Differences between revisions 5 and 6

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 5 as of 2008-12-08 12:38:42 → 
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+   ← Revision 6 as of 2008-12-08 12:50:12 → ⇥
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-Deletions are marked like this.
+Additions are marked like this.
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-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''.
+The image is characterised by a '''probability density function''' (PDF). The PDF describes the probability of the occurrence of the discrete grey levels.
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-{{{#!latex
$\sum_{q=0}^Qf_q=1$
}}}
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+Line 17:
-The following code calculates the PDF {{{pdf}}} for a ''byte'' image {{{img}}} in the intervall {{{[0,255]}}}
+The following code calculates the PDF {{{pdf(q)}}} for a ''byte'' image {{{img}}} in the intervall {{{[0,255]}}}
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+Line 26:
-$\sum_{q=0}^{255}pdf_q=1$
+$\sum_{q=0}^{255}pdf(q)=1$
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+Line 29:
-The anti-derivative of the PDF is the cumulative density function (CDF). It can be approximated from the cumulative sum
+The anti-derivative of the PDF is the  '''cumulative density function''' (CDF). 
{{{#!latex
$cdf(q)=\sum_{q'=0}^{q}pdf(q')$
}}}

The cumulative sum can be calculated using
-Line 38:
+Line 40:
+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
}}}