Image Processing Reference
In-Depth Information
Fig. 11.11. ( Left ) An electrophoresis image of protein expression. ( Right ) the linear symmetry
in log z coordinates ( I 20 ), where red represents presence of circles
images. By contrast, this is possible when (GST) tensor-valued images are correlated
with (GST) tensor valued filters.
Sharing similar integral curves with the generalized structure tensor, the Lie op-
erators of [72, 107] should be mentioned. However, these studies have not provided
tools on how to estimate the parameters of the integral curves, e.g., the direction and
the estimation error.
Granlund and Knutsson first showed that convolving complex images by com-
plex filters can result in detection of intricate patterns [140]. In the follow-up stud-
ies [17, 22, 23, 129] the GST theory, which achieved isocurve detection and iden-
tification by use of analytic models and tensor fields, was introduced. Hansen [94]
studied the structure tensor when it is extended to curves and surfaces in three and
higher dimensional spaces.
We have used a biometric person authentication application to illustrate the the-
ory. A review of biometric recognition with brief summaries of the main schools are
given in [175].
11.13 Appendix
Proof of theorem 11.2 and lemma 11.5
We prove the theorem by induction.
1. Eq. (11.98) holds for p =1.
2. Assume (induction) that Eq. (11.98) is true for p = p 0 , for some p 0
1.
3. Then
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