Halftoning

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Halftoning

• Halftoning problem
• Full-color image
• Few-color display device
• usually 2 colors black and white
• input full color image
• output reduced-color image
• Ancient Solutions!

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Fixed Threshold

• Given
• in (x,y) input image
• S set of output colors, usually S0,1
• out (x,y) c?S, minimum c - in(x,y)
• out(x,y) 0, if in(x,y) lt 1/2
• 1, if gt 1/2

3
Random Threshold

• Given
• in (x,y) input image
• S set of output colors, usually S0,1
• out (x,y) pick r random (0 to 1)
• out(x,y) 0, if in(x,y) lt r
• 1, if gt r

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Comparison

• Fixed Threshold
• sharp edges preserved
• no gray information
• Random Threshold
• noisy edges
• more gray information

5
Color Reduction Problem

• Print continuous-tone images
• image colors gtgt device colors
• eg laser printer colors 2

Continuous tone
Fixed threshold
Random threshold
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Fixed Threshold
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Random Threshold
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1695 Mezzotint
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Basic Principle

• Trade off
• color detail vs spatial detail
• Why?
• eye integrates nearby pixels
• sees gray

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Halftoning

• Two basic types
• Ordered dither
• Bayer, Clustered-dot
• matrices of thresholds
• Error diffusion
• Floyd-Steinberg, Stucki
• process pixel, adjust neighbors

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Halftoning
original
ordered
clustered
Error diffusion
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Ordered Dither

• Many different thresholds
• Threshold depends on (x y)
• Random Threshold
• image looks grainy
• Repeated Thresholds
• visible repeated pattern

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Ordered Dither

• Group pixels into blocks
• eg 2x2 block
• 5 gray levels
• Algorithm
• i x mod n, j y mod ne int( in(x,y)
  n2)out(x,y) 0 if e gt M(i,j)
  1 if e lt M(i,j)

3
1
0
2
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Which Matrix?

• Bad matrices
• linear artifacts
• consecutive entries are nearby in matrix
• Good matrices
• consecutive entries are far apart

0
1
2
3
4
5
6
7
8
0
3
6
1
4
7
2
5
8
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Bayers Matrices

• Bayer 1976
• (2n x 2n)
• Defined recursively

3
1
2x2
0
2
15
7
13
5
3
11
1
9
4x4
12
4
14
6
0
8
2
10
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Ordered Dither Example
input
output
detail
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Dispersed-dot Ordered Dither
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Dispersed-dot synthetic image
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1844 Jacquard Loom Weaving
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Clustered-Dot Dither

• Commercial printing
• cant print tiny isolated dots
• use bigger dots
• less spatial resolution
• spiral threshold pattern

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8
9
15
7
1
2
10
4x4
6
0
3
11
13
5
4
12
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Newspaper Image
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Hand-made Stippling
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Color Clustered Dot

• Square grid of dots very evident
• 45-degree grid less evident
• 4 inks Cyan, Magenta, Yellow, Black
• overlaps moire patterns
• rotate colored inks
• less moires

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Four-Color Printing
yellow
cyan
magenta
black
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Printed Result
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Space-Filling Curve

• Hilbert Curve
• consecutive locations nearby
• like clustered-dot
• dots not round

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Space-Filling Curve

• 4x4 matrix
• 17 gray levels

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Space-Filling Curve
Test Gray Ramps
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Halftoning

• Two basic types
• Ordered dither
• Bayer, Clustered-dot
• matrices of thresholds
• Error diffusion
• Floyd-Steinberg, Stucki
• process pixel, adjust neighbors

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Error Diffusion

• Problem fixed threshold 0 or 1
• info lost error
• overall tone of image incorrect
• Solution use the error
• give it to pixels neighbor

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Floyd-Steinberg
processed

• Algorithm (? ? ? ?1)
• out(x,y) closest to in(x,y)
• e in(x,y) - out(x,y)
• in(x1, y) ? e
• in(x-1,y1) ? e
• in(x, y1) ? e
• in(x1,y1) ? e

?
?
?
?
? 7/16? 3/16? 5/16? 1/16
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Error Diffusion
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Error Diffusion Synthetic image
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Error Diffusion

• Problems
• worms advancing error front
• blurring
• Solutions
• bidirectional processing
• pre-filtering
• wider kernel (spread error further)
• Stucki

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Random-Dot Error Diffusion

• Problem
• Color dot positions known
• Assign colors
• Challenge
• dots not on a grid
• Approach
• error diffusion on graph

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The Approach (2)

• Random-dot F-S
• nearest dot neighbors
• in what order?

• Floyd-Steinberg (F-S)
• pixel neighbors
• scanline order

• Dot-processing orders investigated
• get nearest-neighbor graph, traverse
• depth-first
• breadth first
• error-driven (choose from frontier dots)
• hybrids

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Original Dots
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Points (dot centers)
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Dots and Neighbors (Delaunay Triangulation)
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Pixel Graphs
(Pixels have neighbors too)
used by F-S
4-neighbors
diagonal neighbors
8-neighbors
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The Algorithm

• Visit Dots in some order
• for each dot
• set to closest color in set
• (color changed ? error)
• add error to neighbors

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Visit dots in What order?

• dots nodes in graph
• visit dots traverse graph
• depth-first graph traversal?
• breadth-first?
• other?

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Depth-first graph traversal

• Stack stores advancing front
• push node on stack
• while (stack not empty) node ? top of
  stack process node (color it) for each
  neighbor of node push neighbor on
  stack

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Breadth-first graph traversal

• Queue stores advancing front
• push node on queue
• while (queue not empty) node lt head of
  queue process node for each neighbor of
  node append neighbor on queue

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Error-driven traversal

• Front set of candidate dots
• add node to front
• while (front not empty) choose node from
  front (min, max, or median error)
  process node for each neighbor of node
  add neighbor to front

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Gray Ramp Test (8-neighbor graph)
Gray ramp
Depth-first(DFS)
Breadth-first(BFS)
25 BFS 75 DFS
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Use Hybrid Traversal

• Error-driven traversal looks horrible!
• too ugly to show
• Breadth-first and Depth-first show artifacts
• Hybrid
• queue of pixels, random variable x (0 ? x ?
  1)
• if (x ? 1/4) get pixel from head of queue
• if (x gt 1/4) get pixel from tail of queue

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Pointillism simulation

• Restricted set of colors
• Dots are paint strokes
• Assume dot positions are known

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Seurats Pointillism
Original
Simulation
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Luces Pointillism
Original
Simulation
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Color Stippling
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Application Color Printing

• 4-color offset printing
• dots overlap
• loss of saturation
• random-dot halftoning color stippling
• dots dont overlap
• brighter colors

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Comparison
Standard 4-color printing
Error diffusion on adjacency graph
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Comparison
Sharper, more saturated
4-color printing
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Restricted Mosaic

• Simulated tiles
• tile color underlying image
• many colors possible
• Actual tiles
• limited color choice
• must use halftone
• Dots are square!
• use area voronoi diagram to get graph

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Restricted-color Mosaic
Continuous color
27 tile colors
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Better-chosen colors
2000 tile colors
8 tile colors
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Graph-Order Dither

• Generalizes Bayers Dither
• Pixels in arbitrary positions
• Problem
• Given n points in 2D,find visitation order,with
  consecutive points far apart

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Dispersed-Dot Dither

• Bayer (1976)
• Threshold Matrix (2n x 2n)
• replicated across image
• 4 x 4 matrix
• 0 8 2 10
• 12 4 14 6
• 3 11 1 9
• 15 7 13 5

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Consecutive Separation

• 0 8 2 10
• 12 4 14 6
• 3 11 1 9
• 15 7 13 5
• Consecutive entries (eg 12, 13) far apart
• Yields intermediate tones (grays)
• Avoids clusters

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The Problem

• Matrix Dots on square grid
• Random Dots no grid
• Problem Given dots in 2D, order them so
  that consecutive points are far apart.

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Algorithm

• Color nodes
• Pick one color
• Color only those nodes
• Repeat

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A Graph
node
node
arc
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Colored Graph
arcs join adjacent nodes every two
adjacent nodes get different colors
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Single-Color Separation
red nodes far apart
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Colored Graph
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Single-Color Separation
green nodes far apart
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Colored Graph
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Single-Color Separation
blue nodes too
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Colored Graph
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Single-Color Separation
and yellow too
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Repeat
pick one color
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Repeat
build neighbor graph
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Repeat
color again yellow nodes even farther apart than
before
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Repeat Recursively
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111
1 yellow 2 blue 3 green 4 red
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211
13
31
42
12
23
33
212
41
321
112
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Sort Labels
22
111
lexicographic order 1 111 2 112 3
12 4 13 5 211 6 212 7 22 8
23 9 31 10 321 11 322 12 33 13 41 14
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322
211
13
31
42
12
23
33
212
41
321
112
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Consecutive Separation
7
1
lexicographic order 1 111 2 112 3
12 4 13 5 211 6 212 7 22 8
23 9 31 10 321 11 322 12 33 13 41 14
42
11
5
4
9
14
3
8
12
6
13
10
2
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Algorithm

• label(S) //
  s set of points construct G(S)
  // G(S) neighbor graph color
  vertices with k colors for j 1 to k
  // for each color Sj
  points with color j append j to all
  labels in Sj label (Sj)
• sort all labels in lexicographic order

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Properties

• consecutive points far apart
• many orderings of same points
• each graph has many colorings
• points need not be on grid

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Applications

• Dithering without repetitive artifacts
• use many different dither matrices

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Applications

• Artistic Screening
• usual goal hide artifacts
• repeated matrix repetitive artifact
• artistic goal artifact art
• repeated matrix decorative pattern

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input
dither matrix
detail
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input
26 dither matrices
detail
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Application

• Dither with shapes
• collection of shapes
• can get neighbor graph
• hierarchical coloring

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input
shapes