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# Hessian Affine region detector

The Hessian-Affine region detector is a feature detector used in the fields of computer vision and image analysis. Like other feature detectors, the Hessian-Affine detector is typically used as a preprocessing step to algorithms that rely on identifiable, characteristic interest points.

The Hessian-Affine detector is part of the subclass of feature detectors known as "affine-invariant" detectors: Harris-Affine regions, Hessian-Affine regions, Maximally Stable Extremal Regions, Kadir Brady saliency regios, edge-based regions (EBR) and intensity extrema-based (IBR) regions.

Algorithm Description

The "Hessian-Affine" detector algorithm is almost identical to that of the Harris-Affine detector. In fact, both algorithms were derived by [http://www.robots.ox.ac.uk/~sarchdal/KrystianMikolajczyk.html Krystian Mikolajczyk] and [http://lear.inrialpes.fr/people/schmid/ Cordelia Schmid] in 2002. [http://vasc.ri.cmu.edu/~hebert/04AP/mikolajc_ECCV2002.pdf Mikolajcyk, K. and Schmid, C. 2002. An affine invariant interest point detector. In "Proceedings of the 8th International Conference on Computer Vision", Vancouver, Canada.] ] . See the Harris-Affine entry for a more complete description of the detection algorithm.

How does the Hessian-Affine differ?

The Harris-Affine detector relies on interest points detected at multiple scales using the Harris corner measure on the second-moment matrix. The Hessian-Affine also uses a multiple scale iterative algorithm to spatially localize and select scale & affine invariant points. However, at each individual scale, the Hessian-Affine detector chooses interest points based on the Hessian matrix at that point:

where $L_\left\{aa\right\}\left(mathbf\left\{x\right\}\right)$ is second partial derivative in the $a$ direction and $L_\left\{ab\right\}\left(mathbf\left\{x\right\}\right)$ is the mixed partial second derivative in the $a$ and $b$ directions. It's important to note that the derivatives are computed in the current iteration scale and thus are derivatives of an image smoothed by a Gaussian kernel: $L\left(mathbf\left\{x\right\}\right) = g\left(sigma_I\right) otimes I\left(mathbf\left\{x\right\}\right)$. As discussed in the Harris-Affine article, the derivatives must be scaled appropriately by a factor related to the Gaussian kernel: $sigma_I^2$.

At each scale, interest points are those points that simultaneously are local extrema of both the determinant and trace of the Hessian matrix. The trace of Hessian matrix is identical to the Laplacian of Gaussians (LoG) [http://www.robots.ox.ac.uk/~vgg/research/affine/det_eval_files/mikolajczyk_ijcv2004.pdf Mikolajczykm K. and Schmid, C. 2004. Scale & affine invariant interest point detectors. "International Journal on Computer Vision" 60(1):63-86.] ] :

As discussed in Mikolajczyk et al.(2005), by choosing points that maximize the determinant of the Hessian, this measure penalizes longer structures that have small second derivatives (signal changes) in a single direction [http://www.robots.ox.ac.uk/~vgg/research/affine/det_eval_files/vibes_ijcv2004.pdf K. Mikolajczyk, T. Tuytelaars, C. Schmid, A. Zisserman, J. Matas, F. Schaffalitzky, T. Kadir and L. Van Gool, A comparison of affine region detectors. In IJCV 65(1/2):43-72, 2005] ] . This type of measure is very similar to the measures used in the blob detection schemes proposed by Lindeberg [http://www.nada.kth.se/cvap/abstracts/cvap198.html Lindeberg, Tony. "Feature detection with automatic scale selection", International Journal of Computer Vision, 30, 2, pp. 77-116, 1998.] ] , where either the Laplacian or the determinant of the Hessian were used in blob detection methods with automatic scale selection.

Like the Harris-Affine algorithm, these interest points based on the Hessian matrix are also spatially localized using an iterative search based on the Laplacian of Gaussians. Predictably, these interest points are called Hessian-Laplace interest points. Furthermore, using these initially detected points, the Hessian-Affine detector uses an iterative shape adaptation algorithm to compute the local affine transformation for each interest point. The implementation of this algorithm is almost identical to that of the Harris-Affine detector; however, the above mentioned Hessian measure replaces all instances of the Harris corner measure.

Robustness to Affine and other Transformations

Mikolajczyk et al. (2005) have done a thorough analysis of several state of the art affine region detectors: Harris-Affine, Hessian-Affine, MSER [ [http://www.robots.ox.ac.uk/~vgg/research/affine/det_eval_files/matas_bmvc2002.pdf J.Matas, O. Chum, M. Urban, and T. Pajdla, Robust wide baseline stereo from maximally stable extremal regions. In BMVC p. 384-393, 2002.] ] , IBR & EBR [ [http://www.robots.ox.ac.uk/~vgg/research/affine/det_eval_files/tuytelaars_ijcv2004.pdf T.Tuytelaars and L. Van Gool, Matching widely separated views based on affine invariant regions . In IJCV 59(1):61-85, 2004.] ] and salient [ [http://www.robots.ox.ac.uk/~vgg/research/affine/det_eval_files/kadir04.pdf T. Kadir, A. Zisserman, and M. Brady, An affine invariant salient region detector. In ECCV p. 404-416, 2004.] ] detectors [http://www.robots.ox.ac.uk/~vgg/research/affine/det_eval_files/vibes_ijcv2004.pdf K. Mikolajczyk, T. Tuytelaars, C. Schmid, A. Zisserman, J. Matas, F. Schaffalitzky, T. Kadir and L. Van Gool, A comparison of affine region detectors. In IJCV 65(1/2):43-72, 2005] ] . Mikolajczyk et al. analyzed both structured images and textured images in their evaluation. Linux binaries of the detectors and their test images are freely available at their [http://www.robots.ox.ac.uk/~vgg/research/affine/index.html webpage] . A brief summary of the results of Mikolajczyk et al (2005) follow; see [http://www.robots.ox.ac.uk/~vgg/research/affine/det_eval_files/vibes_ijcv2004.pdf "A comparison of affine region detectors"] for a more quantitative analysis.

Overall, the Hessian-Affine detector performs second best to MSER. Like the Harris-Affine detector, Hessian-Affine interest regions tend to be more numerous and smaller than other detectors. For a single image, the Hessian-Affine detector typically identifies more reliable regions than the Harris-Affine detector. The performance changes depending on the type of scene being analyzed. The Hessian-Affine detector responds well to textured scenes in which there are a lot of corner-like parts. However, for some structured scenes, like buildings, the Hessian-Affine detector performs very well. This is complementary to MSER that tends to do better with well structured (segmentable) scenes.

Software Packages

* [http://www.robots.ox.ac.uk/~vgg/research/affine/ Affine Covariant Features] : K. Mikolajczyk maintains a web page that contains Linux binaries of the Hessian-Affine detector in addition to other detectors and descriptors. Matlab code is also available that can be used to illustrate and compute the repeatability of various detectors. Code and images are also available to duplicate the results found in the Mikolajczyk et al. (2005) paper.

[http://vasc.ri.cmu.edu/~hebert/04workshop/presentations/schmid_sicily04.ppt] - Presentation slides from Mikolajczyk et al. on their 2005 paper.

[http://lear.inrialpes.fr/software] - Cordelia Schmid's Computer Vision Lab

[http://www.robots.ox.ac.uk/~vgg/research/affine/] - Code, test Images, bibliography of Affine Covariant Features maintained by Krystian Mikolajczyk and the [http://www.robots.ox.ac.uk/~vgg/ Visual Geometry Group] from the Robotics group at the University of Oxford.

[http://iris.usc.edu/Vision-Notes/bibliography/twod275.html] - Bibliography of feature (and blob) detectors maintained by USC Institute for Robotics and Intelligent Systems

*Harris-Affine
*MSER
*Scale-space
*Isotropy
*Corner detection
*Interest point detection
*Computer vision

References

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