Alignments in Digital Images

Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 34)

Digital images usually have many alignments due to perspective, human made objects, and so forth. Can they and only they be detected? There should be detection thresholds telling us whether a particular configuration of points is aligned enough to pop out as an alignment. Alignments are not trivial events. They depend a priori on four different parameters, namely the total length l of the alignment, the number k of observed aligned points in it, the precision p of the alignment, and the size of the image N. So a decision threshold function k min (l, p, N) is needed and will be established by the Helmholtz principle. In its weak formulation, this principle commonsensically formulates that k min should be fixed in such a way as to seldom detect any alignment in a white noise image. In the stronger formulation, the Helmholtz principle tells us that whenever a configuration occurs, which could not arise by chance in white noise, this configuration is perceived and must be detected by a Computer Vision algorithm. In Section 5.1, we define and analyze the white noise image a contrario and show how to compute detection thresholds k min (l, p, N) discarding alignments in white noise. Section 5.2 is devoted to the analysis of the Number of False Alarms (NFA) of an alignment and the rest of the chapter considers several estimates of the detection threshold k min . Finally, several consistency problems associated with the definition of meaningfulness are considered. In Section 5.5, the important problem of choosing the precision p is finally addressed.


Digital Image False Alarm Detection Threshold False Alarm Rate Noise Image 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer 2008

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