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Nonparametric Modeling of Images
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
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A Simple Example of Nonparametric Model
Class A: blue square, Class B: red triangle
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Images Live on Manifold
?
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What if we use parametric models?
N(m1,C1) N(m2,C2) Difficult for modeling manifold
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Why Nonparametric? Nonparametric = “Distribution Free”
E.g., we might assume that X1,X2,…,Xn 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)
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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
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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”
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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
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Image Patches For the convenience of implementation, patches are
often taken as square blocks (overlapping is allowed)
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Similarity Distance MSE metric Weighted MSE 2D Gaussian kernel
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Scanning Order Onion-peel scanning
Colored regions denote where synthesis is needed
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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
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Pseudo-Code Implementation
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Image Examples
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Image Examples (Con’d)
More examples can be found at
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Extensions Similarity metric
Cosine distance = normalized Euclidean distance A B
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Extensions (Con’t) B A Sim(A,B) is large but Sim(A,fliplr(B)) is small
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Technical Issue: NN search
Nonparametric sampling heavily rely on the search of data points within -ball in the patch space (or NN/kNN search) Technology before 2000 could not handle such task – conceptually simple but computationally formidable (again due to the curse of dimensionality) Belong to algorithm complexity and computational geometry
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Kd-trees* The kd-tree is a powerful data structure that is based on recursively subdividing a set of points with alternating axis-aligned hyperplanes. The classical kd-tree uses O(dn lgn) precomputation time, O(dn) space and answers queries in time logarithmic in n, but exponential in d. l1 4 7 6 5 1 3 2 9 8 10 11 l5 l1 l9 l6 l3 l10 l7 l4 l8 l2 l2 l3 l4 l5 l7 l6 l8 2 5 4 11 l10 8 l9 1 3 9 10 6 7
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Kd-trees. Construction
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Kd-trees. Query l1 l2 l3 l4 l5 l7 l6 l8 2 5 4 11 l10 8 l9 1 3 9 10 6 7
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Algorithm Presentation
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Scientific Puzzle Behind
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Scientific issues: Locality Revisited
How do we define local neighborhood? If the distance between two patches is defined by their photometric similarity, two “close” points in the patch space could be geometrically distant from each other "Space and time are not conditions in which we live; they are simply modes in which we think.“ – Albert Einstein
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A Short Tour of Neuroscience
Despite all the challenges facing image processing and computer vision, human vision system (HVS) provides a concrete example of beating the curse of dimensionality What is unknown is how HVS did it – i.e., the underlying organizational principle of neurons (best known is so-called Hebbian learning rule)
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Photoreceptors rods cones
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Receptive Fields
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Direction Selectivity
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What do we mean by local? Geometry vs. Topology
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Mountcastle’s Discovery
The theory posits that the remarkably uniform physical arrangement of cortical tissue reflects a single principle of complexity management which underlies all cortical information processing. Columnar organization of directional neurons
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