Status of the ATLAS Muon Status of the ATLAS Muon Spectrometer Alignment Spectrometer Alignment

Rasnik Image Analysis UpgradeRasnik Image Analysis Upgrade

Marc Kea, 2-7-2007Marc Kea, 2-7-2007

(Short) Introduction to Rasnik(Short) Introduction to Rasnik

Rasnik: a 3-point alignment system for the Rasnik: a 3-point alignment system for the ATLAS muon spectrometerATLAS muon spectrometer

Relative displacements between the three Relative displacements between the three components can cause:components can cause:• A translationA translation• A scalingA scaling• A rotationA rotationof the image on sensorof the image on sensor

LED with coded mask (see next slide) Lens

Pixel sensor

The Rasnik coded maskThe Rasnik coded mask

Consists of a fine chessboard pattern (chromium on glass), Consists of a fine chessboard pattern (chromium on glass), block size typically 120µmblock size typically 120µm

To ensure a large dynamic range, every 9 lines and To ensure a large dynamic range, every 9 lines and columns contains an 8-bit binary code which gives the columns contains an 8-bit binary code which gives the vertical and horizontal ‘coarse’ positions, respectivelyvertical and horizontal ‘coarse’ positions, respectively

Camera (4.7x3.5 mm) only ‘looks’ at a small portion of the Camera (4.7x3.5 mm) only ‘looks’ at a small portion of the mask (20x40 mm)mask (20x40 mm)

‘pivot square’, indicates crossing of horizontal & vertical codeline, contains no code info

‘in phase’ with chessboard pattern: bit = 0

‘counter-phase’ to chessboard pattern: bit = 1

Position of this codeline: ‘00100010’ = 34 blocks vertically above (0,0)

Purpose of the Rasnik Image AnalysisPurpose of the Rasnik Image Analysis

Find rotation of the mask on sensorFind rotation of the mask on sensor Find the scale (block size on sensor / Find the scale (block size on sensor /

true block size)true block size) Find the x- and y coordinate in the Find the x- and y coordinate in the

mask system of a reference pixel of mask system of a reference pixel of the camera systemthe camera system

What is attainable?What is attainable? Cramer-Rao lower bound (CRLB) for shift estimation gives a lower Cramer-Rao lower bound (CRLB) for shift estimation gives a lower

limit for resolvable translations for a given imagelimit for resolvable translations for a given image

-> Lower bound of resolution proportional to noise level in image -> Lower bound of resolution proportional to noise level in image and inversely proportional to the gradient energy in translation and inversely proportional to the gradient energy in translation direction direction

For a sharp short-range rasnik image (e.g. from a praxial system): For a sharp short-range rasnik image (e.g. from a praxial system): CRLB ≈ 10nmCRLB ≈ 10nm

For a long range (e.g. projective) system:For a long range (e.g. projective) system:CRLB ≈ 100nmCRLB ≈ 100nm

This is a lower bound, any implementation will have a lower This is a lower bound, any implementation will have a lower resolutionresolution

x2 no ise

2

d Idx

2 y2 no ise

2

d Idy

2

(Very Short) Description of the (Very Short) Description of the Current Rasnik Image AnalysisCurrent Rasnik Image Analysis

Gradient based (edge detection) Gradient based (edge detection) method for finding the fine shift of method for finding the fine shift of the chessboard patternthe chessboard pattern

Intensity based (thresholding) Intensity based (thresholding) method for determining the bit value method for determining the bit value of the codebitsof the codebits

Current Rasnik image analysisCurrent Rasnik image analysis

Line-fitting algorithm fits lines to gradx Line-fitting algorithm fits lines to gradx and grady to determine scale, rotation and and grady to determine scale, rotation and fine translation in x and yfine translation in x and y

Gradient in x of a Rasnik image

(image taken from Kevan Hashemi’s Rasnik analysis page http://alignment.hep.brandeis.edu/Devices/RASNIK/)

Current Rasnik Image AnalysisCurrent Rasnik Image Analysis

Fine translations in x and y, rotation and Fine translations in x and y, rotation and scale can now be used to compare the scale can now be used to compare the intensity of each code square to its intensity of each code square to its neighbors (thresholding) to determine its neighbors (thresholding) to determine its bit valuebit value

Now each codebit has to be judged for Now each codebit has to be judged for reliability; codebits can be obstructed by reliability; codebits can be obstructed by dust, cables, etc. etc. dust, cables, etc. etc.

Be wary of ‘wrong’ code readings: a wrong Be wary of ‘wrong’ code readings: a wrong answer is far worse than no answeranswer is far worse than no answer

Final x and y position = coarse (code) shift Final x and y position = coarse (code) shift + fine shift+ fine shift

2D FFT2D FFT

Computes a 2D spectrum of the Computes a 2D spectrum of the spatial frequencies and their phase in spatial frequencies and their phase in an imagean image

2D FFT: compute and put back the 2D FFT: compute and put back the 1D FFT of each row in an image, then 1D FFT of each row in an image, then compute and put back the FFT of compute and put back the FFT of each column of the resulting imageeach column of the resulting image

2D FFT and Rasnik2D FFT and Rasnik

X

=

Spatial domain

(multiplication)

Frequency domain

(convolution)

*

=

2D FFT

2D FFT

Spectrum of a real Rasnik imageSpectrum of a real Rasnik image

First harmonics

Higher harmonics (the sharper the image, the more higher harmonics!)

DC termLower frequency region: background light variations, large dust, codebits

Higher frequency variations (noise, fine dust)

ωy

ωx

Recovering rotation, scale, Recovering rotation, scale, translation in the Fourier domaintranslation in the Fourier domain

Operation in Operation in spatial spatial domaindomain

Spatial Spatial domaindomain

Fourier Fourier domaindomain

Rotation Rotation through through θθ

f’(x,y)=f(xcosf’(x,y)=f(xcosθθ + ysin + ysin θθ,,

ycosycosθθ - xsin - xsin θθ))F’(F’(ωx, , ωy) = ) =

F(F(ωxcoscosθθ + + ωysin sin θθ, , ωycoscosθθ - - ωxsin sin θθ))

Scaling by a,bScaling by a,b f’(x,y) = f(ax,by)f’(x,y) = f(ax,by) F’(F’(ωx, , ωy) = (1/|a||b|) * ) = (1/|a||b|) * F(F(ωx/a, /a, ωy/b) /b)

Translation by Translation by x0,y0 x0,y0

f’(x,y) = f(x-x0,y-y0)f’(x,y) = f(x-x0,y-y0) F’(F’(ωx, , ωy) = exp(-i() = exp(-i(ωx x0 x0 + + ωy y0)) F( y0)) F(ωx, , ωy) )

In practice:In practice:

ScaleRotation

Translation (within one period of main harmonic) obtained from phase of first harmonics

ImplementationImplementation Select centre 256*256 pixels (for now)Select centre 256*256 pixels (for now) Apply windowing function to reduce edge Apply windowing function to reduce edge

artefactsartefacts Perform 2D FFTPerform 2D FFT Find spectral main harmonic peaksFind spectral main harmonic peaks Fit main harmonic peaksFit main harmonic peaks Look for second harmonics, if they contain Look for second harmonics, if they contain

enough signal power fit them alsoenough signal power fit them also From this, determine scale, rotationFrom this, determine scale, rotation Find accompanying phase and thus fine Find accompanying phase and thus fine

translationtranslation Use this as input for codeline reading!Use this as input for codeline reading!

Progress of implementationProgress of implementation Algorithm to find rotation, scale, fine translation Algorithm to find rotation, scale, fine translation

implementedimplemented In simulations (SNR = 10, rasnik image simulated In simulations (SNR = 10, rasnik image simulated

with 2D sinefield):with 2D sinefield):• Resolution in rotation: 10^-4 radResolution in rotation: 10^-4 rad• Resolution in scale: 10^-4Resolution in scale: 10^-4• Resolution in translation: 10^-4 period (corresponds to Resolution in translation: 10^-4 period (corresponds to

approx. 50nm)approx. 50nm) HOWEVER a real Rasnik image is not a 2D HOWEVER a real Rasnik image is not a 2D

sinefield! More testing will be done using images sinefield! More testing will be done using images from an ultrastable test benchfrom an ultrastable test bench

Algorithm to find and read codelines Algorithm to find and read codelines implemented, but easily disrupted by dust, implemented, but easily disrupted by dust, obstructions, etc. Far less than ideal at the obstructions, etc. Far less than ideal at the moment. moment.

CodelinesCodelines Kevan Hashemi (Brandeis) image analysis routine Kevan Hashemi (Brandeis) image analysis routine

for the endcap muon spectrometer has been tried for the endcap muon spectrometer has been tried and tested for years, proven to be very robustand tested for years, proven to be very robust

Agreed to look into integrating initial fine shift Agreed to look into integrating initial fine shift calculation using 2D FFT with Kevan’s analysis, calculation using 2D FFT with Kevan’s analysis, perhaps even standardising the combined perhaps even standardising the combined analysis for use both in the endcap and barrelanalysis for use both in the endcap and barrel

Thanks to Pierre-Francois Giraud the integration Thanks to Pierre-Francois Giraud the integration of Pascal and c++ will be possibleof Pascal and c++ will be possible

To doTo do Combine fine and coarse shift routinesCombine fine and coarse shift routines Do testing on stable test-bench images to Do testing on stable test-bench images to

compare new and old analysescompare new and old analyses Run the new analysis on a single line in Run the new analysis on a single line in

ATLAS (infrastructure for this is ready ATLAS (infrastructure for this is ready thanks to Robert Hart)thanks to Robert Hart)

If all goes well, implement the new If all goes well, implement the new analysis on lines which cannot be analysed analysis on lines which cannot be analysed using the old analysisusing the old analysis

And then in the entire barrel muon And then in the entire barrel muon spectrometerspectrometer

Bonus: other elegant stuffBonus: other elegant stuff

2D FFT

Filter: keep boxed part, discard rest

2D IFFT

2D FFT

2D IFFT