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.

Multi-dimensional Prewitt operator determines the hyperplane that best fits the estimated function $\textstyle {\hat {p}}(P_{r})$ at point $\textstyle P_{r}$. Such hyperplane can be defined by:

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

It is then found by minimizing the squared error:

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

By differentiating least square error we are able to find $\textstyle a_{0},a_{1}...a_{n}$ which correspond to coefficients of the hyperplane. The data points in the multi-dimensional array that have high gradient values are selected as possible boundary surface elements.

Search for boundary surface elements is performed by sampling a random element and if such element satisfies criteria for being classified as a boundary element, neighbours of such point are evaluated recursively. If there are no more acceptable elements in the neighbourhood of found elements, aggregation process is terminated.

The sampling points that have high gradient values are possible boundary elements. Since mode boundaries are closed hypersurfaces, the boundary extraction procedure must be an omnidirectional aggregation process. The candidate points lying in the neighbourhood of a current boundary point are evaluated on the basis of their satisfying a gradient magnitude criterion for acceptance. To be more specific, let G(P ) ˆ 0 be the estimated magnitude of the gradient at the starting point P0

and let  τ be a tolerance factor, so that the aggregation 

algorithm incorporates only points with gradient magnitude greater than the threshold value G(P ) ˆ . 0 τ . When the algorithm is successful in finding some boundary elements in the neighbourhood of the current point, these elements are added to the currently accepted piece of boundary and become available current points for the next stages of the growth process. If there are no acceptable candidates in the neighbourhoods of all available current points, the aggregation terminates. The aggregation algorithm is then reinitialised from a new starting point, which is the point with the highest gradient value among the points that do not belong to a reconstructed boundary

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.