Differences between revisions 2 and 6 (spanning 4 versions)

Image statistics

The image is characterised by a probability density function (PDF). The PDF describes the probability of the occurrence of the discrete grey levels.

Example

The following code calculates the PDF pdf(q) for a byte image img in the intervall [0,255]

   1 h=histogram(img,bins=256,range=[0,255],normed=True)
   2 pdf,x=h[0],h[1]

The expression normed=True is used for the normalization of the PDF.

latex error! exitcode was 2 (signal 0), transscript follows:

The anti-derivative of the PDF is the cumulative density function (CDF).

latex error! exitcode was 2 (signal 0), transscript follows:

The cumulative sum can be calculated using

   1 cdf=pdf.cumsum()

The probability of the occurence of grey levels in the interval [a,b] can be calculated from the CDF. In the example shown, the probability of grey levels to occur in the interval [12,25] according to the first peak is 0.068. So roughly 7% of the image pixels lie in this grey level interval.

   1 cdf[25]-cdf[12]
   2 0.068

-  ⇤ ← Revision 2 as of 2008-12-08 12:18:31 → 
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+   ← Revision 6 as of 2008-12-08 12:50:12 → ⇥
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  Comment:
-Deletions are marked like this.
+Additions are marked like this.
 Line 9:
-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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+Line 10:
-{{{#!latex
$\sum_{q=1}^Qf_q=1$
+The image is characterised by a '''probability density function''' (PDF). The PDF describes the probability of the occurrence of the discrete grey levels.


== Example ==
{{attachment:landsat_b80.png}}


The following code calculates the PDF {{{pdf(q)}}} for a ''byte'' image {{{img}}} in the intervall {{{[0,255]}}}

{{{#!python
h=histogram(img,bins=256,range=[0,255],normed=True)
pdf,x=h[0],h[1]
-Line 15:
+Line 24:
-== Probability density functions and histograms ==
+The expression {{{normed=True}}} is used for the normalization of the PDF. 
{{{#!latex
$\sum_{q=0}^{255}pdf(q)=1$
}}}
-Line 17:
+Line 29:
-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.
+The anti-derivative of the PDF is the  '''cumulative density function''' (CDF). 
{{{#!latex
$cdf(q)=\sum_{q'=0}^{q}pdf(q')$
}}}
-Line 20:
+Line 34:
-{{attachment:landsat_b80.png}}
+The cumulative sum can be calculated using
{{{#!python
cdf=pdf.cumsum()
}}}

{{attachment:landsat_b80_pdfcdf.png}}

The probability of the occurence of grey levels in the interval {{{[a,b]}}} can be calculated from the CDF. In the example shown, the probability of grey levels to occur in the interval {{{[12,25]}}} according to the first peak is 0.068. So roughly 7% of the image pixels lie in this grey level interval.

{{{#!python
cdf[25]-cdf[12]
0.068
}}}