line simplification algorithm: Visvalingam vs Doug

Question

I am trying to implement a siplification algorithm. The main 2 algorithms I found are Ramer-Douglas-Peucker: https://en.wikipedia.org/wiki/Ramer%E2%80%93Douglas%E2%80%93Peucker_algorithm and Visvalingam-Whyatt: https://bost.ocks.org/mike/simplify/ Currently I am running a few simulations of them on matlab in order to determine which answers my needs better.

The main goal for the algorithm is to simlipfy polygons in a map. My Input is a polygon\polyline and a threshold for mistake- epsilon.

I need the simlified polygon to be as close as possible to the original, and I do not have a requirment for number of points to keep.

I am having difficulties in comparing the two algorithms because: epsilon for R-D-P is a distance while epsilon for V-W is an area. I need help understanding how to compare between the two algorithms. which can give me less points to keep within the threshold?

Answer 1

I need the simlified polygon to be as close as possible to the original, and I do not have a requirment for number of points to keep.

DP method will give you better perceptible fit with lesser number of points - as its control parameter i.e the distance tolerance is captured by your requirement 'as close as possible'.

Having said that, the scale of the overall polygon or point-cloud relative to the pixel dimensions will have a bigger impact for smaller images. The exercise below can give you a 'feel' for how the two algorithms perform.

Here are some comparisons I ran between Visvalingam-Whyatt and Ramer-Douglas-Peucker for a few contours contained in what is around 100x100 bitmap originally. The images are screenshots of a ~ 10x zoom on the contours.

(you might want to download the images to appreciate the differences in performance)

Visvalingam-Whyatt method results : courtesy Zach's implementation on github ported to opencv data types.

VSV simplification - with 0.55(white), 0.4(red), 0.25(magenta), 0.15(cyan) percentage tolerances

VSV - point reductions t : % tolerance . this directly determines n = t*orig/100. n is the final number of points

orig 88: [n=47 for t=0.55], [n=34 for t=0.4], [n=20 for t=0.25], [n=12 for t=0.15]
orig 133: [n=72 for t=0.55], [n=52 for t=0.4], [n=32 for t=0.25], [n=18 for t=0.15]
orig 118: [n=63 for t=0.55], [n=46 for t=0.4], [n=28 for t=0.25], [n=16 for t=0.15]
orig 107: [n=57 for t=0.55], [n=41 for t=0.4], [n=25 for t=0.25], [n=15 for t=0.15]
orig 107: [n=57 for t=0.55], [n=41 for t=0.4], [n=25 for t=0.25], [n=15 for t=0.15]
orig 268: [n=146 for t=0.55], [n=106 for t=0.4], [n=65 for t=0.25], [n=39 for t=0.15]
orig 158: [n=85 for t=0.55], [n=62 for t=0.4], [n=38 for t=0.25], [n=22 for t=0.15]
orig 158: [n=85 for t=0.55], [n=62 for t=0.4], [n=38 for t=0.25], [n=22 for t=0.15]
orig 109: [n=58 for t=0.55], [n=42 for t=0.4], [n=26 for t=0.25], [n=15 for t=0.15]
orig 192: [n=104 for t=0.55], [n=75 for t=0.4], [n=46 for t=0.25], [n=27 for t=0.15]
orig 132: [n=71 for t=0.55], [n=51 for t=0.4], [n=31 for t=0.25], [n=18 for t=0.15]
orig 89: [n=47 for t=0.55], [n=34 for t=0.4], [n=21 for t=0.25], [n=12 for t=0.15]
orig 110: [n=59 for t=0.55], [n=42 for t=0.4], [n=26 for t=0.25], [n=15 for t=0.15]
orig 40: [n=20 for t=0.55], [n=14 for t=0.4], [n=8 for t=0.25], [n=4 for t=0.15]

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DP method results using openCV approxPolyDP

Douglas-Peucker - point reductions t : pixel distance tolerance => no direct correlation to n - the final number of points

orig 88: [n=33 for t=0.1], [n=29 for t=0.5], [n=8 for t=1], [n=6 for t=2]
orig 133: [n=57 for t=0.1], [n=45 for t=0.5], [n=12 for t=1], [n=7 for t=2]
orig 118: [n=50 for t=0.1], [n=40 for t=0.5], [n=15 for t=1], [n=8 for t=2]
orig 107: [n=47 for t=0.1], [n=35 for t=0.5], [n=11 for t=1], [n=6 for t=2]
orig 107: [n=30 for t=0.1], [n=24 for t=0.5], [n=8 for t=1], [n=6 for t=2]
orig 268: [n=126 for t=0.1], [n=110 for t=0.5], [n=32 for t=1], [n=23 for t=2]
orig 158: [n=80 for t=0.1], [n=62 for t=0.5], [n=17 for t=1], [n=11 for t=2]
orig 158: [n=66 for t=0.1], [n=52 for t=0.5], [n=16 for t=1], [n=9 for t=2]
orig 109: [n=50 for t=0.1], [n=38 for t=0.5], [n=12 for t=1], [n=9 for t=2]
orig 192: [n=74 for t=0.1], [n=64 for t=0.5], [n=18 for t=1], [n=15 for t=2]
orig 132: [n=58 for t=0.1], [n=45 for t=0.5], [n=14 for t=1], [n=11 for t=2]
orig 89: [n=37 for t=0.1], [n=31 for t=0.5], [n=7 for t=1], [n=6 for t=2]
orig 110: [n=42 for t=0.1], [n=36 for t=0.5], [n=9 for t=1], [n=7 for t=2]
orig 40: [n=18 for t=0.1], [n=15 for t=0.5], [n=9 for t=1], [n=3 for t=2]

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Summary :

• Both the methods degrade gracefully.
• VSV lets you specify the number of approximated points (in this implementation)
• VSV in this implementation lets you pick multiple approximate polygons in one shot.
• VSV retains a lot of pixel level convexity inflections even for large curvature sections - this could be undesirable in some cases.
• DP follows the convexities better and smooths out inflections more but by sacrificing 'closeness'.
• As a result, DP gives lesser number of points for the same percieved tolerance - which is anyway hard to compare between the two methods
• DP give a better feel for tolerance as its a linear distance specification.

For my application, I prefer the control offered by VWV in this implementation over the possible efficiency of DP method.

But overall I feel openCVs's DP implementation gives a smoother perception.

Though my conclusions of VSV performance are based on Zach's implementation alone, I doubt if other implementations will give characteristically different polygon subsets.

[UPDATE1] Here is the comparison for a worst case scenario. DP is visually more acceptable.