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ECE/OPTI 531 – Image Processing Lab for Remote Sensing Fall 2005

Sensor Models

Reading: Chapter 3

Fall 2005Sensor Models 2

Sensor Models

• LSI System Model• Spatial Response• Spectral Response• Signal Amplification, Sampling, and

Quantization• Simplified Sensor Model• Geometric Distortion

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Overall Sensor Model

• Remote sensors are complex systems of optical,mechanical and electronic components– These components determine the quality of the data

from the sensor– The sensor may be considered a “black-box” that

converts at-sensor radiance to DNs


LSI System Model

• Model the various systems as Linear Shift-Invariant (LSI)– A linear transformation of the input x results in a

similar transformation of the output y• Superposition principle• If T[f1] = g1 and T[f2] = g2 , then T[a1f1 + a2f2] = a1g1 + a2g2

– Shifting the input results in a similar shift of theoutput

• Shift invariance• If T[f(x)] = g(x) , then T[f(x-x0)] = g(x-x0)

• LSI model is generally applicable over thenominal range of operation for these systems– Model will break down as performance limits are

approached (i.e., system response becomes non-linear)

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Instrument Response

• Any signal can be written as a sum of weighteddelta functions using the “sifting” property

• What happens when this input form is put intoa linear system?

• Knowing the transformation of a deltafunction, the “impulse response”, completelycharacterizes the LSI system


Instrument Response (cont.)

• The precision of measurement is determined by theinstrument response, r

• The transformation from the input physical quantity tothe measurement is described mathematically by aconvolution

– where• i(α) is the input signal, a function of time, space, etc.• r(z0-α) is the instrument response, inverted and shifted by z0

• o(z0) is the output signal at z = z0

• W is the range over which the instrument response is significant• Shorthand notation, o(z) = i(z) ∗ r(z), read as “the output

signal is the input signal convolved with the instrumentresponse.”

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Input and Impulse Response

Convolution Operation

Output

1-D Convolution Example

• The measured value at z0 is anaverage of the input signal inthe vicinity of z0, weighted overthe range W by the instrumentresponse


Resolution

• Any instrument that measures a physicalquantity is limited in the amount of detail itcan capture– This limit is referred to as the instrument’s “resolution”– “Resolution” is a term that is widely used, but often

misunderstood• The width W of the instrument response defines

the spatial resolution, or effective GIFOV

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Sensor Models




Spatial Response

• The spectral signal is convolved with the sensorspatial response

– where the spatial response of an imaging system isnow called the Point Spread Function (PSF)

• The net sensor PSF is a convolution of individualresponses from:– optics PSFopt

– image motion PSFIM

– detector PSFdet (defines the geometrical GIFOV)– electronics PSFel

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PSF Properties

• The net sensor PSF is wider than the GIFOVbecause of– PSFopt in both directions– PSFIM

• cross-track for whiskbroom scanners• in-track for pushbroom scanners

– PSFel cross-track for whiskbroom scanners• Reasonable assumption in many cases is that

the PSF is separable in the cross-track and in-track directions


PSF Comparison

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normalized detector PSFdet

AVHRR MSS

SPOT HRV TM

cross-trackin-track

significant “out-of-pixel” response in all four systems

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Sensor Models




Spectral Response

• The at-sensor radiance is transferred to thesensor image plane by the camera equation

– where• τo(λ) is the optics spectral transmittance• N is the optics f-number, given by the ratio of the optical

focal length divided by the aperture stop diameter• the optical magnification is assumed to be one

• The spectral responsivity Rb(λ) weights the imageplane irradiance to yield a signal value

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Spectral Response (cont.)

• Spectral responsivities are not ideal rectangularbands (Fig. 3–8)

0

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400 500 600 700 800 900 1000 1100

Rel

ativ

e R

esponse

wavelength (nm)

AVHRR1 AVHRR2

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ativ

e R

esponse

wavelength (nm)

MSS1 MSS2 MSS3 MSS4

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esponse

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TM1 TM2 TM3 TM4

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ativ

e R

esponse

wavelength (nm)

SPOT1

SPOT2

SPOT3

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2.5 4.5 6.5 8.5 10.5 12.5 14.5

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esponse

wavelength (µm)

AVHRR3 AVHRR4 AVHRR5

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1200 1400 1600 1800 2000 2200 2400

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ativ

e R

esponse

wavelength (nm)

TM5 TM7

AVHRR

TM

SPOT MSS


Hyperspectral Responsivity

• Hyperspectral sensors have relatively constantspectral resolution across a wide spectral range

band center wavelengths linear with

band number

0

500

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2500

8.5

9

9.5

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1 51 101 151 201

center wavelength

bandwidth

cente

r w

avel

ength

(nm

)ban

dw

idth

(nm

)

band

average bandwidth

about 10nm

Spectral band properties for AVIRIS (Fig. 3–9)

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Sensor Models




Signal Amplification

• The electronic signal produced by the detectorsis amplified

• Some sensors have multiple gain settings, e.g.SPOT HRV (Chavez, 1989) and ETM thermal band,to increase signal level for dark objects

signal rangerequired at

A/D input for fullrange DN output

ab

eb

anticipated range

optional high gainb

offsetb

all scenes

standard gainb

of detected signals low radiance scenes

Linear amplification characteristics (Fig. 3–17)

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Sampling and Quantization

• The amplified signal is sampled in time (duringscan) and quantized into Digital Numbers (DNs)

DN

0

2Q – 1

input signal to A/D converter

ab

• Quantization is a low-levelnoise superimposed on thedata values– For Q bits/pixel quantization

there are 2Q integer DNs overthe range [0...2Q-1]

• Radiometric resolution = 2-Q

Linear quantization transfercharacteristics (Fig. 3–18)


Sensor Models



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Effective Sensor Model

• Total measured signal at pixel p in band b

– where• DNpb is the Digital Number at pixel p in band b• Lλ(x,y) is the at-sensor spectral radiance from scene

location (x,y)• Kb is a gain coefficient for band b that includes sensor

gain, detector spectral responsivity and spectral filtertransmittance

• offsetb is the sensor offset coefficient for band b• the gain and offset are effective quantities, averaged

over an effective spectral band


Gain-Offset Model

• The three integrals are over:– the effective spectral response range of band b

(spectral resolution)– the effective spatial response range in-track and cross-

track (spatial resolution)• Assume a band- and space-integrated at-sensor

radiance Lpb at pixel p, band b. Then,

– DNs are linearly proportional to the total at-sensorradiance

– Ignores radiometric quantization and nonuniformresponse within spectral bands and the GIFOV

– Simplifies modeling and radiometric calibration of thesensor

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Sensor Models




Sources of Distortion

• All remote sensing images are distorted relativeto a map– platform motion, especially airborne sensors– scanning distortion of the Ground Sample Interval (GSI)– topography Distortion caused by airplane

motion (ASAS airborne sensor)

cross-track ground distance from nadir (km)in-track

ground distance (km)

01.12.2

3.34.4

5.5

0 200 430 750 1400

1024 20481280 1536 1792

cross-track pixel numberorbital track

Bow-tie distortion in AVHRR data (Fig. 3–23)

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Scanning Distortion

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0 0.2 0.4 0.6 0.8 1

flat earthtrue earth

rela

tive

GS

I

off-nadir scan angle (radians)

!

FOV

nadir

H

"!

f

GSIe(!)

GSIf (!)

line and whiskbroom scanners (Fig. 3–22)

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

flat earthtrue earth

rela

tive

GS

I

off-nadir view angle (radians)

!

FOV

nadir

H

"!

W

f

GSIe(!)

GSIf (!)

pushbroom scanners (Fig. 3–24)


Topographic Relief

• Image offset proportional to elevation abovebase plane, or “datum”

• Stereo pair of images can be used to findelevation

• Imagery corrected for topographic distortion iscalled “orthographic”

AA0

ground point at A actually appears to

come from A0

because of topography

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