Distinctive Image Features from Scale-Invariant Keypoints: Difference between revisions

Distinctive Image Features from Scale-Invariant Keypoints (view source)

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 ==Algorithm==
 ===Scale-space extrema detection===
+\(L(x,y,\sigma)\) is the scale space of an image. This is generated by the convolution of a Gaussian with an input image:
+<math display="block">
+\begin{align}
+L(x,y,\sigma) = G(x, y, \sigma) * I(x,y)
+\end{align}
+</math>
+where <math display="inline">G(x,y,\sigma) = \frac{1}{2 \pi \sigma^2}\exp{-(x^2+y^2)/(2\sigma^2)}</math>.
+This is used to compute the difference-of-Gaussian (DoG) for two scales separated by \(k\):
+<math display="block">
+\begin{align}
+D(x,y,\sigma) &= (G(x,y,k\sigma) - G(x,y,\sigma)) * I(x,y)\nonumber \\
+&= L(x,y,k\sigma) - L(x,y,\sigma)
+\end{align}
+</math>
+For your image, you create multiple octaves by down-sampling the image by 2.
+Within each octave, you have multiple scales with varying \(\sigma\).
+E.g. if you have 5 scales per octave, then you will have 4 DoG images.
+To find local min/max, compare each pixel in each DoG image to its 8 neighboring pixels within the image and 9 neighboring pixels in each neighboring scale. In total, it will be the min/max among (8+9+9=26) neighboring pixels. This is only done on the intermediate DoG images (i.e. excluding the first and last).
+In his paper, Lowe uses \(k=\sqrt{2}=2^{1/s}\) with \(s=2\). This means generating \(s+3=5\) Gaussian blurred images for each octave.
 ===Keypoint localization===
 ===Orientation assignment===
 ===Keypoint descriptor===