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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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| The following code calculates the PDF {{{pdf}}} for a byte image 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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| $\sum_{q=0}^{255}f_q=1$ | $\sum_{q=0}^{255}pdf(q)=1$ |
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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 {{{#!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 }}} |
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]
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.