# Natural measures of alignment

###### Abstract

Natural coordinate system will be proposed. In this coordinate system alignment procedure of a device and a detector can be easily performed. This approach is generalization of previous specific formulas in the field of calibration and provide top level description of the procedure. A basic example application to linac therapy plan is also provided.

Keywords: calibration, continuous groups, group action

## 1 Introduction

There is many approaches to measurement of discrepancy between ideal case and those obtained from real experiment. Short overview of the subject can be found in [Callibration].

This paper redefines the measurement of the quality of alignment using some kind of specific natural coordinates system based on induced group action in the detector space. It uses approach from category theory [Conceptual, Spivak, Spivak2, CategoryForWorking] and abstract algebra [AlgebraChapter0], i.e., the calibrated device is accessible only by the ’morphisms’ - the set of gauges that can be changed in order to adjust the configuration. This description does not go into device details. These gauge degrees can be represented as a group action on the device, which change its configuration. In our approach we focus only on continuous groups - Lie groups [Humphreys], e.g., rotation of device, position change, beam intensity etc. These gauge degrees of freedom projected on the detector give some natural coordinates that can be used to made calibration. The number of ’projected’ coordinates depends on detector geometry and position. It is a ’maximal set’ that can be used for perfect calibration using available gauges only. This is common task in practice as usually no manipulation of the device apart using the gauges is allowed.

This paper is significant generalization of the approach from [Callibration]. The method presented in [Callibration] is in many cases suboptimal as the number of parameters for optimization is selected in most cases arbitrarily. This leads to overdetermined optimization problem. In this context our method presents one specific case, based on geometry of the problem, maximal number of available degrees of freedom and reduction of degrees of gauges by projection. We start from small number of general assumptions on geometry and basic facts from category theory and representation theory of groups. This leads to somehow unique (up to coordinate selection) method that can be extended adding ’new degrees’ of gauges, i.e. group action, and therefore optimization parameters. This approach can be seen as a scheme for generating calibration methods. This abstract description of the calibration process also allows to identify points where the procedure can fail and therefore gives exact and precise suggestions how to avoid these problematic points.

The paper is organized as follows: In the next section general description of natural coordinate system in detector space is presented. Then the idea of alignment measure is given. Finally in the last section some specific example applied to the cancer therapy plan is presented.

## 2 General considerations

In this section general considerations on the device and detector setup using Lie groups approach [Humphreys] will be presented in two consecutive subsections. The presentation uses continuous groups, however discrete groups [AlgebraChapter0] can be also used in analogous way.

### 2.1 The device

It is assumed that there is an imaging device or generally a device which output can be registered by some detector. The device has some degrees of freedom and can be moved along some directions using gages. This movement can be described by the action of some Lie group (continuous group) on the space of parameters of the device, see [Humphreys] for review of Lie groups, and [AlgebraChapter0] for group action.

The device can be seen as a set of parameters in some space that uniquely gives configuration at a given instant of time. These parameters can have either geometrical meaning like position or angle, or intrinsic meaning like the intensity of the imaging beam. Therefore, we have the first observation

###### Observation 1

There is iso mapping from the device to the parameter space