Computer Vision News - January 2017

Morphological reconstruction for grayscale images has proved extremely valuable for a variety of image analysis tasks. This month, our new section “ Focus on… ” will demonstrate how morphological reconstruction works and show its practical application in identifying the healthy tissue of the liver and kidneys. Morphological reconstruction for grayscale images is a useful but little-known method for extracting meaningful information about objects within an image. The objects could be just about anything: letters in a scanned text document, fluorescently stained cell nuclei or galaxies in a far-infrared telescope image. This process takes two images. The original 2D grayscale image is called the mask and is denoted here by X. A second grayscale image, whose peaks identify the location of objects in the mask that one wants to emphasize, is called the marker and is denoted here by Y. Before introducing morphological reconstruction, let’s define a few concepts: Geodesic dilation: This is a function that takes as input two binary images: the mask image X and the marker image Y, Y ⊆ X (that is, every pixel of Y has a value smaller or equal to the equivalent pixel of X), and a distance n≥0. The geodesic dilation will be the set of pixels of X whose geodesic distance from Y is smaller or equal to n. ( ( = { ∈ | ( , ≤ } Binary Morphological reconstruction: The binary morphological reconstruction of the original image X from the mask Y ⊆ X is obtained by iterating elementary geodesic dilations of Y inside X until stability. This can be formulated as: Threshold: The k-value threshold of an arbitrary 2D grayscale image I is simply all pixels of I with grayscale value greater than k. Formally this can be defined as: ( = { ∈ | ( ≥ } Now we are ready to define the morphological reconstruction for grayscale images: let Y be the original grayscale image and X be the marker image; the grayscale reconstruction ( of X from Y is given by: ∀ ∈ , ( ( = { ∈ [0, − 1] | ∈ ( ( ( } 14 Computer Vision News Focus on… Morphological Reconstruction Focus on…