mod13_part2 probability and staticistics
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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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