KAZE

• Traditional methods like SIFT/SURF use Gaussian blurring (a linear scale space), which blurs both boundaries and noise equally. This weakens edges. KAZE uses a nonlinear diffusion filter to:

- Preserve edges (boundaries)

- Suppress noise

This leads to better keypoint localization, especially around object edges.

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KAZE constructs a nonlinear scale space using the following equation:

\begin{equation}\frac{\partial L}{\partial t}=\mathrm{div}(c(x,y,t)\cdot\nabla L)\end{equation}

• L: Image at time t
• ∇L: Gradient (edge direction)
• c(x,y,t): Conductivity function – controls the diffusion speed
  • High ∣∇L∣ (strong edge): small c → less blurring
  • Low ∣∇L∣ (flat region): large c → more blurring

This way, KAZE blurs noise but preserves edges.

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Instead of Gaussian pyramids (like in SIFT), KAZE builds scale space with increasing diffusion time . The figure shows blurred images at increasing time steps .

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Feature	SIFT (Scale-Invariant Feature Transform)	SURF (Speeded Up Robust Features)	KAZE (Nonlinear Scale Space)
Scale Space	Gaussian pyramid (linear scale space)	Gaussian pyramid (linear scale space)	Nonlinear diffusion (preserves edges and boundaries)
Detector	Difference of Gaussian (DoG)	Hessian matrix (approximated by box filters)	Determinant of Hessian matrix in nonlinear space
Descriptor	Gradient orientation histograms	Haar wavelet responses	First-order image derivatives
Rotation Invariance	Yes	Yes	Yes
Illumination Invariance	Normalized gradient magnitude	Normalized Haar responses	Normalized first-order derivatives
Computation Speed	Slower	Faster than SIFT	Slower (due to nonlinear PDE solving)
Robustness	Good for scale and rotation	Good, faster alternative to SIFT	Better at preserving edge structures and boundaries
Descriptor Dimension	128	64	64
Key Innovation	Scale-space extrema + gradient histograms	Integral image + box filters for speed	Nonlinear scale space via anisotropic diffusion