In a previous post [Link], we were able to measure a distance between two points or the length of a sample by counting the number of pixels thanks to

`Analyze > Analyze Particles...`

. However, the result is directly dependent of the pixel connectivity... #### 1- How is a line drawn ?

Drawing a line with a computer consists of converting the formula f(x)=a *x + b (a function existing in a continuous/analog world) in a set of pixels (a discrete system) leading to an approximation of the drawing line as shown in Fig. 1.Fig.1: The red line drawn between the two red points is converted in a series of black pixels in an image Thus, this is only an approximation of the red line. |

*chessboard*" or 8-connectivity corresponding to the move of the king in chess.

Fig.2: 4-connectivity |

In Fig.2, the pixels are only connected by their edges. This is a "Manhattan","city-block", "North-South-East-West", or 4-connectivity. This is summarized in Fig. 3.

The main parameter is the pixel connectivity which can be 4, 8, or mixed.

[TODO]

**Fig 3**: 4-, 8-, and mixed-connectivity.

Depending of the pixel connectivity used in the drawing algorithm, the number of pixels required to go from the starting point A(0,0) to the end point B(20,5) is different.Note: The Bresenham algorithm [Wikipedia] is one of the drawing algorithms.

Euclidean distance: sqrt( (20-0)^2+(5-0)^2) = 20.61

8-connectivity: 21 pixels

4-connectivity: 26 pixels

The 8-connectivity system gives a good estimation of length in our example especially because the line is short. However, for longer distances, it will underestimate the real length...

Note: In mathematics,

The euclidean distance corresponds to the L2 norm of the vector AB.

In 4-connectivity [Wikipedia], the distance - L1-norm - is equal to:

- $D=\sqrt{{\left({x}_{0}-{x}_{1}\right)}^{2}+{\left({y}_{0}-{y}_{1}\right)}^{2}}$

In 8-connectivity [Wikipedia], the formula is:

- D = |x0 – x1| + |y0 - y1|

- D = max(|x0 – x1|,|y0 - y1|)

Now, the question is:

How can we weight the pixel count to get a good evaluation of the length?

<< TOC : PreviousNext: Weighting factors >>

## No comments:

## Post a Comment