User:Gintsasn/Multi-Dimensional Edge Detection

Multi-Dimensional Edge Detection is a topic in computer vision which is discussed in CVonline

Here is an example formula: $\textstyle \sum _{n=1}^{\infty }1/n^{2}=\pi ^{2}/6$.

Background

In many advanced imaging applications we are dealing with three dimensional images. This is particularly prominent is medical imaging, where a medical scanner, such as MRI, will acquire multiple parallel image planes, effectively producing a three dimensional image. Detecting surface planes in such image helps to reconstruct and model scanned three-dimensional objects.

Besides the obvious application in processing three dimensional imaging, multidimensional edge detection is used in range of other fields, such as analysing seismic data and finding stratigraphic or lithological boundaries.

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Multi-dimensional edge detection by hypersurface fitting

One method to define operators for detecting edges in multidimensional arrays of data is to fit a hypersurface to a neighbourhood of each array data point and take the magnitude of the gradient of the hypersurface as an estimate of the rate of change of data value in the array at that point. This corresponds to a multi-dimensional generalisation of using Prewitt operator for edge detection in two dimensional images.

Hyperplane can be defined by:

$\Pi (x_{1},x_{2}...x_{n})=a_{0}+\sum _{n=1}^{N}a_{i}x_{i}$.

It is found by minimizing the squared error:

$E_{e}rr=\int _{V_{\delta (P_{r})}}(Pi(x_{1},x_{2}...x_{n})-p(x_{1},x_{2}...x_{n}))^{2}dx_{1}dx_{2}...dx_{n}$

Bibliography

• Morgenthalter, David (1981). "Mutlidimensional Edge Detection by Hypersurface Fitting". IEEE Transactions of Pattern Analysis and Machine Intelligence. 3 (4): 482–486. doi:10.1109/TPAMI.1981.4767134. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Chittineni, CB (1983). "Edge and Line Detection in Multidimensional Noisy Imagery Data". IEEE Transactions on Geoscience and Remote Sensing. 21 (2): 163–174. doi:10.1109/TGRS.1983.350485.

• Khotanzad, A (1989). "Unsupervised Segmentation of Textured Images by Edge-Detection in Multidimensional Features". IEEE Transactions of Pattern Analysis and Machine Intelligence. 11 (4): 414–421. doi:10.1109/34.19038.