ultrasound. For this case, the points can be derived from the surfaces resulting from image segmentation. As for the type of transformation between the source and the target dataset, the following discussion is limited to Euclidean transformations, as it is the most widely adopted model in CAS [4].

As it was shown earlier in this section, the parameters of a Euclidean transformation between two sets of points can be found with a closed-form solution. However, for the solution to be applicable, two general conditions need to be met:

1. The object from the source set needs to be present in the target set.

2. Each point from the source set needs to have an exact counterpart in the target set, a point, that satisfies Equation 1.1.

In recognition and matching, the difficulty lies in the fact that none of these assumptions is necessarily true. As a result, the only viable solution is the hypothesize-and-test approach, that is generating hypothetical transformations, testing them against some criteria and choosing the one that fits the best. The complexity of model-based recognition and matching can be reduced by operating on entities that include more context information about the surroundings of the point. The context helps to reduce the number of possible solutions. Two common examples of context-based entities are oriented points and keypoints. The oriented point is a point assigned with a unit vector, perpendicular to the object’s surface at that point’s location. The keypoints form a more general class of entities that are composed of a point and a local descriptor that does not necessarily relate to geometric properties.6By combining two oriented points together, one obtains oriented pair, an entity that is particularly useful for the task, since it is sufficient to generate a unique 3D transformation hypothesis. The oriented pair is composed of two oriented points and the Euclidean distance between them.

The model-based recognition and matching procedure starts with extraction of context-based entities from both model and input datasets. The least number of entities required to generate a transformation hypothesis is called a minimal sample. For bare 3D points, the minimal sample to generate a Euclidean transformation consists of three points. For oriented pairs, the size of minimal sample is one. The matching strategy is similar to that known from the Generalized Hough Transform (GHT) [22]. The minimal samples are taken pairwise from the model and the input sets, and a transformation hypothesis is generated for each pair. The transformation casts a vote in a feature space, whose dimensions correspond to the transformation’s parameters. Each of the dimiesions is quantized with some small step, so that the space is effectively an array of discrete bins. Once all possible minimal samples are tested, the bin with the largest number of votes is chosen as the one that corresponds to the optimal transformation.

The exhaustive matching strategy outlined above may be time-consuming and inefficient because a large number of samples must be tested before the winning hypothesis is chosen. As a consequence, many transformation hypotheses need to be generated. This limitation can be addressed by using a technique called geometric hashing [23]. The idea of geometric hashing is to map the entities drawn from both datasets to a lower dimensional, pose-independent descriptors. An example of such descriptor was used by Papazov et al. [24]. They proposed to assign an oriented pair (a pair of points,

6_{For an exhaustive review of local descriptors for model-based recognition and matching, the reader is referred to the} work of Guo et al. [21]

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p1and p2and their corresponding normals, n1and n2) with a 4-element vector:

f =[||d|| ∠(n₁, d) ∠(n2,−d) ∠(n1, n2)

]⊤

(1.19) where d = p2−p1,||d|| is the length of b, and ∠ is the angle between a pair of vectors. The method of

calculating the descriptor is illustrated in Fig. 1.4. The transformation hypotheses are generated only for entities with similar descriptors which speeds up the matching procedure, particularly when the transformation is computationally costly to estimate.

m 1 m 2 n 1 n 2 d (n2, - )d (n1, )d (n1,n2)

Figure 1.4:Oriented pair of points with normals.

The time of model-based matching can be also shortened by limiting the number of samples to be tested, using a Random Sample Consensus (RANSAC) scheme [25]. Similarly to the exhaustive search, RANSAC is an iterative procedure, but the tested pairs of samples obtained from the model and the input datasets are drawn at random. The transformation hypothesis based on each pair is generated and verified immediately, so that no accumulation table is necessary. The hypothesis is verified by transforming the model dataset to the input dataset and testing the transformed model for some matching criteria. For instance, in standard RANSAC scheme the hypothesis is assumed to meet the criteria if a sufficiently large fraction of model points falls within a tolerance margin built around the points of the input dataset. Once the matching criteria are met, the hypothesis is accepted as a solution and the procedure is terminated. The maximum number of iterations is determined in terms of probability of drawing an entity that belongs to the object of interest, which is especially important when the input dataset includes a number of background points.

The above matching techniques have some limitations that can lead to inaccurate outcomes. For GHT approach, the parameters of the estimated transformation are constrained to a discrete set of values, which results in quantization errors. In RANSAC, the resulting transformation is based on a randomly selected minimal sample, which introduces some random errors. Thus, the result obtained with these methods can serve as a starting point to further optimization. The parameters of the initial transformation can be fine-tuned using Iterative Closest Point (ICP) algorithm [26]. ICP minimizes the distance between two sets of points, even if the correspondence between these sets is unknown. In successive iterations each point of the model set is assigned with its closest counterpart in the input set. Then, the best-fit algorithm is applied to find the optimal transformation that maps the corresponding points. The transformation is then applied to the input set, and the procedure is repeated. The algorithm terminates when the minimized objective function reaches the level below some threshold value. The convergence of ICP to the local minimum can be proven mathemetically, but without a guarantee, that the optimization will stop in the global minimum. Therefore, starting with an appropriate initial

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transformation is of great importance to the optimization outcomes [26]. Many versions of ICP have been presented in the literature, varying in methods of finding the closest points, weighting schemes, and the initial selection of points that are used for optimization [27].

1.4 Intreoperative measurement techniques

Intraoperative measurements in CAS rely on mechanical probing, contactless sensing, and kinematics. The two former methods yield sets of 3-dimensional surface points of anatomical structures. The former are typically done either by touching the target structure with the tip of a hand-held pointer or by running the tip over the target’s surface. The location of the pointer is simultaneously measured with an external tracking system, so that all the points digitized with the tip can be expressed in a common coordinate frame. For most of the contactless methods, the points are acquired optically, by means of artificial visual patterns, fixed to (or projected on) the measured surface. The patterns can be physical markers, light points [28] or some more complex light shapes [29].

The kinematic techniques are useful in orthopedic navigation, where the operated structures are rigid solids. They are used to determine the axis of a hinge joint (such as the knee7), the rotation center of a ball-and-socket joint (e.g. the hip joint) and the main anatomical planes of patient’s body. The kinematic measurements are based on the trajectory that is described by each point on the bone during a typical movement in the examined joint, i.e. an arc and a sphere for hinge and ball joint, respectively. During the measurement, a tracking device is fixed at distal part of the bone, and its trajectory is measured as the bone moves in the joint. Based on the trajectory, the center or the axis of the joint is determined using analytical geometry, by fitting a sphere or a circle to the acquired points. The kinematic measurements are less accurate than direct mechanical probing, but are the only available measurement technique if the object of the measurement lies within the volume of the bone and has no distinctive representation in its morphology.

Another source of spatial information in CAS is intraoperative medical imaging, such as CT, MRI, fluoroscopy or ultrasonography. The imaging techniques allow for measuring the points located both on the surface and within the volume of the organs. The measurements may be based on fiducials implanted into the tissues that facilitate the image registration. The intraoperative imaging provides the most complete spatial information among all the methods presented in this section, but its practical sig-nificance in CAS is limited for a few reasons. Depending on the technology, the imaging devices may be incompatible with other equipment in the operating room or with the surgical workflow, because of the space and time constraints, respectively. Some of the imaging modalities still suffer from image distortions, that make the accuracy of image-based measurements insufficient for CAS applications. Also, the X-ray based techniques have to be used sparingly, to avoid unnecessary exposition to ionizing radiation.

Most of the intraoperative measurement techniques require a tracking system called a localizer. In direct and kinematic methods, the localizer is an essential measurement tool. In image-based measurements, it plays an auxiliary role. For example, in tomographic techniques, it allows to reduce the number of intraoperative scans needed for navigation. The localizer tracks the current location of tools and the CAS system displays the location in the image. The scan can be used for navigation as long as some major change of anatomy occurs, such as a resection of a tumor. Localizers also

7_{The knee is actually a modified hinge joint, but for the purpose of kinematic measurements it is simplified to the hinge} model.