Digital Image Processing

• The top hat transformation is used as a simple tool for segmenting objects in gray-scale images that differ in brightness from background, even when the background is of uneven gray-scale. • The top hat transform is superseded by the watershed segmentation for more complicated backgrounds. • Assume a gray-level image X and a structuring element K. The residue of opening as compared to original image X ∖ (X ∘ K) constitutes a new useful operation called a Top hat transformation. • The top hat transformation is a good tool for extracting light objects on a dark but slowly changing background. Those parts of the image that cannot fit into structuring element K are removed by opening. • Subtracting the opened image from the original provides an image where removed objects stand out. • The actual segmentation can be performed by simple thresholding (Fig 18). • If an image were a hat, the transformation would extract only the top of it, provided that the structuring element is larger

• Hit-or-miss transformation is the morphological operator for finding local patterns of pixels, where local means the size of the structuring element. • It is a variant of template matching that finds collections of pixels with certain shape properties. • Structuring element B, Tested points X, operation denoted by a pair of disjoint sets B = (Bl, B2), called a composite structuring element. • The Hit-or-miss transformation ⊗ is defined as • Finding local patterns in image Bl tests objects, B2 background (complement), Useful for finding corners, for instance. • The hit-or-miss transformation operates as a binary matching between an image X and the structuring element(Bl, B2). It may be expressed using erosions and dilations as well Opening and closing • Erosion and dilation are not inverse transformation—if an image is eroded and then dilated, the original image is not re-obtained. • Erosion followed by dilation is called opening. The opening of an image X by the structurin

1. Compatibility with translation: Let the transformation l/J depend on the position of the origin 0 of the co-ordinate system, and denote such a transformation by l/J0 . If all points are translated by the vector −ℎ , it is expressed as l/J-h . The compatibility with translation principle is given by If l/J does not depend on the position of the origin, then the compatibility with translation: principle reduces to invariance under translation 2. Compatibility with change of scale: Let AX represent the homothetic scaling of a point set X (i.e., the co-ordinates of each point of the set are multiplied by some positive constant A). This is equivalent to change of scale with respect to some origin. Let l/JA denote a transformation that depends on the positive parameter A (change of scale). Compatibility with change of scale is given by If l/J does not depend on the scale A, then compatibility with change of scale reduces to invariance to change of scale 3. Local knowledge: The loc

• Mathematical Morphology is based on the algebra of non-linear operators operating on object shape and in many respects supersedes the linear algebraic system of convolution. • It performs in many tasks – pre-processing, segmentation using object shape, and object quantification – better and more quickly than the standard approach. • Mathematical morphology tool is different from the usual standard algebra and calculus. • Morphology tools are implemented in most advanced image analysis. • Mathematical morphology is very often used in applications where shape of objects and speed is an issue—example: analysis of microscopic images, industrial inspection, optical character recognition, and document analysis. • The non-morphological approach to image processing is close to calculus, being based on the point spread function concept and linear transformations such as convolution. • Mathematical morphology uses tools of non-linear algebra and operates with point sets, their connectivi

Convex hull • A region R is convex if and only if for any two points x1, x2 from R, the whole line segment defined by its end-points x1, x2is inside the region R. • The convex hull of a region is the smallest convex region H which satisfies the condition R is a subset of H. • The convex hull has some special properties in digital data which do not exist in the continuous case. For instance, concave parts can appear and disappear in digital data due to rotation, and therefore the convex hull is not rotation invariant in digital space. • The convex hull can be used to describe region shape properties and can be used to build a tree structure of region concavity. Fig 23: Convex hull • A discrete convex hull can be defined by the following algorithm which may also be used for convex hull construction. o This algorithm has complexity O(n2) and is presented here as an intuitive way of detecting the convex hull. Algorithm: Region convex hull construction 1. Find all pixels of a reg