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Hybrid Texture Synthesis

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Hybrid Texture Synthesis

Andrew NealenMarc AlexaDiscrete Geometric Modeling Group (DGM)Technische Universität Darmstadt

Eurographics Symposium on Rendering 2003

nxm Input Texture

NxM Output Texture

- The goal: Synthesize an output texture which isperceptually similarto the input texture. Also ensure that the result contains sufficient variation.

2D Texture Synthesis

- Existing (and Impressive) Technology
- Pixel-Based:
- Non-parametric Sampling [Efros and Leung 1999]
- Tree-structured Vector Quantization [Wei and Levoy 2000]
- Image Analogies [Hertzmann et al. 2001]

- Patch-Based:
- Patch-Based Sampling [Guo et al. 2001]
- Image Quilting [Efros and Freeman 2001]

- Pixel-Based:

Wei/Levoy Algorithm:

Can occur when using structured

Textures with rich histograms.

This is due to the L2 norm.

Possible Drawbacks ?

- Loss of global structure
- Loss of scale
- Boundary mismatch
- Blurring

Image Quilting Algorithm:

boundary mismatch artifacts are

noticeable

Possible Drawbacks ?

- Loss of global structure
- Loss of scale
- Boundary mismatch
- Blurring

Patch-Based Sampling Algorithm:

Can occur when texture features are not

well-alignedacross patch boundaries

Possible Drawbacks ?

- Loss of global structure
- Loss of scale
- Boundary mismatch
- Blurring

- Combine best aspects from other approacheswhile avoiding (or improving over) known pitfalls:
- Patch-based algorithms
- Are good at preserving global structure
- Can introduce artifacts along patch boundaries

- Pixel-based algorithms
- Preserve local coherence (MRF model: local and stationary)
- Possibly fail to preserve global structure
- This is especially problematic for textures with rich histograms and many high frequency features, due to the smoothing nature of the distance metric (L2norm)

- Patch-based algorithms

- Hybrid, two-fold approach to texture synthesis
- Adaptive patch sampling [Soler et al. 2002]
- Start with a uniform quadrilateral grid of patches (the output)
- Recursively split a patch if the best fit taken from the input texture is not good enough (a user-defined tradeoffmax)

- Overlap re-synthesis(novel overlap repair strategy)
- Mark invalid pixels in the overlap region
- Validitymap is defined by the error surface E(x)=(xsrc – xdst)2 and a user-defined tolerance parametermax
- Define an ordering for the invalid pixels and re-synthesize them using a per-pixel strategy

- Adaptive patch sampling [Soler et al. 2002]

1: HYBRIDSYNTHESIZE(T,P,ov,Δmax,δmax,R) : R

2: for all patches pi ∈P do

3: [Pi,Δi]← FINDBESTPATCH(T,Ri−1, pi,ov)

4: if (Δi < Δmax or ISSINGLEPIXEL(pi)) then

5: Si ← ERRORSURFACE(Pi,Ri−1)

6: Pi ← MARKINVALIDPIXELS(Pi,Si,δmax)

7: Mi ← BUILDTRAVERSALMAP(Pi)

8: Ri,composite ← COMPOSE(Pi,Ri−1)

9: Ri ← OVERLAPRESYNTHESIS(T,Ri,composite,Mi)

10: else

11: Ps ← SPLITPATCH(pi)

12: ovs ←max(3, ov/2)

13: Ri ← HYBRIDSYNTHESIZE(T,Ps,ovs,Δmax,δmax,Ri−1)

14: end if

15: end for

16: return R

1: FINDBESTPATCH(T,Ri−1, pi,ov) : [Pi,Δi]

2: Ii, Ji ← BUILDIMAGEMASK(Ri−1, pi,ov)

3: Ei ← ERRORIMAGE(T, Ii, Ji)

4: Ei,trim ← TRIMINVALIDREGIONS(Ei, Ji, pi)

5: [Pi,Δi]← BESTPATCH(Ei,trim,T)

6: return [Pi,Δi]

Synthesize Black Patch i

- Grow patch by overlap (toroidally)

Synthesize Black Patch i

- Extract image mask (Ii) and binary support (Ji), and circularly shift to upper left corner

Synthesize Black Patch i

Image Mask (Ii)

Binary Support (Ji)

* Compute error Ei(x0) between Ii and T for each circular shift x0=(x,y) of the input texture T

Input Texture (T)

Image Mask (Ii)

Binary Support (Ji)

C={R,G,B}

(both with RGB color values in [0,1])

WR,G,B = {0.299,0.587,0.114}

X0: circular shift of T

Input Texture (T)

Error Image(Ei)

(1)

Input Texture (T)

Error Image(E)

Selected Patch (P)

(2)

- Given that Ji(x) Ii(x) = Ii(x),and the cross correlation between two images (functions) f ◇ g is defined as ( f ◇ g )(x0) = Σx f (x)g(x+x0)
- The correlation f◇ g between two functions can be computed in O(N logN)
- in fourier space as f ◇g = F-1(F( f )∗F(g))
- (N :number of pixels in the input texture)
- In implementation it’s can be pre-compute the fourier transform for Tand need only recompute the fourier transforms of Iiand Jifor each new patch pi.

- Note: error image can be computed efficiently in the Fourier domain. Complexity: O(n log n)per patch.

Input Texture (T)

Error Image(E)

Selected Patch (P)

- Essentially: Hierarchical Pattern Mapping applied to the plane[Soler et al. 2002]

Synthesize White Patch

i > max Split Patch

i < max

- Trade-off between preserving global structure and avoiding detail artifacts

max = 0.01

max = 1.0

max = 0.04

- Trade-off between preserving global structure and avoiding detail artifacts

max = 0.01

max = 1.0

max = 0.04

- Trade-off between preserving global structure and avoiding detail artifacts

max = 0.01

max = 1.0

max = 0.04

Patch and Image Mask

Selected Patch (P)

Image Mask (I)

Selected Patch (P)

Image Mask (I)

Selected Patch (P)

Error Surface (S)

- If(Ji(x) !=0 and Si(x)> δmax )
as invalid and in need of per-pixel re-synthesis.

else if( Ji(x) != 0 and Si(x) < δmax )

are preserved and a stepwise alpha mask (feathering) is applied during Compositing.

(Trade-off: user-defined maxis the pixel error tolerance. Setting to 1 results in pure feathering.)

Patch and Image Mask

P with Invalid Pixels (blue)

Error Surface (S)

P OVER Ri-1

Intermediate Result

Patch with Invalid Pixels

- Simple scanline re-synthesis: possibly insufficient causal neighborhood within valid pixelregion
- Solution: alternative ordering (Pixel Traversal Map)

Composited Result

Patch with Invalid Pixels

1: BUILDTRAVERSALMAP(Pi) : Mi

2: level = 1

3: Dlevel ←binary image of size Pi initialized to 0

4: Dlevel ←set all valid pixels ∈ Pi to 1

5: Mi ←Dlevel

6: while (∃pixel ∈ Dlevel ∧ pixel = 0) do

7: level = level+1

8:Dlevel ← DILATE(Dlevel−1,Disk)

9: Mi ←Mi +(Dlevel ∧Dlevel−1) ∗level

10: end while

11: return Mi

- Pixel Traversal Map: repeated morphological dilation of the binary support for valid pixels with a euclidian disk of radius 1 (city-block distance transform).

Composited Result

Patch with Invalid Pixels

Pixel Traversal Map (M)

- Pixel Traversal Map:step 1

Composited Result

Patch with Invalid Pixels

Pixel Traversal Map (M)

- Pixel Traversal Map: step 2 ...

Composited Result

Patch with Invalid Pixels

Pixel Traversal Map (M)

- Pixel Traversal Map: ... step n

Composited Result

Patch with Invalid Pixels

Pixel Traversal Map (M)

- Art Restorer Analogy: stepwise restoration of the hole from the boundary of the existing image.

Composited Result

Patch with Invalid Pixels

Pixel Traversal Map (M)

Synthesize red pixel from valid (or already re-synthesized) pixels

Intermediate Result

Pixel Traversal Map (M)

Extract image mask (Ij) and binary support (Jj) for best-pixel search in the input texture. Complexity: O(n log n) per pixel.

Image Mask (Ij)

Binary Support (Jj)

Intermediate Result

Intermediate Result

Pixel Traversal Map (M)

Intermediate Result

Shifted

Pixel Traversal Map (M)

Intermediate Result

with Traversal Map

Pixel Traversal Map (M)

Overlap Re-synthesis

Step 1

Pixel Traversal Map (M)

Overlap Re-synthesis

Step 2

Pixel Traversal Map (M)

Overlap Re-synthesis

Step 3

Pixel Traversal Map (M)

Overlap Re-synthesis

Result

Pixel Traversal Map (M)

No Repair

IQ

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No Repair

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Input

Efros/Leung

Wei/Levoy

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Input

Efros/Leung

Wei/Levoy

IQ

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Input

Efros/Leung

Wei/Levoy

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Input

Efros/Leung

Wei/Levoy

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Input

Efros/Leung

Wei/Levoy

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Input

Efros/Leung

Wei/Levoy

IQ

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Input

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presence of high frequency structure cause the error metric (L2 Norm)

- Improve Computational Complexity
- Pixel neighborhoods of patch and pixel stage are not known a priori, so precomputation is not straightforward
- We can solve this in the pixel stage by employing k-coherence search [Tong et al. 2002] [Ashikhmin 2001]

- Improve Error Metric
- Still using the L2 norm due to its simplicity
- Develop a metric which takes feature mismatch into account