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Diffusion Curves. Dr. Scott Schaefer. Diffusion Curves. Vector graphics-based representation for 2D images Images are piecewise smooth with discontinuities represented as curves. Image taken from “Diffusion Curves: A Vector Representation for Smooth-Shaded Images”.

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Diffusion Curves

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Diffusion curves
Diffusion Curves

Dr. Scott Schaefer


Diffusion curves1
Diffusion Curves

  • Vector graphics-based representation for 2D images

  • Images are piecewise smooth with discontinuities represented as curves

Image taken from “Diffusion Curves: A Vector Representation for Smooth-Shaded Images”


Representation
Representation

  • Bezier curves represent discontinuities

  • Give each curve a color function on left/right side of curve

    • Original paper uses linear color change

    • Modification: Control points can have any color

Image taken from “Diffusion Curves: A Vector Representation for Smooth-Shaded Images”


Image construction
Image Construction

Use curves as boundary constraints

Find a harmonic function that interpolates those boundary constraints

Image taken from “Diffusion Curves: A Vector Representation for Smooth-Shaded Images”


Implementation simplified
Implementation (Simplified)

Define a left/right color for points

Use a background color to represent no data (black)

Draw curve as a polygon made of quads (thick) with smooth shading


Implementation simplified1
Implementation (Simplified)

  • Colored pixels are constraints

  • Find a harmonic function satisfying constraints

    • Harmonic function has Laplacian zero everywhere


Implementation simplified2
Implementation (Simplified)

1

0

0

-4

1

1

0

0

  • Colored pixels are constraints

  • Find a harmonic function satisfying constraints

    • Harmonic function has Laplacian zero everywhere

1


Implementation simplified3
Implementation (Simplified)

1

0

0

-4

1

1

0

0

  • Colored pixels are constraints

  • Find a harmonic function satisfying constraints

    • Harmonic function has Laplacian zero everywhere

    • Each value is average of its neighbors

1


Implementation simplified4
Implementation (Simplified)

  • Draw curves as quads

  • Read pixel buffer back from OpenGL

  • Repeat a lot

    • For all pixels whose initial value was black, replace with average color of its neighbors from previous iteration



Problem1
Problem

Final Result


Problem2
Problem

100 Iterations


Problem3
Problem

200 Iterations


Problem4
Problem

400 Iterations


Problem5
Problem

800 Iterations


Problem6
Problem

1600 Iterations


Problem7
Problem

6400 Iterations


Problem8
Problem

Infinity Iterations


Problem9
Problem

256


Simple multi grid
Simple Multi-Grid

  • Create power of 2 down-sampled images

    • Average value of all non-black pixels

  • For each level, starting at second to last

    • Up-sample previous level

      • Copy pixel value to black high-res pixels (non-black pixels are constraints)

    • For some number of iterations

      • For each non-constrained pixel, replace with color of its neighbors from previous iteration


Example
Example

Original 512x512


Example1
Example

Down-sampled 256x256


Example2
Example

Down-sampled 128x128


Example3
Example

Down-sampled 64x64


Example4
Example

Down-sampled 32x32


Example5
Example

Down-sampled 16x16


Example6
Example

Down-sampled 8x8


Example7
Example

Down-sampled 4x4


Example8
Example

Down-sampled 2x2


Example9
Example

Down-sampled 1x1


Example10
Example

Up-sampled


Example11
Example

Smoothed


Example12
Example

Up-sampled


Example13
Example

Smoothed


Example14
Example

Up-sampled


Example15
Example

Smoothed


Example16
Example

Up-sampled


Example17
Example

Smoothed


Example18
Example

Up-sampled


Example19
Example

Smoothed


Example20
Example

Up-sampled


Example21
Example

Smoothed


Example22
Example

Up-sampled


Example23
Example

Smoothed


Example24
Example

Up-sampled


Example25
Example

Smoothed


Example26
Example

Up-sampled


Example27
Example

Smoothed



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