Loading in 2 Seconds...

Wedgelets: A Multiscale Geometric Representation for Images

Loading in 2 Seconds...

- 317 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'wedgelets: a multiscale geometric representation for images' - Sophia

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### Wedgelets: AMultiscaleGeometricRepresentation for Images

Michael Wakin

Rice University

dsp.rice.edu

Joint work with Richard Baraniuk, Hyeokho Choi, Justin Romberg

Computational Harmonic Analysis

- Representation
- Analysis study through structure of should extract features of interest
- Approximation exploit sparsity of

coefficients

basis, frame

Nonlinear Approximation

few big

sorted index

- -term approximation: use largestindependently
- Greedy / thresholding

1-D Piecewise Smooth Signals

- smooth except for singularities at a finite number of 0-D points

Fourier sinusoids: suboptimal greedy approximation and extraction

wavelets: optimal greedy approximation

extract singularity structure

2-D Piecewise Smooth Signals

- smooth except for singularities along a finite number of smooth 1-D curves
- Challenge: analyze/approximategeometric structure

geometry

texture

texture

Wavelet-based Image Processing

- Standard 2-D tensor product wavelet transform

Wavelet Challenges

- Current wavelet methods suboptimal for 2-d piecewise smooth functions

- Geometrical info not explicit
- Inefficient -large number of significant WCs cluster around edge contours, no matter how smooth

2-D Wavelets: Poor Approximation

13

26

52

- Even for a smooth C2 contour, which straightens at fine scales…
- Too many wavelets required!

-term wavelet approximation

not

Response: A Pursuit of Sparsity

I. Anisotropic, Directional Frames

- curvelets
- contourlets
- bandelets
- etc…

II. Geometric Tilings / Dyadic Partitions

- wedgelets
- beamlets
- platelets
- etc…

Geometry Model for Cartoons

C2 boundary

- Toy model: flat regions separated by smooth contours
- Goal: representation that is
- sparse
- simply modeled
- efficiently computed
- extensible to texture regions (RichB -> Saturday)

2-D Dyadic Partition

- Multiscale analysis
- Zoom in by factor of 2 each scale
- Represent image on each block
- Partition/tiling

not a basis/frame

2-D Dyadic Partition = Quadtree

- Each parent node has 4 children at next finer scale
- Pruning reflects partitioning
- Leaf nodes decorated with atoms

Wedgelet Representation

r

- Build a cartoon using wedgelets on dyadic squares
- Quad-tree structuredeeper in tree finer curve approximation
- Decorate leaves with (r,q)

parameters

Wedgelet Representation

- Prune wedgelet quadtree to approximate local geometry (adaptive)
- New problem:

How to find the proper approximation?

Wedgelet Inference

- Find representation / prune tree to balance a fidelity vs. complexity trade-off

Wedgelet Inference

- Find representation / prune tree to balance a fidelity vs. complexity trade-off
- For Comp(W) –measure the complexity of the wedgelet representation(quadtree + (r,q))
- Donoho: Comp(W) = #leaves

NLA-like solution

Wedgelet Inference

- Find representation / prune tree to balance a fidelity vs. complexity trade-off
- Exponential computational cost to solve brute force
- Dynamic programming to the rescue!

distortion

complexity

Top-Down: Greedy

3

3

1

1

2

1

1

3

4

4

1

- Label each node of the tree with
- Move down from tree root and prune when

15

15

6

4

6

4

3

Bottom-Up: Dynamic Programming

1

1

3

2

1

3

1

4

1

4

3

2

3

2

3

1

1

1

- Label each node of the tree with
- Move up from the leaves and prune whenelse

15

15

6

4

6

4

3

3

Bottom-Up: Dynamic Programming

1

3

1

1

3

1

4

1

3

4

1

1

3

2

2

- Label each node of the tree with
- Move up from the leaves and prune whenelse

15

15

6

4

5

4

3

Bottom-Up: Dynamic Programming

1

3

1

1

3

1

4

1

3

4

1

1

3

2

2

- Label each node of the tree with
- Move up from the leaves and prune whenelse

15

9

6

4

5

4

3

Summary: Dynamic Programming

- Simple recursive bottom-up algorithm
- Efficient: O(N) for N-node tree
- Finds optimal solution provided cost is additivewith independent components

Wedgelet Inference

- Find representation / prune tree to balance a fidelity vs. complexity trade-off
- Optimal approximation
- Near optimal rate distortion decay
- Near optimal estimation [Donoho]

#Leaves Complexity Penalty

- Accounts for wedgelet partition size, but not wedgelet orientation

#leaves

#leaves

“smooth”

“simple”

“rough”

“complex”

Multiscale Geometry Model (MGM)

- Decorate each tree node with orientation (r,q) and then model dependencies thru scale

Multiscale Geometry Model (MGM)

- Decorate each tree node with orientation (r,q) and then model dependencies thru scale
- Insight: Smooth curve Geometric innovations small at fine scales
- Model: Favor small innovations over large innovations (statistically)

Multiscale Geometry Model (MGM)

- Wavelet-like geometry model: coarse-to-fine prediction
- model parent-to-child transitions of orientations

small

innovations

large

MGM

- Wavelet-like geometry model: coarse-to-fine prediction
- model parent-to-child transitions of orientations
- Markov-1 statistical model
- state = (r,q) orientation of wedgelet
- parent-to-child state transition matrix

MGM

- Markov-1 statistical model
- Joint wedgelet Markov probability model:
- Complexity = Shannon codelength == number of bits to encode

MGM Inference

- Find representation / prune tree to balance the fidelity vs. complexity trade-off
- Efficient solution via dynamic programming

Wedgelet Coding of Cartoon Images

- Choosing wedgelets = rate-distortion optimization

Shannon code length

to encode

Predictive Wedgelet Coding

C2contour (1-d)

Optimal rate-distortion performance

compared to

for leaf-only encoding

Proof: Start with leaf-encoding Consider a wedgelet at scale j Block size is 2-jx2-j

O(2-2j)

2-j

C2 curve contained in strip with width ~ length2

A discrete wedgelet exists that deviates from curve by only O(2-2j); L2 distortion on this block = O(2-3j)

O(2-2j)

2-j

Since curve is C2, there are O(2j) such blocks

Total distortion D~O(2-2j)

Number of bits (bitrate) to encode tick marks = O(j) bits per block; thus, total rate R ~ O(j2j)

O(2-2j)

2-j

Combining: for leaf-only encoding

Can reduce bitrate by predicting wedgelets down through scale (MGM); consider scale j+1…

O(2-2(j+1))

2-(j+1)

2-(j+1)

Finer scale wedgelets within O(2-2(j+1)) of curve

Coarse scale wedgelet within O(2-2j) of curve

O(2-2(j+1))

2-(j+1)

2-(j+1)

Given the parent, there are only O(1) possible wedgelets on each finer block. Thus, rate is O(1) bits per block, or R~O(2j) bits total. This gives the optimal D(R)~R-2.

Summary: Wedgelets

- Adaptive dyadic partition
- Fit to data using fidelity/complexity optimization
- Improve using MGM statistical model for parent-to-child wedgelet orientation correlations
- MGM parent-to-child modeling akin to “wavelets” for wedgelets
- Optimal approximation and R/D rates

[Dynamic programming as a useful tool]

- Only suited for cartoon images

n-D Surflet Representation [VC,MW,DB,RB]

- Generalize wedgeletconcept to
- arbitrary dimension n
- arbitrary (n-1)-D manifold smoothness k
- requires >linear elements

Joint Texture/Geometry Modeling

- Dictionary D = {wavelets} U {wedgelets}
- Representation tradeoff: texture vs.geometry
- Test case: approximation / compression

Download Presentation

Connecting to Server..