1. Nonparametric Modeling of Textures Outline Parametric vs. nonparametric Image patches and similarity distance Efros-Leung’s texture synthesis by non- parametric sampling Next week Application into image inpainting Application into image quilting Demos and discussions

2. A Simple Example of Nonparametric Model Class A: blue square, Class B: red triangle

3. What if we use parametric models? N(m 1,C 1 ) N(m 2,C 2 )

4. Why Nonparametric? Nonparametric = “Distribution Free” E.g., we might assume that X 1,X 2,…,X n are independent identically distributed (iid) but we do not know its specific distribution – this is particularly useful for handling data in high-dimensional space Advantage: the resulting inferential statements are relatively more robust than those from parametric models Disadvantage: limited application because it is difficult, and often impossible to build into the model more sophisticated structures based on our scientific knowledge (i.e., purely data-driven)

5. Examples Regression analysis: predict the stock market value based on the history Parametric regression: use AR model to fit the observation data Nonparametric regression: use heuristics – e.g., if the value of stock A increases, then the value of stock B is likely to increase (or decrease) Texture synthesis: Parametric: two images will look similar if they have similar first-order/second-order statistics Nonparametric: two images will look similar if they form similar “clouds” in high-dimensional patch space

6. Nonparametric Sampling in Natural Language I took a walk in town one day And met a cat along the way. What do you think that cat did say? Meow, Meow, Meow I took a walk in town one day And met a pig along the way. What do you think that pig did say? Oink, Oink, Oink I took a walk in town one day And met a cow along the way. What do you think that cow did say? Moo, Moo, Moo - cited from “Wee Sing for Baby”

7. Efros-Leung’ Scheme (1999) Image patches Look at a group of pixels instead of individual one Similarity distance Are two patches visually similar? Scanning order Which pixel to synthesize first? Nonparametric sampling

8. Image Patches For the convenience of implementation, patches are often taken as square blocks (overlapping is allowed)

9. Similarity Distance MSE metric Weighted MSE 2D Gaussian kernel

10. Scanning Order Colored regions denote where synthesis is needed Onion-peel scanning

11. Putting Things Together ? 1. Form an inquiry patch 2. Find best matched patches 3. Obtain the histogram of center pixels in all matched patches 4. The ? intensity value is given by sampling the empirical distribution

12. Pseudo-Code Implementation http://graphics.cs.cmu.edu/people/efros/research/NPS/alg.html

13. Image Examples

14. Image Examples (Con’d) http://graphics.cs.cmu.edu/people/efros/research/EfrosLeung.html More examples can be found at

15. Extensions Similarity metric Cosine distance = normalized Euclidean distance A B

16. Extensions (Con’t) A B Sim(A,B) is large but Sim(A,fliplr(B)) is small

17. Scientific Puzzle Behind

18. Photoreceptors cones rods

19. Receptive Fields

20. Direction Selectivity