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--- knowledge:dip:morphological-image-analysis [2016/11/02 13:26]
pkleczek
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-====== Morphological Image Analysis ======
-//Morphological Image Analysis -- Principles and Applications//, Pierre Soille
-===== Erosion =====
-The first question that may arise when we probe a set with a structuring element is //"Does the structuring element fit the set?"// The eroded set is the locus of points where the answer to this question is affirmative.
-The eroded value at a given pixel $x$ is the **minimum** value of the image in the window defined by the structuring element when its origin is at $x$:
-$$[\varepsilon_{B}(f)](x) = \min_{b \in B} f(x+b)$$
-FIXME 81, 82
-===== Dilation =====
-The dilation is the dual operator of the erosion and is based on the following question: //"Does the structuring element hit the set?"// The dilated set is the locus of points where the answer to this question is affirmative.
-the dilated value at a given pixel $x$ is the **maximum** value of
-the image in the window defined by the structuring element when its origin
-is at $x$:
-$$[\delta{B}(f)](x) = \max_{b \in B} f(x+b)$$
-FIXME 83, 84
-===== Basic morphological gradients =====
-Only symmetric structuring elements containing their origin are considered. By doing so, we make sure that the arithmetic difference is always nonnegative.
-  - arithmetic difference between the dilation and the erosion (//Beucher gradient//): $\rho_B = \delta_B - \varepsilon_B$ (= maximum variation of the grey level intensities within the neighbourhood)
-  - arithmetic difference between the dilation and the original image (//half-gradient by dilation//, //external gradient//): $\rho_{B}^{+} = \delta_B - id$
-  - arithmetic difference between the original image and its erosion (//half-gradient by erosion//, //internal gradient//): $\rho_{B}^{-} = id - \varepsilon_B$
-The choice between internal or external gradient depends on the geometry and relative brightness of the objects to be extracted. For instance, an external gradient applied to a two or one pixel thick dark structure will provide a thin edge following the structure whereas an internal gradient will output a double edge (one on each side of the structure).
-FIXME 102
-If the size of the SE is greater than 1, morphological gradients are referred
-to as //thick gradients//: $$\rho_{nB} = \delta_{nB} - \varepsilon_{nB}$$. \\
-Thick gradients give the maximum variation of the function in a neighbourhood
-of size $n$. If the size $n$ equals the width $e$ of the transition between regions of homogeneous grey level, the thick gradient will output the contrast value $h$ between these regions. These gradients are therefore recommended when the transitions between objects are smooth. However, thick gradients output thick edges.
-FIXME 88
-FIXME A combination of thick gradients of increasing size avoiding thick edges is presented in Sec. 4.6.

Joanna Jaworek-Korjakowska

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