Generic Sensor Modeling

1
Generic Sensor Modeling

• Presented by
• Dennis Helder, Jason Choi
• Image Processing Laboratory
• Electrical Engineering Department
• South Dakota State University
• April, 2003

2
Outline

• Introduction
• Procedures
• Generic Sensor Model
• PSF Modeling
• MTF Estimators
• Result and Analysis
• Parameter Selection for Sensor Model
• Noise Free Image Generation
• Interpolation Method Simulation
• Noise Simulation
• Edge Angle Simulation
• Aliasing Simulation by Sine Inputs
• Aliasing Simulation by GSD Change
• Conclusions

3
Introduction

• A satellite sensor model was described by optics,
  detector, motion, and electronics in spatial
  domain.
• A IKONOS like sensor was developed by choosing
  appropriate parameters.
• Possible MTF estimators were simulated in various
  situations such as SNR levels, edge angles and
  aliasing effect by changing GSD.

4
Procedures

• Generic Sensor Model
• A generic sensor model could be developed by
  estimating point spread function (PSF).
• PSF is net convolution of optics, detector,
  motion, and electronics PSF.
• Atmospheric blurring was ignored.

5

• Block diagram of the model

Generic Sensor Model

• Ground input
• I(x,y)
• Continuous
• 2-D function

Atmosphere blurring HAtm(wx, wy)
Optical blurring HOpt(wx, wy)
Detector blurring HDet(wx, wy)
Motion blurring HMot(wx, wy)
Electronics blurring HElec (wx, wy)

• Output image
• g(m,n)
• Discrete
• 2-D function

A / D converter

Noise ?(x,y)
6

• PSF Modeling
• Optical PSF
• The optical blurring will be modeled as a spatial
  point distribution in the image.
• A common model is the 2-D Gaussian function.
• a and b are cross and along-track optical
  blurring
• parameters

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• Detector PSF
• The detector blurring is caused by the non-zero
  spatial area of each detector in the sensor
• The size of the detector rectangle is a variable
  for a sensor
• Detector PSF can be modeled by using rect
  function

GIFOV_X and GIFOV_Y are detector width and
length.
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• GIFOV is Ground-projected Instantaneous Field Of
  View, and the rect function is

Fig 1. The rect function plot
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• Motion PSF
• The satellite moves while it takes images, which
  causes motion blurring.
• The sensor speed and line integration time are
  key factor of this model.
• where S is
• S sensor speed integration time.

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• Electronics PSF
• Electronics blurring effects was modeled by a
  simple low-pass filter.
• The first order Butterworth low pass filter (LPF)
  was defined in frequency domain.
• The LPF was inverse transformed to get spatial
  electronic PSF.
• The first order Butterworth LPF equation is
• where n is 1.

11

• MTF Estimators
• Edge Method
• Sub-pixel edge locations were found by Gaussian
  function fit.
• A least-square error line was calculated through
  the edge locations.
• One interpolation method was applied.
• The filtered profile was differentiated to get
  LSF
• MTF calculated by applying Fourier transform on
  LSF.

Fig 2. Edge Method
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• Pulse Method
• A pulse input is given to a sensor model.
• Output of the system is the resulting image.
• Edge detection and interpolation was applied to
  get output profile.
• Take Fourier transform of the input and output.
• MTF is calculated by dividing output by input.

Figure 3. Pulse method
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• Interpolation Methods
• Spline
• Cubic splines were used to interpolate the
  aligned edge data.
• Twenty values were interpolated between two
  actual data points to build a pseudo-continuous
  line.
• Every usable line of the test image was splined.
• Average profile was calculated by averaging all
  sub-pixel points in the figures.

Figure 4. Cubic Spline interpolation
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• Interpolation Methods
• Sliding Window
• One pixel width window was selected in the figure
• A least square error line was found.
• Slide the window one sub-pixel (0.05 pixel) to
  the right in the figure.
• Find the least square error line again.

Figure 5. Sliding window interpolation
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• Interpolation Methods
• Sliding Window
• All the least square error lines were averaged
  vertically in the Figure 6.
• Sub-pixel points were uniformly spaced.
• Average profile was shown in Figure 7.

Fig 6. Least square error lines
Fig 7. Average profile
16

• Interpolation Methods
• Savitzky-Golay Helder-Choi (SGHC) filtering
• Not like S-Golay filtering SGHC filter is
  applicable to randomly spaced input.
• By using the original concept, best fitting 4th
  order polynomial was calculated within 1-pixel
  window using Matlab fmeansearch.m
• One middle filtered output was evaluated by the
  polynomial.
• The next value was evaluated by the shifted
  window with the sub-pixel resolution.
• The shifting step determines output resolution

Figure 8. SGHC filtering
17
Result and Analysis

• Parameter Selection
• An arbitrary parameters were chosen to be PSF
  similar to IKONOS sensor.
• Optical PSF
• Normalized PSF is shown in Figure 1 when a 0.39
  and b 0.39

Figure 9. Optical PSF
18

• Parameter Selection for Sensor Model
• Detector PSF
• GIFOV_X and Y are 1 meter in along-track and
  cross- track direction.

Figure 10. Detector PSF
19

• Parameter Selection
• Motion PSF
• S is 0.25 meter along-track only.
• One dimensional blurring.

Figure 11. Motion PSF
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• Parameter Selection
• Electronics PSF
• Cutoff frequency range is 14 cycle per pixel in
  frequency domain.
• The LSF filter was inversely transformed in
  spatial domain.

Figure 12. Electronics PSF in frequency domain
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• Spatial domain electronics PSF is shown in Figure
  13.

Figure 13. Electronics PSF in spatial domain
22

• Parameter Selection
• The net MTF is

Cross-track LSF
Along-track LSF
Figure 14. The net PSF
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• Noise Free Synthetic Image Generation
• The edge angle was determined to 6 degrees from
  the true North
• The original PSF and the synthetic images were
  finely sampled down to 20 times than the original
  Ground Sample Distance (GSD).
• The synthetic edge was convolved with PSF.
• A noise free output image was generated by
  resampling the convolved image to the desired
  GSD.
• Synthetic pulse target was generated by the same
  processes.

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• Synthetic Edge

Convolution

Synthetic image
Net PSF
Resampling by GSD
Figure 15. Synthetic edge generation
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• Synthetic Pulse

Convolution
Three-pixel wide pulse

Synthetic image
Net PSF
Resampling by GSD
Three-pixel wide pulse
Figure 16. Synthetic pulse generation
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• Interpolation Method Simulation
• By using edge method, the three interpolation
  methods were tested.
• Spline interpolation did not follow the original
  LSF shape with under/overshoots.
• SGHC filtering was superior to other two methods

Figure 17. Interpolation methods test
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• Noise Simulation
• Relationship between signal-to-noise (SNR) ratio
  and MTF was found.
• Ground and system noise sources were defined.
• System noise was caused by the noise present in
  the imaging system.
• Ground noise was caused by ground intensity
  variation in ground target areas.

Generic Sensor Model

• Ground input
• I(x,y)
• Continuous
• 2-D function
• Ground Noise

Atmosphere blurring HAtm(wx, wy)
Optical blurring HOpt(wx, wy)
Detector blurring HDet(wx, wy)
Motion blurring HMot(wx, wy)
Motion blurring HMot(wx, wy)

• Output image
• g(m,n)
• Discrete
• 2-D function

A / D converter

System Noise ?(x,y)
Figure 18. Ground and system noise
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• SNR Calculation for an Edge

DNbright
s
DNdark
Figure 19. SNR calculation
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• System Noise SNR
• System noise was changed to get various SNR
  levels.
• Edge method was applied to the synthetic image
  with the noise.
• MTF values were investigated to find
  relationships with SNR.

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• System noise SNR result
• MTF values were getting noisy below 100 SNR.

Figure 20. SNR vs. MTF plot by system noise
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• Ground Noise SNR
• Six levels of noise images (100 variance) were
  added to generate noise image.
• The noisy image was added to synthetic images
  with changing DN difference level.
• The noisy synthetic image was convoluted with
  sensor PSF.
• Edge method was applied to the synthetic image
  with the noise.
• MTF values were investigated to find
  relationships with SNR.

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• Six levels of noise images (100 variance) were
  added to generate noise image.

5 cm sub-pixel
10 cm sub-pixel
20 cm sub-pixel
25 cm sub-pixel
50 cm sub-pixel
1 m sub-pixel
Figure 21. SNR vs. MTF plot by system noise
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• The noisy image was added to synthetic images
  with changing DN difference level.

500
1000
Convolution

Resampling
Edge method MTF estimator
Figure 22. Synthetic image with ground noise
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• Ground noise SNR result
• MTF values were getting noisy below 100 SNR.

Figure 23. SNR vs. MTF plot by system noise
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• Edge Angle Simulation
• Synthetic edges with angles from 2 to 12 with 0.1
  steps generated.
• The edges were convolved with a PSF.
• The convoluted images were resampled to desired
  GSD (1m).
• The resampled images processed by the MTF
  estimators using parametric edge detection and
  SGHC filtering.

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• Edge Angle Simulation Results
• Fermi function edge detection algorithm was
  superior to the 3rd order polynomial fitting edge
  detection.
• MTF values at Nyquist from Fermi function edge
  detection was very close to the forced angle
  curve.
• Any angle deviations caused degradations of MTF
  value at Nyquist.

Original MTF 0.2684
Figure 24. Using 3rd order polynomial fitting
Figure 25. Using Fermi function fitting
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• Aliasing Simulation by Sine Inputs
• Any input signal above the Nyquist frequency,
  fN, will be aliased down to a lower frequency.
• After aliasing, the original signal can never be
  recovered.

Figure 26. Signals above Nyquist frequency are
aliased down to the base band.
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• Generate high frequency input images (over
  Nyquist frequency in x-axis direction) .
• Convolve the input image with the PSF.
• Compare the input and output images with Fourier
  transform analysis.

0.8 cycle / pixel
0.3 cycle / pixel
Generic Sensor Model
Mapped to a base band frequency
Higher than Nyquist freq.
Figure 27. An example of aliasing.
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• Frequency 0.1 cycle per pixel

Figure 28. Spatial and frequency domain plots at
0.1 cycle/pixel.
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• Frequency 0.2 cycle per pixel

Figure 29. Spatial and frequency domain plots at
0.2 cycle/pixel.
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• Frequency 0.3 cycle per pixel

Figure 30. Spatial and frequency domain plots at
0.3 cycle/pixel.
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• Frequency 0.4 cycle per pixel

Figure 31. Spatial and frequency domain plots at
0.4 cycle/pixel.
43

• Frequency 0.5 cycle per pixel

Figure 33. Spatial and frequency domain plots at
0.5 cycle/pixel.
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• Frequency 0.6 cycle per pixel

Figure 34. Spatial and frequency domain plots at
0.6 cycle/pixel.
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• Frequency 0.7 cycle per pixel

Figure 35. Spatial and frequency domain plots at
0.7 cycle/pixel.
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• Frequency 0.8 cycle per pixel

Figure 36. Spatial and frequency domain plots at
0.8 cycle/pixel.
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• Frequency 0.9 cycle per pixel

Figure 37. Spatial and frequency domain plots at
0.9 cycle/pixel.
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• Sub-sample profile doesnt show any aliasing
  effect.
• Aliasing started at Nyquist frequency in pixel
  sample profile.
• Aliasing effect was decreasing by higher
  frequency smeared into the base band.

Figure 38. Aliasing plot by pixel and sub-pixel
sampling
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• Aliasing Simulation by GSD Change
• Generate a PSF blur is approximately 1 meter.
• The optical blurring parameters were a 0.1 and
  b 0.1.
• The motion blurring parameter was s 0.1.
• GIFOV_X and GIFOV_Y were 0.5 and 0.5.
• The Butterworth filter cutoff frequency was 5 for
  electronic blurring.
• The final PSF had almost 1 m blur in x and y
  directions.

Cross track
Figure 39. PSF with 1meter blurring
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GSD 2
GSD 1.75
GSD 1.5
GSD 1.25
GSD 1
Nyquist frequency occurred at circle points
Figure 40. MTF with increasing GSD
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Nyquist frequency occurred at circle points
GSD 1
GSD 0.8
GSD 0.6
GSD 0.4
Figure 41. MTF with decreasing GSD
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• As GSDs changed, the Nyquist positions moved
  probably on the similar patterned plot.
• Shape degradation was observed with increased
  GSD.
• High frequency components were observed with
  finer sampling over 1m GSD.
• ???? Strong question marks on these comments.

53
Conclusions

• With Interpolation Method Simulation, Fermi
  function edge detection and SGHC filtering
  perform as an excellent MTF estimator with in 6
  of error.
• System and noise simulation provided a minimum
  confidence level of 100 SNR.
• Any angel error introduced decreased MTF value.
• MTF value at Nyquist was quadratically decreasing
  from angle 2 to 12 degrees.
• Sine input over the Nyquist frequency was
  decreasingly aliased down to a base band
  frequency which was the mirror point of Nyquist
  frequency.