Christoph Florian Schaller - FU Berlin, FB MI
Christoph Florian Schaller - FU Berlin, FB MI
Christoph Florian Schaller - FU Berlin, FB MI
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<strong>Christoph</strong> <strong>Schaller</strong> - STORMicroscopy 39<br />
B<br />
Assigning the matching background values<br />
We briey proof that the χ 2 -sum is minimized by assigning the background values to the dierences<br />
ordered by size. This is easily seen by the repeated application of the following formula.<br />
Claim: m 1 > m 2 ∧ l 1 > l 2 =⇒ (m 1 − l 1 ) 2 + (m 2 − l 2 ) 2 < (m 1 − l 2 ) 2 + (m 2 − l 1 ) 2<br />
Proof: (m 1 − l 1 ) 2 + (m 2 − l 2 ) 2 < (m 1 − l 2 ) 2 + (m 2 − l 1 ) 2<br />
⇐⇒ m 2 1 − 2m 1 l 1 + l 2 1 + m 2 2 − 2m 2 l 2 + l 2 2 < m 2 1 − 2m 1 l 2 + l 2 2 + m 2 2 − 2m 2 l 1 + l 2 1<br />
⇐⇒ m 1 l 1 + m 2 l 2 − m 1 l 2 − m 2 l 1 > 0<br />
⇐⇒ (m 1 − m 2 ) · (l 1 − l 2 ) > 0<br />
} {{ } } {{ } □<br />
>0<br />
>0
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