“is this a compressed sensing application?” · 2015-08-11 · “is this a compressed sensing...
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“Is this a compressed sensing application?”
Team 2 – Progress Report
Abishek Agarwal, Dimitrios Karslidis, Byong Kwon, Kevin Palmowski, Shant Mahserejian, Xuping Xie
Industry Mentor: John Hoffman, PhD
CyberOptics Corporation
IMA/PIMS Math Modeling in Industry XIX
August 10, 2015
• Background and motivation • Current approach • New ideas and future directions
Outline
How tall is each component on this printed circuit board?
Motivating Question
https://en.wikipedia.org/wiki/File:Surface_Mount_Components.jpg
Example: CyberOptics SQ3000™
http://cyberoptics.com/eai_products/sq3000/
Industrial profilometry
• Profilometry: measuring the profile of a three-dimensional object
• Industrial profilometry is a “BRUTALLY COMPETITIVE” field
• Speed and accuracy are key
How do you quickly and accurately acquire a height map for an object?
• Background and motivation • Current approach • New ideas and future directions
Outline
Current setup
• 4 cameras – compensate for shadows and blocked components
• Field of view is 2000 x 2000 pixels
Simplified model
• Surfaces are diffuse reflectors of light • No multipath effects
(multipath - not modeled)
Simplified model
• Camera above object to be profiled • Lens allows camera to only receive
perpendicular light beams
Gray code fringes
• Project sequence of Gray codes onto scene
• Determine which sent pixel maps to which received pixel for entire scene
• Issue: lenses introduce edge blur
Sinusoidal fringes
• Project fringe patterns generated by sine waves onto scene
• Advantage: lenses preserve sinusoidal nature
http://cyberoptics.com/pdf/AOI/SQ3000/2015-SMTA-SEA-Presentation.pdf
Approach
• General model at each pixel:
• Fringe frequency λ known • Reflectance r, modulation (amplitude) m, and
phase offset Δϕ unknown • 3 fringe phase shifts {Pi} are used – can solve
for unknowns • Δϕ is linearly related to height
Approach
• Issue: Δϕ only determined mod 2π/λ • Use multiple fringe frequencies {λk} to resolve
the true value of Δϕ at each pixel • Theory: 2 unknowns → 2 fringe frequencies • Practice: 4 or 5 fringe frequencies used to
compensate for available resolutions
Problem statement
• Faster performance → more money • Current hardware takes 90 images per site
• Sampling far too much data compared to
what is theoretically required
How can we reduce the amount of time needed to acquire a profile while
maintaining high accuracy?
(3 phases) x (5 fringe frequencies) x (3 light levels) x (2 directions)
• Background and motivation • Current approach • New ideas and future directions
Outline
Idea 1: Fewer fringe frequencies
• Goal: reduce number of fringe frequencies λk
• Use information from neighboring pixels to inform maximum height
• Fewer fringe frequencies → fewer images → faster
• Proposed optimization problem to globally solve for Δϕ (as a vector)
• Minimize f, defined as follows:
where
Idea 2: Minimization problem
Idea 2: Minimization problem
• Issue: |x-y|S1 is not convex
Idea 3: Single-pixel camera approach
• Utilizes digital micromirror device (DMD) • Randomly-generated masks point each mirror
on DMD toward or away from a photodiode • Photodiode sums photons to yield a single
voltage reading for each mask • Huge savings: n2 pixels → 1 voltage value
http://dsp.rice.edu/publications/new-compressive-imaging-camera-architecture-using-optical-domain-compression
Idea 3: Single-pixel camera approach
• Sensed image is sparse in wavelet domain • Wavelet domain image recovered via basis
pursuit (BP), a compressed sensing technique
• BP can be formulated as a linear program • Question: Can we use these tools?
Future directions We are still thinking about… • Can we modify the proposed minimization
problem to make it convex? • Can we use single-pixel camera architecture? • “Is this a compressed sensing application?”
We would also like to look into… • Compressive depth map acquisition methods • Time-of-flight methods • Optical frequency comb profilometry • Fourier transform profilometry
Thank you for listening!
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