# Characterization of Prints Based on Microscale Image Analysis of Dot Patterns
## 3 Measurement, Modelling and Analysis at the Microscale of Printed Dots
### Computation of Dot Descriptors (Step 6)  
We defined six categories of dot patterns (see Figure 10):  
Figure 9: Illustration of the results obtained step by step. In this example yellow dots are extracted, measured and analyzed. (**a**) CMY image, (**b**) Step 1: extraction of the Yellow channel, (**c**) Step 1: Y image after removal of Magenta shadow, (**d**) Step 2: binarization, (**e**) Steps 3.1 to 3.4: removal of noise and irrelevant areas, (**f**) Step 4: labelling of regions of interest, (**g**) Step 5: removal of outliers, (**h**) Step 7: extraction of dots with concavity (\(\in\) ID1/ID2/ID3) or an irregular shape (\(\in\) ID4) defined by a solidity upper than 0.75, (**i**): extraction of dots with a convex shape (\(\in\) ID5) with a circular equivalent shape, or an elliptic equivalent shape, or another pattern shape. * ID1 narrow doughnut (with a hole inside the convex hull of the dot) with a symmetric shape;
* ID2 concave shape (with a large hole) with a symmetrical shape;
* ID3 concave shape (with a hole inside the convex hull of the dot) with an asymmetric shape;
* ID4 elliptic/circular shape with a hole inside the dot;
* ID5 elliptic/circular shape;
* ID6 random shape;  
The ID6 pattern is similar to the ID23 pattern defined in [5]. To analyze each individual dot, we propose computing the following shape descriptors:  
* Centroid (X, Y coordinates in \(\upmu\)m);
* A Area of the dot surface in \(\upmu\)m\({}^{2}\);
* A\({}_{\rm h}\) Area of the hole in the dot in \(\upmu\)m\({}^{2}\);
* A\({}_{\rm c}\) Area of the convex hull of the dot in \(\upmu\)m\({}^{2}\);
* A\({}_{\rm e}\) Area of the elliptic shape bounding the dot in \(\upmu\)m\({}^{2}\);
* Solidity = A/A\({}_{\rm c}\);
* Perimeter of the dot surface in \(\upmu\)m;
* P\({}_{\rm c}\) Perimeter of the convex hull of the dot in \(\upmu\)m;
* P\({}_{\rm e}\) Perimeter of the elliptic shape bounding the dot in \(\upmu\)m;
* Convex = P\({}_{\rm c}\)/P;
* Circularity= sqrt(4\(\pi\)A/P\({}^{2}\));
* Roundness = sqrt(4\(\pi\)A/P\({}_{\rm e}\)\({}^{2}\));
* DF\({}_{\rm min\,max}\) Pert diameters in \(\upmu\)m;
* Aspect ratio = DF\({}_{\rm min}\)/DF\({}_{\rm max}\);
* Orientation = the angle between the \(x\)-axis and the major axis of the ellipse that has the same second-moments as the region;
* Major and minor axis;
* A\({}^{\rm x1}\) Area of the dot upper the major axis;
* A\({}^{\rm x2}\) Area of the dot lower the major axis;
* A\({}^{\rm y1}\) Area of the dot at left of the minor axis;
* A\({}^{\rm y2}\) Area of the dot at right of the minor axis;
* Eccentricity = length of the minor axis / length of the major axis;
* Elongation along X axis = width of the elliptic bounding box/ width of the highest inscribed ellipse;
* Symmetry relatively to X axis, to Y axis = A\({}^{\rm x1}\)/A\({}^{\rm x2}\), = A\({}^{\rm y1}\)/A\({}^{\rm y2}\), respectively. Only a few of these descriptors were also used by [PERSON] and [PERSON] [5,6]. To categorize a dot in one of the six categories of dot patterns, we defined the following rules:  
* Concave shapes belonging to category ID1 are defined by an elongation upper than 1.75, a symmetry value A\({}^{\rm x1}\)/A\({}^{\rm x2}\) upper than 0.85 and lower than 1.15, and a solidity upper than 0.8. Figure 10: Categories of dot patterns (convex hulls of dots are highlighted in green, major/minor axis are highlighted in blue, convex hulls of holes are highlighted in red). * Concave shapes belonging to category ID2 are defined by a solidity upper than 0.75 and a symmetry value A\({}^{\text{x1}}\)/A\({}^{\text{x2}}\) upper than 0.85 and lower than 1.15. * Concave shapes belonging to category ID3 are defined by a solidity upper than 0.75 and a ratio A\({}^{\text{ij}}\)/A upper than 0.15 (assuming that the concavity is in the region R\({}^{\text{ij}}\)). * Dots belonging to category ID4 have a ratio A\({}_{\text{h}}\)/A upper than 0.15. Below this value, holes are insignificant.