Binary Morphology and Image Processing Techniques

Reconstruction of Binary Objects

• Create a marker image R0 with seed pixels from selected objects in image I.
• Iteratively compute Ri = (Ri−1 ⊕ S) ∩ I until no change occurs (Ri remains constant).
• For removing partly visible objects, use boundary pixels B as seeds and subtract the resulting reconstruction from the input image.
• To fill holes in objects, compute the reconstruction of the complement image Ic using boundary pixels as seeds, then take its complement Rc.

Distance Transform of Binary Images

• Compute object pixel distances to the background by iteratively dilating the input image I, maintaining non-merging constraints.
• The result is known as Voronoi (Dirichlet) tessellation.

Skeletonization of Binary Images

• Find the representative centerline of objects through iterative conditional erosion (thinning) that preserves connectivity.
• The final structure obtained is a one-pixel thick skeleton of the object.

Binary Morphology in nD Images

• Concepts extend to any dimensionality, such as 3D with volumetric pixels (voxels).
• Algorithms include 3D dilation, opening, erosion, and closing.

Gray-scale Mathematical Morphology

• Treat nD gray-scale images as (n+1)D binary images for processing.
• The umbra of an image represents the landscape surface with volume beneath it.

Gray-scale Dilation and Erosion

• Dilation is defined as I ⊕ S = U⁻¹[U(I) ⊕ U(S)], where U(I) is the umbra of the gray-scale image.
• Erosion is defined as I S = U⁻¹[U(I) U(S)], mapping the binary erosion back to gray-scale.
• Both operations can be interpreted as local max-filtering (dilation) and min-filtering (erosion) for suitable structuring elements.

Opening and Closing of Gray-scale Images

• Opening is defined as I ⊖ S followed by dilation: I ° S = (I S) ⊕ S.
• Closing combines dilation followed by erosion: I ● S = (I ⊕ S) S, smoothing structures within the grayscale image.

Morphological Operations for Image Smoothing

• Gray-scale opening removes high-valued structures, while closing suppresses low-valued structures.
• Demonstrates advanced nonlinear filtering capabilities beyond linear filtering techniques.

Morphological Gradients and Laplacean

• Gradient is the difference between dilated and eroded images: D = I ⊕ S, G = D - E.
• Laplacean combines outer and inner gradients: L = D + E - 2I, yielding features important for edge detection.

Top-hat Filtering

• A technique using closing followed by subtraction to enhance or extract specific features: O = I - (I ⊕ S).

Summary of Mathematical Morphology

• Fundamental toolbox for image segmentation involving gray-scale and binary morphology.
• Pre-processing includes gray-scale noise removal, background shading removal, and eliminating unwanted structures.
• Post-processing functions include closing holes, detecting outlines, separating touching objects, and extracting shapes.

Ultimate Erosion and Dilation

• Ultimate erosion identifies representative center points by computing distance transforms and retaining local maxima.
• To separate touching objects, ultimate erosion is performed followed by a constrained reconstruction to prevent merging.
• Ultimate dilation computes background points equidistant to objects through iterative dilation while enforcing non-merging constraints.