Lecture 13: Remote Estimation of the Temperature and Size of an Unresolved Object in

Space

Part II: Imaging Sensor Measurement Model

Object Radiance  Assume  that  there  are  2  measurement  bands  B1  

and  B2.    B1  corresponds  to  the  wavelength  range  of  λ11  to  λ12  .  B2  corresponds  to  the  wavelength  range  of  λ21  to  λ22.    Assume  that  the  point  object  looked  at  by  the  sensor  has  an  physical  area  A  and  emissivity  εi.      

 The  emissivity  of  an  object  is  a  number  between  0  and  1  and  described  what  frac@on  of  the  possible  energy  the  object  will  radiate.    Black  object  have  emissivi@es  approaching  1  while  white  or  shiny  objects  have  values  approaching  0.  

• The emissivity of an object is a number between 0 and 1

•  It describes what fraction of the possible energy the object will radiate

• Black objects have emissivities approaching 1 while white or shiny objects have values approaching 0 o  This is why woodstoves are black

Object Emissivity

Object Radiance The  “in-­‐band”  radiance  for  band  i  is  Ji  having  units  of  WaCs/Steradian  (W/sr).    It  is  given  by  the  following  formula:  

2

1

5

2 1

1( , , )

i

i

i i hckT

i i

hcJ A d

ef T A B

λ

λ λ

ε λλ

ε

=

−=

∫

Planck  Func*on  Object  Emissivity  

Object  Physical  Area  

•  h    is  Planck’s  Constant  •  k  is  Boltzmann’s  Constant  •  c  is  the  speed  of  light  •  λ  is  the  wavelength  of  light  •  T  is  the  temperature  of  the  object  

(Kelvin)  

The  Planck  func@on  is  integrated  over  the  spectral  range  of  the  measurement  band.  A  MATAL  func@on  will  be  provided  to  compute  this.  

The “Inverse-Square Law”

The  radiant  intensity  (J  or  S  as  shown  in  the  diagram  on  the  right)  of  a  point  source  falls  off  as  1/r2  where  r  is  the  radial  distance  from  the  point.  

r  J  

dx  

dy  dA   The  power  density  at  a  distance  r  from  the  source  is  given  by  J/r2  having  units  W/m2.    The  power  intercepted  by  an  area  dA  is  P=J  dA/r2.  

Detector Array

rij  =  J/r2  +  nij  

“Measurement”   “Addi@ve  Noise”  Assume  that  the  point  source  is  in  some  pixel  and  that  each  pixel  can  make  measurements  in  2  bands  B1=[λ11,λ12],  B2=[λ21,λ22].    The  signal  measurement  model  is  given  by:  

“Pixel’  

11 1 1 1 2

22 2 2 2 2

;

;

Jr s n s

rJ

r s n sr

= + =

= + =

Problem Statement A  sensor  capable  of  making  2-­‐band  measurements  is  located  at  a  distance  r  from  a  point  source.    The  sensor  makes  noisy  measurements  of  the  emiCed  energy  according  to  the  specified  measurement  model.    Assume  that  that  noise  terms  are  independent  between  bands  with  each  Gaussian  N(0,σ2).    Also  assumes  that  the  sensor  precisely  knows  the  range  r  though  in  prac@ce  this  is  an  addi@onal  Bayesian  parameter  that  should  be  included.    Develop  a  joint  MAP  (Bayesian)  es@mator  for  the  temperature  T  and  the  emissive-­‐area  εA  of  the  source  object.  

Part III: Bayesian Problem Formulation

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