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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”

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

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)

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 (Simplified)

- Colored pixels are constraints
- Find a harmonic function satisfying constraints
- Harmonic function has Laplacian zero everywhere

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 (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 (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

Problem

Final Result

Problem

100 Iterations

Problem

200 Iterations

Problem

400 Iterations

Problem

800 Iterations

Problem

1600 Iterations

Problem

6400 Iterations

Problem

Infinity Iterations

Problem

256

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

- Up-sample previous level

Example

Original 512x512

Example

Down-sampled 256x256

Example

Down-sampled 128x128

Example

Down-sampled 64x64

Example

Down-sampled 32x32

Example

Down-sampled 16x16

Example

Down-sampled 8x8

Example

Down-sampled 4x4

Example

Down-sampled 2x2

Example

Down-sampled 1x1

Example

Up-sampled

Example

Smoothed

Example

Up-sampled

Example

Smoothed

Example

Up-sampled

Example

Smoothed

Example

Up-sampled

Example

Smoothed

Example

Up-sampled

Example

Smoothed

Example

Up-sampled

Example

Smoothed

Example

Up-sampled

Example

Smoothed

Example

Up-sampled

Example

Smoothed

Example

Up-sampled

Example

Smoothed

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