Random variables

The measurements that form an image show statistical fluctuations. The image is characterised by a probability density function (PDF). The PDF f describes the probability of the occurrence of a discrete grey level q in the range of grey levels Q.

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Probability density functions and histograms

The PDF of an image can be calculated and displayed using the pylab.hist function or it can be calculated using the scipy.histogram function.

ASAR image of sea ice

The received power (intensity) of a radar system is proportional to the (normalized) radar backscatter coefficient

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as a function of frequency and incidence angle. The backscatter coefficient describes how much of the transmitted energy is backscattered from the surface media.

A (calibrated) ASAR image img(y,x)=I is made of the measured intensities I which can be directly related to the backscatter coefficient. A value of zero means that no energy is reflected from the surface whereas a value of one means the total reflection.

A useful representation of the intensity is the logarithmic transformation to the decibel unit

   1 img_dB=10*log10(img)

The PDF of the image above can be estimated from the number of occurrence of grey levels in the B intervals between q and q+dq and displayed using

   1 hist(img,bins=50)

with the resolution B for 50 bins (intervals).

Source code and data

Estimation of the PDF parameters

   1 from scipy import *
   2 from pylab import *
   3 import scipy.optimize as opti
   4 import scipy.stats as stats
   5 
   6 
   7 def my_pdf(p,x):
   8     """A linear mixture of two gamma functions"""
   9     pdf1=stats.gamma.pdf(x,p[0],loc=p[1],scale=p[2])
  10     pdf1/=sum(pdf1)
  11     pdf2=stats.gamma.pdf(x,p[3],loc=p[4],scale=p[5])
  12     pdf2/=sum(pdf2)
  13     pdf=pdf1*p[6]+pdf2*p[7]
  14     return pdf
  15 
  16 def my_cost(p,x,y):
  17     """The cost (error) function"""
  18     f=my_pdf(p,x)
  19     return y-f
  20 
  21 img=reshape(fromfile('ASAR_seaice_mixed_20080421_f32_1000x1000.dat',dtype=float32),(1000,1000))
  22 
  23 close('all')
  24 h=histogram(img,bins=500)
  25 y,x=h[0].astype(float64)/sum(h[0]),h[1].astype(float64)
  26 
  27 
  28 plot(x,y)
  29 p0=[7.0,0.003,0.005,7.0,0.1,0.005,0.5,0.5 ] # initial guess
  30 
  31 p,success = opti.leastsq(my_cost, p0[:], args = (x, y))
  32 y_fit=my_pdf(p,x)
  33 plot(x,y_fit)
  34 axis([0,0.3,0,0.023])
  35 legend(('Observed PDF','Optimized model PDF'),'upper right')
  36 xlabel('Backscatter coefficient')
  37 ylabel('Probability')
  38 savefig('model_stat.png',dpi=100)
  39 show()