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Image Super-resolution Using Statistical Learning

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Image Super-resolutionUsing Statistical Learning

Preliminary Exam Presentation

Karl Ni

Professor Truong Nguyen

Professor Nuno Vasconceles

Professor William Hodgkiss

- Problem Description: Image super-resolution
- Background Information
- Non-statistical Techniques
- Statistical Techniques
- Classification-based Approaches

- Contributions
- Regression-based Approach and Rationale
- Spatial Domain SVM Superresolution
- Frequency Domain SVM Superresolution
- Results

- Conclusion

- Low-resolution image transferred to high resolution
- Addition of pixels
- Single-frame image superresolution is a way to fill in the missing information for a larger image, specifically what values these pixels take on.

Low Freq

Coefficients

Zero-pad

Low Freq

Coefficients

All Zeros

Frequency Domain

- B-Splines Methods
- Bilinear
- Bicubic

- Cosine Domain Upscaling: Zero Padding.

- Instead of blindly guessing or filling in information, we can use prior knowledge included in a training set.
- Nearest Neighbor: Freeman, Jones, Pasztor
- Expectation Maximization: Atkins and Bouman

- Image estimation lowers MSE

- There’s some relationship between every even and every odd components, just as there is some relationship between high frequencies and low frequencies.

Knowledge Base

Decision

Operations

Informed Decision

Observations

- Prior knowledge of the values and locations of the missing information.
- Exploit this knowledge as a relationship between known and unknown

- Would like pattern recognition for application of knowledge base to observation data
- Call input random variables X, the data.
- Call associated labels for input random variables Y.
- Have pairs:
{ (x1, y1), (x2, y2), …, (xN, yN), … }

- Two types of variables:
- X : vector of observations
(features) in the world

- Y : state (class) of the
world

- X : vector of observations
- X, Y are related by a
f: function (unknown)

Knowledge Base:

{ (x1, y1), (x2, y2), …, (xN, yN)}

Decision

Operations based on

Cost Function

h(xobs)

Informed Decision = ydec

Observations = xobs

- Goal: Make h(x) = f(x) given training data as knowledge base

Pixel Locations

- What is the feature set x?
- Can be a single pixel
- Can be a vector of all the pixels
- Can be linear transformation of pixels
- Can be kernelized transformation of pixel values
- Can be anything! (reasonable)

- What is the h(x) that we are trying to learn?
- Can be filter coefficients with input x = original pixels
- Can be actual pixel values
- Can be anything! (reasonable)

C.B. Atkins, et. al., 1998, “Classification Based Methods in

Optimal Image Interpolation”, PhD Dissertation, Purdue

University, West Lafayette, IN, USA

- We wish to learn the relationship of f(x) = y, and model it the best we can with h(x)
- “0-1” loss function:
0, if y = h(x, α)

1, if y ≠ h(x, α)

- Minimize the risk, defined as expected loss:
R(α) = EX,Y{ L[ y, h(x, α)] }

= ∫ PX,Y (x, y) L[y, h(x, α)] dx dy

= 0 • PX,Y[y= h(x, α)] + 1 • PX,Y[y ≠ h(x, α)]

= PX,Y[y ≠ h(x, α)]

L[ y, h(x, α)] =

- What function h(x, α) minimizes the risk?
h* = argminhR (α, x)

= argminhEX,Y{ L[ y, h(x, α)] }

= argminh PX,Y[y ≠ h(x)]

h*(x) = argminh PY|X[y ≠ h(x) | x]

= argminh 1 – PY|X[y = h(x) | x ]

= argmaxhPY|X[ h(x) | x ]

= argmaxiPY|X[ i | x ]

- In other words, the optimal value of h(x) = i, given an observation x, is the value which maximizes the posterior PY|X(i | x)

- Determine the value i that maximizes PY|X(i | x)
- All methods must assume a model
- Two Different Philosophies:
- Generative Method
- Model p(x,y) and use Bayesian rules to calculate p(y|x).
- Possibly biased due to the unknown assumptions

- Discriminant Method
- Model p(y|x) directly and map accordingly.
- Possibly highly variable with few data points
- Relatively new field in the past decade, universally applied

- Generative Method

- Underlying concept is to use discriminant functions to estimate a boundary or regression in feature space.
- Use a hyperplane
to create boundary

of classes:

wTx + b = 0.

- A correct decision
is given by

y•g(x) = y•(wTx + b)

> 0 ?

wTx + b = 0 divides hyperspace into two subspaces.

Distance to origin is

b/||w||, where ||w|| is the norm of w.

Distance to closest point is:

wTxi + b

γ = mini

||w||

wTxi + b

γ = mini

||w||

- Recall the minimum distance to nearest point is:
- This is called the margin.
- It is natural that we would wish to maximize the margin. (Maximize this minimum distance.)
- The SVM classifier that maximizes the margin under some normalization is

- Perhaps data is not well behaved. Not all data can be separated.
- Introduce an extra “slack variable”.

- Classification has a tendency of “discretizing” our output
- Can think of regression as kind of like a continuous version of classification, which would need infinite number of classes
- Will approximate the function that given the known image information
- Function is the relationship between known and unknown elements

- Soft Margin SVMs for Classification:
Wish to minimize { ||w||2 + C Σξi } subject to

yi (wTxi + b) ≥ 1 – ξi

ξi ≥ 0, for all i

- Soft Margin SVMs for Regression:
Wish to minimize { ||w||2 + C Σ( ξi+ + ξi- ) } subject to

-ξi- ≤ | yi - (wTxi + b) | – ε ≤ ξi+

ξi-, ξi+ ≥ 0

- Rearranging, the Lagrangian can be written:
L(w,b,ξ+,ξ-) = wTw + Σαi-((wTxi+b)-yi-ε+ξi-)

+ Σαi+(yi-(wTxi+b)i-ε+ξi+) + Σri-ξi- + Σri+ξi+

- The optimization is, for an ε and C chosen a priori:
max W(α+, α-) = -εΣ (α+ + α-) + Σi (αi+ - αi-)yi - ½ Σi Σk (αi+ - αi-) (αk+ - αk-) k ( xi ,xk)

subject to

0 ≤ α+, α-≤ C, Σi (αi+ - αi-) = 0

- The regression estimate will then have the form of:
f(x) = Σi (α+ - α-) k (x, xi) + b

- Unsupervised learning using SVM
- Application of Support Vector Regression to Superresolution Problem
- Direct Application (i.e. pixel prediction)
- Indirect Application (i.e. filter coefficients)

- Use additional equations to add structure and better results.

FILTER COEFFICIENTS

c11

c12

c13

c21

c22

c23

c31

c32

c33

- Regression to find spatial domain filters
- Direct regression actually works better

f

Regression

- DCT Domain for statistical purposes (regression)
- A downsampled version of an image is all the even samples.
- From the DCT-II of a downsampled image, can we reconstruct the DCT-II of the original image?

K. R. Rao, P. Yip 1988, “Discrete Cosine Transform: Algorithms, Advantages, Applications”, San Diego, CA, USA: Academic Press

- We can rewrite decimation in time for DCT as a linear combination of the time/spatial domain terms corresponding to even and odd samples
- Decimation in Time
DCT-II(x) = k(m) (DCT-I(xeven) + DCT-I(xodd) + DCT-II(xeven) + DCT-II(xodd))

- Decimation in Time
DCT-II(x) = k(m) { X1 + X2 + X3 + X4 }

- Overall Idea
DCT-II(x) = Input Signal + f (Input Signal) + g (Remaining Terms),

f is exactly known and g is to be estimated

- Decimation in Time
- DCT-II(X2N) = DCT-II(XN) + Known + Estimated.

- DCT-II(X2N) = DCT-II(XN)+Known+Estimated.
- Given DCT-II(XN), can we determine the estimated terms that will give us DCT-II(X2N)?
- Our regression is thus,
Estimated= Σi (αi+ – αi-) K ( xi, DCT-II[XN] ) + b

- This is done for all lower coefficients.
- LibSVM & LSSVM, Regression package

Bilinear Filtering

SVR Spatial Filtering

SVR Filtering

Bilinear

Small training set (10 frames)

Bilinear Interpolation

SVR Frequency Regression

Small training set (10 frames)

PSNR Values

Method

PSNR

Bilinear

23.301

Bicubic

22.209

Spatial

25.995

Frequency

26.843

Small training set (10 frames)

Bilinear Interpolation

SVR Frequency Regression

4 x 4 features

8 x 8 features

Structured Frequency Regression

- Direct Frequency Regression

- Apply superresolution to the error residual in video
- Denoising algorithms using Support Vector Regression
- Markov Random Fields, or some interrelationship between predicted values
- Motion Prediction Values