This presentation was given as the foundation for the MIT Lincoln Lab's Student Working Group problem lab. Working group members used this information to begin researching their problem and later went on to present their findings to their peers.

1. John Peach
July 17 2018
Light Reflections
2018 Industrial Math/Stat Modeling Workshop for Graduate Students
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2. IMSM Light Reflections- 2
JRP 07/16/18
• MIT Lincoln Laboratory conducts research and development in
support of national security
• Principal core competencies:
– Sensors
– Information extraction
– Communications
– Integrated sensing
– Decision support
Laboratory Mission

3. IMSM Light Reflections- 3
JRP 07/16/18
• Purpose: Build a mathematical model of the properties of an
object which faithfully captures the relevant optical phenomena
to:
– Support pre-mission sensor predictions
– Validate post-mission data
– Perform computer based trade studies
– Provide a framework for development of new concepts and
algorithms
Optical Modeling

4. IMSM Light Reflections- 4
JRP 07/16/18
Optical Response Equation
( )
( )
( )
2 2
Power on a differential area
Irradiance
Differential area perpendicular to incident rays
Transmitted power
Received power
Optical Cross Sec
cos
S
S
dA
W
TX sr
TX
RX
STX TX
dA
TX
dAJ dA P
P
P
J
dA
P W
P W
R R
θ
σ
⊥
⊥
=
=
=
=
=
=
= =
Ω
( )
( )
( )
( )
( )
2
2
tion
Solid angle (beam divergence)
Range
Angle between incident ray and local surface normal
Surface differential area
System efficiency
TX
S
SYS
m
sr
R m
rad
dA m
θ
η
Ω =
=
=
=
=
Iθ
N
SdA
IR
Rθ
N
SdA
RXP
RR2
4
cos
4
TX S
RX SYS
TX
P dA
P
R
σ
θ η
π
=
Ω∫
TXP
( )
( )
2
Target reflectivity
Target area
Solid angle of scattering
A
A m
sr
ρ
σ
ρ
=
Ω
=
=
Ω =
Scattering function is estimated by:
Target properties are contained in 𝝈𝝈. All
other parameters depend on the
transmitter and receiver.

5. IMSM Light Reflections- 5
JRP 07/16/18
0 20 40 60 80 100
Elevation angle (deg)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
MRDF
Specular
Diffuse
• The optical response of a material is the sum of the specular and
diffuse components
• The response is aspect angle and wavelength dependent
• The response is characterized as either
– MRDF: Mono-static Reflectance Distribution Function – Incident and
reflected light rays are concurrent
– BRDF: Bi-static Reflectance Distribution Function – Source and receiver are
in significantly different positions
Material Properties and Measurements
Example MRDF
Specular
Reflection
Detector
Φ Angle
Out-of-plane Measurements
Incident
Θ Angle
Incident
Beam
Surface
Normal
Incident
Φ Angle
Detector
Θ Angle
• MRDF / BRDF data obtained
from coupon samples
• Source and detector positions
moved in hemisphere to obtain
reflectance data

6. IMSM Light Reflections- 6
JRP 07/16/18
Example BRDFs
MoonLambertian SphereBilliard BallsEscher
• Highly specular materials
(mirror-like) have sharp
specular peak
• Lambertian surfaces reflect
diffusely scattering light
equally in all directions
• Lunar regolith reflects better
at glancing incidence angles,
known as “opposition surge”

7. IMSM Light Reflections- 7
JRP 07/16/18
Scattering function calculation
Facet
𝑵𝑵
𝑰𝑰 𝜽𝜽
0 20 40 60 80 100
Elevation angle (deg)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
MRDF
𝜽𝜽
For each facet in the model:
• Find angle between incident ray
and facet normal, 𝜽𝜽
• Calculate 𝝆𝝆 = MRDF 𝜽𝜽
• Return 𝝈𝝈 = 𝝆𝝆𝝆𝝆
• 𝑨𝑨 = 𝑭𝑭𝑭𝑭 cos 𝜽𝜽 where 𝑭𝑭𝑭𝑭 is the
facet area
• Facets may be on back side of
object
• Some facets may be hidden by
other facets
• Light may reflect between
multiple facets in non-convex
regions before returning to
receiver
Calculation steps:
• Obtain CAD model representation
• For each facet calculate reflectance from
incident angle, material properties
• Sum over all visible facets to obtain Optical
Cross Section (OCS)
CAD Model
MRDF

8. IMSM Light Reflections- 8
JRP 07/16/18
Example Optical Cross Section
0 5 10 15 20 25 30 35 40 45
Aspect angle (deg)
0
1
2
3
4
5
6
7
8
9
10
OpticalCrossSection(m
2
)
Unit Sphere OCS, diffuse = 1
OCS = 8.3625
Analytic OCS =
𝟖𝟖
𝟑𝟑
𝝅𝝅 ≈ 𝟖𝟖. 𝟑𝟑𝟑𝟑𝟑𝟑𝟑𝟑
θ
N
rdθ
r′
( )( )
2
2
2 sin
2 sin
ringdA r rd
r rd
r d
π θ
π θ θ
π θ θ
′=
=
=
2
2 2
2
3
2 2 2
2
2
0
2
2
0
0
0
2
4 cos
cos 8
8
3 3
cos si
4 2 cos si
n
3
n
8
d
D
d
d dA
r r
R d
d r d
R r
π
π
ππ
σ σ πρ
σ πρ θ
θ
π ρ π ρ
ρ θ θ θ πρ
σ π
π θ θ θ
=− =
=
=
=
=
= ∫
∫ ∫
∫


9. IMSM Light Reflections- 9
JRP 07/16/18
Multipath Problem
for Non-Convex Regions
1F
2F
3F
4F
1Ν
41i
1Si
31i
21i
Source
Element Power Source Power
Multipath calculation:
• Determine relative angles between
facets
• Use BRDF to calculate reflectance
in non-convex region
• Calculate fraction of power
returned to receiver
( )
( )( )
2
1
1
2
1 1
2
1
2 2
2
22
2
cos
cos
where
cos
cos
cos
cos
n
j
RX j jR j
j jR
R
R
R
R
n nR
nR
P
P r dA
R
r
I E E S dA
R
r
R
r
r
R
R
r
R
ρ
ρ
ρ
ρ
ρ
ρ
=
−
=
 
= − +  
 
 
 
 
 
 =  
 
 
 
  
∑

Radiance back
to source
( ) ( )
( ) ( )
( ) ( )
( )
( )
12 2
1 1 121 1 22 2
121 1
1
21 1
2 2 212 2 222 2
212 2
1 1 2 2
1 22 2
1 2
cos0 cos cos
coscos 0 cos
0
cos cos 0
REFn
n n TX
Sn
TX Sn
TXn n
Sn
TX Sn
n
TX
nn n
n n
P PP
P i dAi i
RR R
dA
PPP
P i dAi i dA
RR RP
dA
PP P
Pi i
R R
ρ σρ
σρρ
ρ ρ
 
=  Ω
   
    =    Ω+   
   
   
  =
  



  
 ( )2
cos
REF
n
Sn n
TX Sn
n
i dA
R
E dA S
σ
 
 
 
 
 
 
 
 
 
 Ω 
= +
( ) ( )1 1 1 12 2
11 1
cos cos
j jTX
dA S j
jTX S j
PP
P i i dA
R R
σ
≠
 
= +  Ω 
∑

10. IMSM Light Reflections- 10
JRP 07/16/18
• Develop analytic models to replace facetized CAD models
• Convert MRDF / BRDF look-up tables to continuous functions
• Provide analytic OCS function
• Extend discrete multipath solution to continuous domain
Project Goals

11. IMSM Light Reflections- 11
JRP 07/16/18
OpenSCAD
• OpenSCAD is open source software for creating solid 3D CAD models
• Uses Constructive Solid Geometry to build complex shapes
• Complex shapes obtained from simple basis shapes and Boolean algebra
• May be useful for experimenting with CSG, determining basis shapes
http://www.openscad.org/index.html
Constructive Solid Geometry

12. IMSM Light Reflections- 12
JRP 07/16/18
Chebfun Toolbox
http://www.chebfun.org/
• Chebfun is open source software for numerical computing with functions
• Chebfun has extensive capabilities for dealing with linear and nonlinear differential and integral
operators
• Chebfun2 extends Chebfun to two variables, Chebfun3 recently released
• Several basic shapes pre-defined, others may be written in functional form
• Available as Matlab and Python toolboxes

13. IMSM Light Reflections- 13
JRP 07/16/18
Parametric Surfaces
Define a parametric surface by:
Tangent plane:
Unit normal:
Fundamental Form:
Area:

14. IMSM Light Reflections- 14
JRP 07/16/18
Rvachev Functions
• f and g are implicitly defined surfaces of unit spheres
• Can be combined using constructions representing Boolean operators
Intersection:
𝑓𝑓 + 𝑔𝑔 − 𝑔𝑔2 + 𝑓𝑓2𝑓𝑓 ∧ 𝑔𝑔
Union:
𝑓𝑓 + 𝑔𝑔 + 𝑔𝑔2 + 𝑓𝑓2𝑓𝑓 ∨ 𝑔𝑔
Difference:
𝑓𝑓 ∧ -𝑔𝑔
𝑓𝑓 + (−𝑔𝑔) − (−𝑔𝑔)2+𝑓𝑓2
V. L. Rvachev (1963), ‘On analytical description of some geometric objects’
• Solves the inverse problem of constructing equations for geometric objects
• Uses Boolean algebra to build objects

15. IMSM Light Reflections- 15
JRP 07/16/18
R-Function CSG
Isosurface of R-functionConstructive Solid Geometry Tree to
generate more complex model
(Cube ∧ Sphere)-(Cylinder ∨ Cylinder ∨ Cylinder)

16. IMSM Light Reflections- 16
JRP 07/16/18
Example Problem
Object Construction
Multipath
Hidden region
• Given the observation vector, find
• Regions where internal reflections occur (multipath)
• Regions hidden by nearer surfaces
• Local surface normals
• Integral of surface area × 𝒄𝒄𝒄𝒄𝒄𝒄(𝜽𝜽)

17. IMSM Light Reflections- 17
JRP 07/16/18
• The proposed project is expected to
– Provide more accurate calculations of Optical Cross Sections
– Run more quickly than the current method
– Use Constructive Solid Geometry instead of CAD models
– Reduce code complexity making maintenance simpler
– Reduce model storage requirements
• The project may extend the current method to include multi-
path conditions
• CAD point cloud models will be converted to solid objects
making definition cleaner
Summary