A Regularized Solution to Edge Detection
We consider edge detection as the problem of measuring and localizing changes of light intensity in the image. As discussed by Torre and Poggio (1984), edge detection, when defined in this way, is an ill-posed problem in the sense of Hadamard. The regularized solution that arises is then the s...
| Main Authors: | , , |
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| Language: | en_US |
| Published: |
2004
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| Online Access: | http://hdl.handle.net/1721.1/5618 |
| _version_ | 1826215309692895232 |
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| author | Poggio, Tomaso Voorhees, Harry Yuille, Alan |
| author_facet | Poggio, Tomaso Voorhees, Harry Yuille, Alan |
| author_sort | Poggio, Tomaso |
| collection | MIT |
| description | We consider edge detection as the problem of measuring and localizing changes of light intensity in the image. As discussed by Torre and Poggio (1984), edge detection, when defined in this way, is an ill-posed problem in the sense of Hadamard. The regularized solution that arises is then the solution to a variational principle. In the case of exact data, one of the standard regularization methods (see Poggio and Torre, 1984) leads to cubic spline interpolation before differentiation. We show that in the case of regularly-spaced data this solution corresponds to a convolution filter---to be applied to the signal before differentiation -- which is a cubic spline. In the case of non-exact data, we use another regularization method that leads to a different variational principle. We prove (1) that this variational principle leads to a convolution filter for the problem of one-dimensional edge detection, (2) that the form of this filter is very similar to the Gaussian filter, and (3) that the regularizing parameter $lambda$ in the variational principle effectively controls the scale of the filter. |
| first_indexed | 2024-09-23T16:22:57Z |
| id | mit-1721.1/5618 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T16:22:57Z |
| publishDate | 2004 |
| record_format | dspace |
| spelling | mit-1721.1/56182019-04-12T13:39:42Z A Regularized Solution to Edge Detection Poggio, Tomaso Voorhees, Harry Yuille, Alan We consider edge detection as the problem of measuring and localizing changes of light intensity in the image. As discussed by Torre and Poggio (1984), edge detection, when defined in this way, is an ill-posed problem in the sense of Hadamard. The regularized solution that arises is then the solution to a variational principle. In the case of exact data, one of the standard regularization methods (see Poggio and Torre, 1984) leads to cubic spline interpolation before differentiation. We show that in the case of regularly-spaced data this solution corresponds to a convolution filter---to be applied to the signal before differentiation -- which is a cubic spline. In the case of non-exact data, we use another regularization method that leads to a different variational principle. We prove (1) that this variational principle leads to a convolution filter for the problem of one-dimensional edge detection, (2) that the form of this filter is very similar to the Gaussian filter, and (3) that the regularizing parameter $lambda$ in the variational principle effectively controls the scale of the filter. 2004-10-01T20:17:15Z 2004-10-01T20:17:15Z 1985-04-01 AIM-833 http://hdl.handle.net/1721.1/5618 en_US AIM-833 22 p. 1655960 bytes 1300701 bytes application/postscript application/pdf application/postscript application/pdf |
| spellingShingle | Poggio, Tomaso Voorhees, Harry Yuille, Alan A Regularized Solution to Edge Detection |
| title | A Regularized Solution to Edge Detection |
| title_full | A Regularized Solution to Edge Detection |
| title_fullStr | A Regularized Solution to Edge Detection |
| title_full_unstemmed | A Regularized Solution to Edge Detection |
| title_short | A Regularized Solution to Edge Detection |
| title_sort | regularized solution to edge detection |
| url | http://hdl.handle.net/1721.1/5618 |
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