“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!