SURF: Speeded Up Robust Features: Difference between revisions
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Our features will be regions in the image where the determinant of the Hessian are local maxima. | Our features will be regions in the image where the determinant of the Hessian are local maxima. | ||
[[File:surf_fig_1.png | thumb | 500px | Figure 1 from the paper. Each region can be computed using a summed area table/integral image.]] | |||
* The Hessian matrix: | * The Hessian matrix: | ||
<math>\mathcal{H}(\mathbf{x}, \sigma) | <math>\mathcal{H}(\mathbf{x}, \sigma) | ||
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\end{bmatrix}</math> | \end{bmatrix}</math> | ||
* Each entry is a convolution of a the Gaussian second order derivative with the image at <math>\mathbf{x}</math> | * Each entry is a convolution of a the Gaussian second order derivative with the image at <math>\mathbf{x}</math> | ||
* These convolutions are approximated using box filters on an integral image. | * These convolutions are approximated using box filters on an integral image (Fig 1). | ||
*: The approximations are denoted as <math>D_{xx}, D_{yy}, D_{xy}</math> | *: The approximations are denoted as <math>D_{xx}, D_{yy}, D_{xy}</math> | ||
* The determinant of the hessian is then <math>D_{xx}D_{yy} - (0.9*D_{xy})^2</math> | * The determinant of the hessian is then <math>D_{xx}D_{yy} - (0.9*D_{xy})^2</math> | ||
Revision as of 12:28, 23 April 2020
Feature Extraction
Fast-Hessian Detector
Our features will be regions in the image where the determinant of the Hessian are local maxima.
- The Hessian matrix:
\(\displaystyle \mathcal{H}(\mathbf{x}, \sigma) = \begin{bmatrix} L_{xx}(\mathbf{x}, \sigma) & L_{xy}(\mathbf{x}, \sigma)\\ L_{xy}(\mathbf{x}, \sigma) & L_{yy}(\mathbf{x}, \sigma) \end{bmatrix}\)
- Each entry is a convolution of a the Gaussian second order derivative with the image at \(\displaystyle \mathbf{x}\)
- These convolutions are approximated using box filters on an integral image (Fig 1).
- The approximations are denoted as \(\displaystyle D_{xx}, D_{yy}, D_{xy}\)
- The determinant of the hessian is then \(\displaystyle D_{xx}D_{yy} - (0.9*D_{xy})^2\)
- 0.9 is a correction term for the approximation
- \(\displaystyle \frac{|L_{xy}(1.2)|_{F}}{|L_{xx}(1.2)|_{F}}\frac{|D_{xx}(9)|_{F}}{|D_{xy}(9)|_{F}} = 0.912\)
- 0.9 is a correction term for the approximation
- Interest points are local extrema of the determinant and trace of the Hessian
Scale-space representation
- They can increase (e.g. double) the filter size for their approximation and to get representations at multiple scales.
- They apply a "non-maximum suppression in a \(\displaystyle 3 \times 3 \times 3\) neighborhood" to "localise interest points in the image and over scales"
- Non-maximum suppression is a filtering technique to remove duplicates
- Basic idea: Let B be a set of regions. Let D be the filtered set we want to output.
- Pick the max confidence region from set B to D. Remove it from B.
- For each region in B, delete it if the IOU with selected is > threshold.
- See non-maximum suppression
- Non-maximum suppression is a filtering technique to remove duplicates
SURF Descriptor
Orientation Assignment
- Sample Haar-wavelet responses in x and y-direction at points around each feature
- Using integral images, only 6 operations are need to compute in x or y direction
- We have 6 distinct corners so we need 5 fma operations in total for each direction.
- Using integral images, only 6 operations are need to compute in x or y direction
- Using a 360-degree (pivoting) sliding window with radius \(\displaystyle \frac{\pi}{3}\), calculate the sum of all horizontal and vertical responses yielding vector. Note the window moves in increments of \(\displaystyle \frac{\pi}{3}\)
- Pick the direction with the largest vector.
Descriptor Components
- Create square regions positioned at feature points and oriented using the calculated orientation
- ...
OpenCV
SURF is available in OpenCV