Logarithmic Mathematical Morphology: theory and applications

TL;DR

The paper tackles the challenge of lighting variations in grey-level mathematical morphology by introducing Logarithmic Mathematical Morphology ($LMM$), which employs the $LIP$ additive law to make the structuring function’s amplitude depend on image intensity. It formalizes $LIP$-dilation/erosion, logarithmic openings/closings, and dualities, and connects these operators to functional MM via a lattice isomorphism, enabling both standard MM operations and new $LIP$-based variants. A key contribution is the Map of Asplund distances under the $LIP$ framework, including tolerance to extrema and a novel operator based on the $LIP$-difference between $LIP$-erosions, plus extensions of top-hat techniques. The authors validate robustness to uniform and non-uniform lighting, demonstrating superior performance in vessel segmentation on eye-fundus images under varying exposure and illumination conditions, and show potential for integration with neural-network based methods for robust image analysis in uncontrolled lighting contexts.

Abstract

In Mathematical Morphology for grey-level functions, an image is analysed by another image named the structuring function. This structuring function is translated over the image domain and summed to the image. However, in an image presenting lighting variations, the amplitude of the structuring function should vary according to the image intensity. Such a property is not verified in Mathematical Morphology for grey level functions, when the structuring function is summed to the image with the usual additive law. In order to address this issue, a new framework is defined with an additive law for which the amplitude of the structuring function varies according to the image amplitude. This additive law is chosen within the Logarithmic Image Processing framework and models the lighting variations with a physical cause such as a change of light intensity. The new framework is named Logarithmic Mathematical Morphology (LMM) and allows the definition of operators which are robust to such lighting variations.

Figures (15)

• Figure 1: Comparison between usual methods and LMM to detect a spiral in (a) a simulated image also containing confounders : two close curves and a lighting drift. (b) This image is converted in LIP-greyscale (i.e. the inverted greyscale). (d) The top-hat with a flat disk extracts the spiral and both curves. However, it is sensitive to the lighting drift. (e) The LIP-top-hat with a flat disk extracts the spiral and both curves without the lighting drift. (f) The LMM operator is a LIP-difference between two LIP-openings: the first LIP-opening is done by (c) a probe composed of a ring and a Gaussian and the second by a probe composed of the same ring. The LMM operator successfully extracts the spiral and strongly attenuates both close curves.
• Figure 2: In an image $f$ (represented in the LIP-scale), comparison between functional MM and LMM for: (a) the erosions $\varepsilon_b(f)$, $\varepsilon_b^{ \mathbin{ \ooalign{$△$\crcr \hidewidth \hbox{$\vstretch{0.45}{\hstretch{0.45}{\bm{+}}}$}\hidewidth}} }(f)$, (b) the dilations $\delta_b(f)$, $\delta_b^{ \mathbin{ \ooalign{$△$\crcr \hidewidth \hbox{$\vstretch{0.45}{\hstretch{0.45}{\bm{+}}}$}\hidewidth}} }(f)$, (c) the openings $\gamma_b(f)$, $\gamma_b^{ \mathbin{ \ooalign{$△$\crcr \hidewidth \hbox{$\vstretch{0.45}{\hstretch{0.45}{\bm{+}}}$}\hidewidth}} }(f)$ and (d) the closings $\varphi_b(f)$, $\varphi_b^{ \mathbin{ \ooalign{$△$\crcr \hidewidth \hbox{$\vstretch{0.45}{\hstretch{0.45}{\bm{+}}}$}\hidewidth}} }(f)$. (a) and (b) For both image peaks, the structuring function $b$ is represented (after an horizontal translation) for the erosions $\varepsilon_b(f)$, $\varepsilon_b^{ \mathbin{ \ooalign{$△$\crcr \hidewidth \hbox{$\vstretch{0.45}{\hstretch{0.45}{\bm{+}}}$}\hidewidth}} }(f)$ and the dilations $\delta_b(f)$, $\delta_b^{ \mathbin{ \ooalign{$△$\crcr \hidewidth \hbox{$\vstretch{0.45}{\hstretch{0.45}{\bm{+}}}$}\hidewidth}} }(f)$.
• Figure 3: In the LIP-scale, (a) an image $f$ is analysed by a probe $b$ from above and below. (b) The mlub$c_{1_b} (f)$, the mglb$c_{2_b} (f)$ and the map of Asplund distances $Asp_b^{ \mathbin{ \ooalign{$△$\crcr \hidewidth \hbox{$\vstretch{0.45}{\hstretch{0.45}{\bm{+}}}$}\hidewidth}} } (f)$ between the image and the probe. Both arrows point towards the minima of the map of Asplund distances.
• Figure 4: In the LIP-scale, comparison between the morphological gradient $\varrho_B(f)$, the LIP-morphological gradient $\varrho^{{ LIP}}_B(f)$ and the map of Asplund distances $Asp_b^{{ \mathbin{ \ooalign{$△$\crcr \hidewidth \hbox{$\vstretch{0.45}{\hstretch{0.45}{\bm{+}}}$}\hidewidth}} }}(f)$ of the image $f$. The flat structuring element $B$ has the same domain $D_B$ as the one of the structuring function $b$ in Fig. \ref{['fig:signal_mapAspAdd:sgn']}. Both arrows point towards the regional minima of the gradients.
• Figure 5: (a) $f^n$ is an image with a Gaussian white noise, with a standard deviation of 20.0 grey levels and a density of 0.08. The mlub $c_{1_b}(f^n)$ and the mglb $c_{2_b}(f^n)$ (without tolerance) are compared to the mlub $c_{1_b,p}(f^n)$ and the mglb $c_{2_b,p}(f^n)$ with a tolerance of $p= 85%$. $b$ is the probe. (b) The map of Asplund distances $Asp^{ \mathbin{ \ooalign{$△$\crcr \hidewidth \hbox{$\vstretch{0.45}{\hstretch{0.45}{\bm{+}}}$}\hidewidth}} }_{b}(f^n)$ (without tolerance) is compared to the map of Asplund distances with tolerance $Asp^{ \mathbin{ \ooalign{$△$\crcr \hidewidth \hbox{$\vstretch{0.45}{\hstretch{0.45}{\bm{+}}}$}\hidewidth}} }_{b,p}(f^n)$. The LIP-scale is used to represent grey-levels.

Theorems & Definitions (30)

• Definition 1
• Definition 2
• Proposition 1
• Definition 3
• Definition 4
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• proof : Proof of proposition \ref{['prop:LMM:base_operators']}