Proposal for precise geometry parametrization of planar imaging detectors.

Content

Coordinate system for setup

Axes definition

Cartesian coordinate system of setup is defined by three mutually orthogonal right-hand-indexed axes with origin in the interaction point (IP) of the photon beam with target:

X – pointing from IP to the north,
Y – pointing from IP to the top, and
Z – pointing from IP along the photon beam propagation direction,

as shown in the plot:

Each photon with sample collision happens in different place... IP is an abstract position of the crossing point of the "beam line" and "target", which both are not well defined... Is it really good definition?

Detector position

Detector coordinate system may have a translation and rotation with respect to the setup, which can be defined by the 3 vectors in the setup frame:

P – translation vector pointing from IP to the detector origin,
e_x – unit vector along the detector frame x axis,
e_y – unit vector along the detector frame y axis,

as shown in the plot:

Third unit vector e_z – is assumed to be right-hand triplet component,

e_z=[e_x × e_y].

Components of these three unit vectors form the rotation matrix

/ e_x1 e_y1 e_z1 \

R_ij = | e_x2 e_y2 e_z2 |

\ e_x3 e_y3 e_z3/

where indexes "i" and "j" corresponds to the vector components in the setup and detector local coordinate frames, respectively. Within this definition 3-d pixel coordinate "c" in the detector frame can be transformed to the setup coordinate "C" using equation

C_i=R_ij·c_j + P_i.

Parameters for detector position and orientation

The list of parameters, which defines detector position and orientation in the setup (lab) frame consists of vector components:

P1	P2	P3	ex1	ex2	ex3	ey1	ey2	ey3

assuming e_z=[e_x × e_y], and |e_x|=|e_y| =|e_z|=1.

Alternatively, directions of the three unit vectors e_x, e_y, and e_z of the detector frame can be defined in terms of three Euler angle rotations with respect to the setup (lab) frame:

P1	P2	P3	alpha	beta	gamma

with classic relations between (e_x,e_y,e_z) and (alpha, beta, gamma) presentations.

Access to these parameters should should be provided at least two methods of the calibration store object:

(P1, P2, P3, ex1, ex2, ex3, ey1, ey2, ey3) = calib.get( type, detector_name )

calib.put(type, detector_name, (P1, P2, P3, ex1, ex2, ex3, ey1, ey2, ey3))

or similar methods for Euler angles.

Coordinate system of the detector

In this note we assume that:

the detector image is produced by a set of tiles (a.k.a., segment, section, 2x1, sensor, Si-pixel matrix, etc),
the tiles form almost flat (x-y) surface around z=0 within precision of production technology and installation,
the tile rotation angles in (x-y) are N*90 degrees, where N=0,1,2,or 3,
detector local coordinate system is defined in this plane, as explained below.

These assumptions are not necessary for most of algorithms. But, in many cases it may be convenient to use small angle approximation or ignore angular misalignment for image mapping between tiles and 2-d image array.

Tile ideal geometry

Each tile has well defined by design geometry of pixels, which does not need in separate calibration.
Pixel center (x,y) coordinates at z=0 for each tile can be defined as a look-up table in its "natural" matrix-style coordinate system.
For example, CSPAD 2x1 tile pixel geometry is defined as

Regular pixel size is 109.92 x 109.92 µm²,
Pixel size in two middle columns 193 and 194 is 274.80 x 109.92 µm².

For any type of tile there should be a class or method returning pixel coordinates in this ideal geometry

(x, y, z) = get_pixel_xyz(row,column),

or look-up tables for consecutive pixel indexes

(xarr, yarr, zarr) = get_pixel_xyz_arrays().

Memory model

It is assumed that each tile is presented in DAQ or offline data by a consecutive block of memory. Uniform matrix-type geometry of pixel array is preferable, but other geometry can be handled. For example, for CSPAD 2x1 sensor pixel index in the tile memory block of size (185,388) can be evaluated as

i = column + row*388.

Detector data record consists of consecutive tile-blocks, in accordance with numeration adopted in DAQ. For effective memory management, some of the tile-blocks may be missing due to current detector configuration. Available configuration of the detector tile-blocks should be marked in a bit-mask word in position order (bit position from lower to higher is associated with the tile number in DAQ).

Optical measurements

Optical measurements provide 3-d coordinates of 4 corners for all tiles in the detector in some arbitrary microscope plane which coincides with detector imaging array (tiles) plane within precision of installation.
Tile corner coordinates do not necessarily coincide with corner pixel centers or edges.
Numeration of tiles should not necessarily coincide with their numeration in DAQ, but it should be done in fixed order or with indication of tile numbers:
Optical measurements provide a list of coordinates for all sensor points:
Tile #
Point # X[µm] Y[µm] Y[µm]

Quality check

Quality check of optical measurement may include for each tile:

equity of opposite sides
equity of diagonals
equity of tilt angles for all sides
tile flatness in 3-d

Position of tiles in the detector

Detector pixels' geometry in 3-d can be unambiguously derived from

the list of coordinates obtained in optical measurements,
tile ideal geometry, and
designed orientation of sensors in the detector.

The tile location and orientation can be defined by the table of records (see example of file: geometry)

Parent name	Parent index	Object name	Object index	X0[µm]	Y0[µm]	Y0[µm]	Rotation Z [°]	Rotation Y [°]	Rotation X[°]	Tilt Z[°]	Tilt Y[°]	Tilt X[°]

where

Parent name - name and version of the parent object; all translation and rotation pars are defined w.r.t. parent Cartesian frame
Parent index - index of the parent object
Object name - name and version of the object inserted in parent frame
Object index - index of the object
X0, Y0, and Z0 [µm] – object frame origin coordinates in the parent frame
Rotation X, Y, and Z [degree] – object ideal (design) rotation angle around axes Z, Y, and X of the parent frame
Tilt X, Y, and Z [degree] – tilt angles for correction of the object rotation angles around axes Z, Y, and X of the parent frame, respectively

Tile center coordinates are defined as an average over 4 corners.

Tilt angles are projected angles of the tile sides on relevant planes. Each angle is evaluated as an averaged angle for 2 sides.

Origin of the detector local frame is arbitrary. There are few preferable options for choosing coordinate frame origin:

use the same coordinate system as in optical measurements,
apply minimal offset (translation) to optical frame in order to make all coordinates positive (see left plot below),
use averaged coordinate of all tiles (see right plot below),
use user-defined translation, for example set origin in the evaluated point of beam crossing with the detector plane.

Pixel coordinate reconstruction in the detector frame uses

tile ideal geometry,
tile center positions,
tile N·90 rotation and tilt angles:

(x,y,z)_{in detector} = (Xc,Yc,Zc) + R(N·90+tilt) × (x,y,z)_{in tile},

where rotation matrix R is a product of three rotations around appropriate axes,

Pixel coordinates in the detector should be accessed by method like

(x, y, z) = get_pixel_xyz_in_detector(tile_number,row,column),