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# Super-Resolution

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Super-Resolution

Barak Zackay
Yaron Kassner
Outline
• Introduction to Super-Resolution
• Reconstruction Based Super Resolution
– An Algorithm
– Limits on Reconstruction Based Super Resolution
• Example Based Super Resolution
– Halucination
– Example Based
– Single Image Super Resolution
• Summary
Introduction to
Super Resolution
Definition of the Problem
• Super-resolution is the process of combining
multiple low resolution images to form a higher
resolution one.
• No cheating!
– Resulting image should represent reality better than
all the input images.
The Imaging Process
Physical Properties
• Each camera suffers from some inherent optical
issues:
– Finite size of the aperture - generates blur, modeled
– Noise
Mathematical Model
• Each pixel in the resulting image is given by:

• Loi(m) – the i-th LR image in
pixel m.
• Ei (x) – total photon count
from the direction x
• PSFi – Point Spread Function
Deresolution
• Given HR image
• Project to LR image
• Each LR pixel is a
linear combination         LR
of HR pixels

HR    HR   HR
Reconstruction-based
Super Resolution
• Reconstruct hidden HR pixels out of
known linear combinations.

LR         LR

HR   HR   HR    HR
LR    HR    HR
LR    HR
Example-Based
Super Resolution
• Use prior knowledge to reconstruct a HR
image.

Prior Knowledge of faces
Reconstruction Based
Super Resolution
from
Improving Resolution by Image Registration

Michal Irani and Shmuel Peleg
Basic Idea
• The HR image should create the LR
images when deresoluted.
Notation
•    gk :     The kth observed LR image.
•    f  n : The approximation to the HR image after n
iterations.
•        
g : The LR image obtained by applying the
k
n

simulated imaging process to f n.
•   h PSF : The point spread function of the imaging blur.
•    x : a HR pixel

•    y : a LR pixel influenced by x

•   z y : The center of the receptive field of y.
Problem Formulation
• Find a HR image   f n    , that gives   g n   g   .

–
Algorithm Overview
• Register the LR images.
• Guess the HR image f   .      0
n

• Iteration n:
n 
– Simulate the imaging process to create                gk

from f n .
 
– Compare         gk
n
and g .
k

– Correct      f  n  in the direction of the error.

• output    f n 
Registration
LR        LR
LR         LR

LR        LR
LR         LR

LR          LR

HR   HR       HR
HR LR        HR
HR LR      HR
Iteration
• Take the current guess.
n 
• Decrease its resolution to get g k
• Update each HR pixel x according to the error in all LR pixels (y) it
influences.c is a constant normalizing factor.

– c is a constant normalizing factor.
– Yk,x is the group of all pixels y that are influenced by x.

– hxy BP is a back-projection kernel applied on x  z y that represents the
way the HR pixel x should be updated from y.
Wasach
One of three input images

Initial guess (average of input images)

Output
Debluring

Original Image     Blurred Image   Restored Image
Wasach

Blurred Image    Initial Guess   Restored Image
Limits on Reconstruction
Based Methods
from
Limits on Super-Resolution and How to
Break Them
Large Magnification Factor is
Problematic
• Large magnification factor causes:
– Overly smooth HR image
– Fine details are not recovered

• An explanation is needed.
Evil Example
• Suppose we want to increase the
resolution by exactly M=2.
• Lets look on a checkboard like scene,
where each pixel is either white or black.

LR

HR   HR   HR
Information is Inherently Missing
• The resulting image would be grey
independently from the offset of the LR
image!
• Conclusion: some information is inherently
missing on our LR images!
When M is not an Integer

LR

HR   HR   HR
Limits of Super-Resolution
• Size of LR images: N pixels.
• Size of HR image: NM 2 pixels.
• Each HR pixel can be added noise of amplitude smaller
than M 2 which wont change the LR image!
• Volume of possible HR solutions: O(M 2N) 1
• It can be shown that under practical considerations the
effective magnification factor (M) is bounded by 1.6, no
matter how many LR images are taken2.

1 Limits on Super-Resolution and How to Break Them, Simon Baker and Takeo Kanade
2 Fundamental Limits of Reconstruction-Based Superresolution Algorithms under Local Translation,
Zhouchen Lin, and Heung-Yeung Shum
Break
• Introduction to Super-Resolution
• Reconstruction Based Super Resolution
– An Algorithm
– Limits on Reconstruction Based Super Resolution
• Example Based Super Resolution
– Halucination
– Example Based
– Single Image Super Resolution
• Summary
Example Based
Super Resolution
Introduction to Example-Based
Super Resolution
• Reconstruction constraints are not
enough.
• One has to use prior knowledge of the
image to break the reconstruction limits.
• The following algorithms will use priors
learned from databases of example
images.
Recogstruction or
Hallucination
from
Limits on Super-Resolution and How to Break Them

General Idea
• Find a HR image Su that satisfies two
kinds of constraints:
– Reconstruction constraints: When projected to
the LR dimensions, the image is similar to the
observed input images.
– Recognition constraints: The pixels of Su
should resemble pixels from images in the DB
that where found to have similar features to
the observed LR images’ features.
MAP formulation
• To solve the problem, given the LR
images, we need to find the HR image
that maximizes

- Su: the HR image    Reconstruction Constraints
- Lo: the LR images
Recognition Constraints
Reconstruction Constraints
• The probability of the LR images given the HR
image can be computed from the distance
between the deresoluted HR image and the LR
images.

–        : the noise variance
– ri z : The pixel in Lo that corresponds to pixel z in Su.
– m: a LR pixel index
Recognition: LR features
• We use “Parent Structures” to describe LR
features.
Recognition: Choosing the
Pixels from the DB

PS = Parent Structure
F = Features – like First deriviative, or Laplacian
Formulation of Recognition
Constraints
• Instead of estimating the probability of the
HR image, Su, we estimate its probability
given each pixel’s “recognition”.

H0 – Horizontal derivative
V0 – Vertical derivative.
- Variance of the recognition errors.
T - the training images.
BI – best images for the pixels of the LR images.
BP – best pixel indices in the best images for the pixels of the LR images.
Ci,BP,BI – Class of all images that would have the Best corresponding Images BI,
and the Best corresponding Pixels BP in the db.
- The function that fits a LR pixel index to the corresponding HR pixel index.
2k – the ratio between the HR image scale and the LR image scale.
Maximization

• Note that the function we need to
maximize is quadratic with the HR image’s
pixels.

Algorithm Summary
• Preliminary work:
– Take a training set of images.
– Build a DB that matches parent structures to HR
pixels.
• Compute the reconstruction constraints.
• For each LR image:
– For each HR pixel index:
• Search for the corresponding parent structure in the DB.
• Find the HR image that fits best both the
reconstruction constraints and the HR pixels
extracted from the database.
Comparison
Comparison
Best and Worst Image
Noise Effect
Image Size
Hallucination
Hallucination
Results on Text
Example Based
Super Resolution

William T. Freeman, Thouis R. Jones
and Egon C. Pasztor
Algorithm Overview
• Construct a DB of matching LR-HR patches

• Algorithmically find the most coherent patch
assignment to generate a good image
Constructing the DB
• Given a DB of images
• Make a table from LR patches to HR patches.
• Each image in the DB is treated as follows:
– Take each 7x7 patch from the image and deresolute
into a 5x5 patch
– Normalize the 5x5 patches to have the same mean
and relative contrast.
– Arrange the DB by the low frequencies of the LR
patches
Local Patch Matching
• Match a LR patch to a HR patch from the DB,
using low frequencies.
• Get an estimation to the unknown (high)
frequencies, based on the match.
• Remaining problem: match between neighboring
overlapping patches.
Global Patch Matching
• Run over patches from left to right and from top
to bottom
• Match each patch its nearest neighbor in the DB
using the predetermined patches as additional
constraints.
Algorithm
Wasach
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Wasach

Cubic-spline   Super-resolution   True high-resolution image
Wasach
Complete Failure
Priors are definitely used!
Super Resolution From a
Single Image

Daniel Glaser, Shai Bagon and
Michal Irani
Patch Redundancy in a Single
Image
Employing in-scale Patch
Redundancy
Employing Cross-scale Patch
Redundancy
• Build a cascade of decreasing resolution
images from the LR image.
• For each patch in the LR image, search for
its Nearest Neighbour in the even lower
resolution image.
• Take the found neighbour’s parent in the
original LR image and copy it to be the HR
image.
Combining the Methods
Wasach

Bicubic interpolation   Unified single-image SR (x3)   Ground truth image
Wasach

Bicubic interpolation       Unified single-image SR (x3)
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Summary
• We have presented two basic approaches for super
resolution:
– Reconstruction-based – which simply tries to reverse the
imaging process
– Example-based – which uses example images to reconstruct the
original image.
• We have shown that there are limits to reconstruction
based methods, which are independent of the number of
LR images we use.
• We have presented an algorithm that combines both
approaches to achieve SR from a single image.
Questions?

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