Computational Mathematics
Mathematical morphology
Binary image
Some connected set \(B\) that is a subset of some Euclidean space \(E=\mathbb{R}^2\) or \(E=\mathbb{Z}^2\)
Structuring element
Connected set \(B\) that characterises operations on binary images
Erosion Erosione 侵食
Operator of a binary image \(A\) and a structure element \(B\) that gives the effect that an image 'erodes'
This is the result of all points that structure element \(B\) can be translated to whilst being a subset of \(A\)
Intuitively, this is the result of the points that are in all translation of its binary image by the nevative of some point in the structure element, giving the visual effect that the binary image is eroding
Definitions
\( A \ominus B = \{ z \in E : B_{z} \subseteq A \} \)
- \(A\) is the binary image set
- \(B\) is the structure element set
- \(z\) is a translation vector
- \( B_z = \{b + z : b \in B \} \) is the structure element set with all elements translated by \(z\)
\(A \ominus B = = \bigcap_{b \in B} A_{-b} \)
- \(A\) is the binary image set
- \(B\) is the structure element set
- \(A_{-b}\) is the binary image set with all elements negatively translated by some element of \(B\)
Dilation Dilatazione
Operator of a binary image \(A\) and a structure element \(B\) that gives the effect that an image 'dilates outwards'
Intuitively, this is the result of the points that are in any translation of its binary image by some point in the structure element, giving the visual effect that the binary image is being dilated
Definitions
\( A \oplus B = \bigcup_{b \in B} A_{b} \)
- \(A\) is the binary image set
- \(B\) is the structure element set
- \(A_{b}\) is the binary image set with all elements translated by some element of \(B\)
- \(\)
\( A \oplus B = \{ z \in E : (B^s)_z \bigcap A \neq \nullset \} \)
- \(A\) is the binary image set
- \(B\) is the structure element set
- \( B^s = \{x \in E : -x \in B \} \) is the symmetric structuring element to \(B\)
- \(z\) is a translation vector
- \( (B^s)_z = \{b + z : b \in B^s \} \) is the symmetric structuring element set with all elements translated by \(z\)
Opening Apertura 開放
\( A \circ B = (A \ominus B) \oplus B\)
Closing Chiusura 閉鎖
\( A \bullet B = (A \oplus B) \ominus B\)
Otsu's method 大津の二値法
Thresholding algorithm for converting grayscale images into binary images by employing probability and optimisation techniques to identify the threshold demonstrating the largest outer-class variance. Assume the image has greyscale points with intensity taking discrete calues\([0,n]\)
- For each level of intensity \(t \in [0,n]\), assume it is the foreground cutoff and execute the following:
- Calculate \(\omega_{0}(t) = \sum^{t}_{i=0} \text{Pr}(i \text{ is the pixel's intensity})\) and \(\omega_{1}(t) = \sum^{n}_{i=t+1} \text{Pr}(i \text{ is the pixel's intensity})\). These weights are the probabilities that a pixel is in the background \(\omega_{0}(t)\) or foreground \(\omega_{1}(t)\) respectively.
- Calculate \(\mu_{0}(t) = \frac{ \sum^{t}_{i=0} i\text{Pr}(i \text{ is the pixel's intensity}) }{\omega_{0}(t) }\) and \(\mu_{1}(t) = \frac{ \sum^{n}_{i=t+1} i\text{Pr}(i \text{ is the pixel's intensity}) }{\omega_{1}(t) }\). These are the weighted averages
- Calculate \(\sigma^2(t) = \omega_{0}(t)\omega_{1}(t)[\mu_{0}(t) - \mu_{1}(t)]^2\)
- Elect the foreground cutoff as $\mathrm{argmax}_{t\in [0,n]} \{ \sigma^2(t) : t \in [0,n] \} $
Top-hat transform Trasformazione top-hat トップハット変換
White
\(T(f) = f - f \circ B\)
Black
\(T(f) = f \bullet B - f\)
Posterization
Shrinking the grayscale intensity range of an image, using some algorithm such as Otsu's to find the optimum translation to this new intensity range
Coding theory
Algebraic code
Substitution system
Binary system