Status of the ATLAS Muon Spectrometer Alignment Rasnik Image Analysis Upgrade Marc Kea, 2-7-2007

(Short) Introduction to Rasnik LED with coded mask (see next slide) Pixel sensor Lens Rasnik: a 3-point alignment system for the ATLAS muon spectrometer Relative displacements between the three components can cause: A translation A scaling A rotation of the image on sensor

The Rasnik coded mask ‘in phase’ with chessboard pattern: bit = 0 ‘pivot square’, indicates crossing of horizontal & vertical codeline, contains no code info ‘counter-phase’ to chessboard pattern: bit = 1 Position of this codeline: ‘00100010’ = 34 blocks vertically above (0,0) Consists of a fine chessboard pattern (chromium on glass), block size typically 120µm To ensure a large dynamic range, every 9 lines and columns contains an 8-bit binary code which gives the vertical and horizontal ‘coarse’ positions, respectively Camera (4.7x3.5 mm) only ‘looks’ at a small portion of the mask (20x40 mm)

Purpose of the Rasnik Image Analysis Find rotation of the mask on sensor Find the scale (block size on sensor / true block size) Find the x- and y coordinate in the mask system of a reference pixel of the camera system

What is attainable? Cramer-Rao lower bound (CRLB) for shift estimation gives a lower limit for resolvable translations for a given image -> Lower bound of resolution proportional to noise level in image and inversely proportional to the gradient energy in translation direction For a sharp short-range rasnik image (e.g. from a praxial system): CRLB ≈ 10nm For a long range (e.g. projective) system: CRLB ≈ 100nm This is a lower bound, any implementation will have a lower resolution

(Very Short) Description of the Current Rasnik Image Analysis Gradient based (edge detection) method for finding the fine shift of the chessboard pattern Intensity based (thresholding) method for determining the bit value of the codebits

Current Rasnik image analysis Gradient in x of a Rasnik image Line-fitting algorithm fits lines to gradx and grady to determine scale, rotation and fine translation in x and y (image taken from Kevan Hashemi’s Rasnik analysis page http://alignment.hep.brandeis.edu/Devices/RASNIK/)

Current Rasnik Image Analysis Fine translations in x and y, rotation and scale can now be used to compare the intensity of each code square to its neighbors (thresholding) to determine its bit value Now each codebit has to be judged for reliability; codebits can be obstructed by dust, cables, etc. etc. Be wary of ‘wrong’ code readings: a wrong answer is far worse than no answer Final x and y position = coarse (code) shift + fine shift

2D FFT Computes a 2D spectrum of the spatial frequencies and their phase in an image 2D FFT: compute and put back the 1D FFT of each row in an image, then compute and put back the FFT of each column of the resulting image

2D FFT and Rasnik X = Spatial domain (multiplication) 2D FFT 2D FFT = * Frequency domain (convolution)

Spectrum of a real Rasnik image Lower frequency region: background light variations, large dust, codebits DC term First harmonics Higher harmonics (the sharper the image, the more higher harmonics!) Higher frequency variations (noise, fine dust) ωy ωx

Recovering rotation, scale, translation in the Fourier domain Operation in spatial domain Spatial domain Fourier domain Rotation through θ f’(x,y)=f(xcosθ + ysin θ, ycosθ - xsin θ) F’(ωx, ωy) = F(ωxcosθ + ωysin θ, ωycosθ - ωxsin θ) Scaling by a,b f’(x,y) = f(ax,by) F’(ωx, ωy) = (1/|a||b|) * F(ωx/a, ωy/b) Translation by x0,y0 f’(x,y) = f(x-x0,y-y0) F’(ωx, ωy) = exp(-i(ωx x0 + ωy y0)) F(ωx, ωy)

In practice: Rotation Scale Translation (within one period of main harmonic) obtained from phase of first harmonics

Implementation Select centre 256*256 pixels (for now) Apply windowing function to reduce edge artefacts Perform 2D FFT Find spectral main harmonic peaks Fit main harmonic peaks Look for second harmonics, if they contain enough signal power fit them also From this, determine scale, rotation Find accompanying phase and thus fine translation Use this as input for codeline reading!

Progress of implementation Algorithm to find rotation, scale, fine translation implemented In simulations (SNR = 10, rasnik image simulated with 2D sinefield): Resolution in rotation: 10^-4 rad Resolution in scale: 10^-4 Resolution in translation: 10^-4 period (corresponds to approx. 50nm) HOWEVER a real Rasnik image is not a 2D sinefield! More testing will be done using images from an ultrastable test bench Algorithm to find and read codelines implemented, but easily disrupted by dust, obstructions, etc. Far less than ideal at the moment.

Codelines Kevan Hashemi (Brandeis) image analysis routine for the endcap muon spectrometer has been tried and tested for years, proven to be very robust Agreed to look into integrating initial fine shift calculation using 2D FFT with Kevan’s analysis, perhaps even standardising the combined analysis for use both in the endcap and barrel Thanks to Pierre-Francois Giraud the integration of Pascal and c++ will be possible

To do Combine fine and coarse shift routines Do testing on stable test-bench images to compare new and old analyses Run the new analysis on a single line in ATLAS (infrastructure for this is ready thanks to Robert Hart) If all goes well, implement the new analysis on lines which cannot be analysed using the old analysis And then in the entire barrel muon spectrometer

Bonus: other elegant stuff 2D FFT 2D IFFT Filter: keep boxed part, discard rest 2D FFT 2D IFFT