Give the formula for calculating D4 and D8 distance.

In image processing and computer vision, D4 and D8 distances refer to distance measures based on different types of connectivity in a grid or pixel space. These measures are often used in algorithms such as pathfinding, skeletonization, and distance transforms. Here are the formulas for calculating D4 and D8 distances:

D4 Distance:

D4 distance (4-connectivity distance) considers movement only along the horizontal and vertical directions, where movements between adjacent pixels are weighted equally.

For two pixels p=(xp,yp)p = (x_p, y_p)p=(xp?,yp?) and q=(xq,yq)q = (x_q, y_q)q=(xq?,yq?) in a 2D grid, the D4 distance d4(p,q)d_4(p, q)d4?(p,q) is calculated as:

d4(p,q)=?xp−xq?+?yp−yq?d_4(p, q) = |x_p - x_q| + |y_p - y_q|d4?(p,q)=?xp?−xq??+?yp?−yq??

This formula sums the absolute differences in the xxx and yyy coordinates of the two pixels, reflecting the number of horizontal and vertical steps required to move from pixel ppp to pixel qqq under 4-connectivity.

D8 Distance:

D8 distance (8-connectivity distance) considers movement along both horizontal, vertical, and diagonal directions, where movements between adjacent pixels are weighted equally.

For two pixels p=(xp,yp)p = (x_p, y_p)p=(xp?,yp?) and q=(xq,yq)q = (x_q, y_q)q=(xq?,yq?) in a 2D grid, the D8 distance d8(p,q)d_8(p, q)d8?(p,q) is calculated as:

d8(p,q)=max?(?xp−xq?,?yp−yq?)d_8(p, q) = \max(|x_p - x_q|, |y_p - y_q|)d8?(p,q)=max(?xp?−xq??,?yp?−yq??)

This formula calculates the maximum of the absolute differences in the xxx and yyy coordinates of the two pixels, reflecting the shortest path in terms of steps under 8-connectivity.

Explanation:

• D4 Distance: Uses Manhattan distance, where movement is restricted to horizontal and vertical steps.
• D8 Distance: Uses Chebyshev distance, allowing movement in horizontal, vertical, and diagonal directions.

These distance measures are essential in various image processing tasks where the spatial relationships between pixels (or voxels in 3D) need to be quantified or optimized based on connectivity constraints. They provide a basis for algorithms such as distance transforms, pathfinding, and spatial analysis in digital images and volumes.