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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
by the Point-Spread-Function (PSF).
– 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 of HR pixels
HR HR HR
LR
Reconstruction-based Super Resolution
• Reconstruct hidden HR pixels out of known linear combinations.
HR HR HR HRHR HR HR HR HR HR HR
LRLR
LRLR
LR
LR
LR
LR
LRLR
LRLR
LR
LR
LR
LR
Example-BasedSuper Resolution
• Use prior knowledge to reconstruct a HR image.
Prior Knowledge of faces
Reconstruction Based Super Resolution
fromImproving Resolution by Image Registration
Michal Irani and Shmuel Peleg
Basic Idea
• The HR image should create the LR images when deresoluted.
Notation
• : The kth observed LR image.• : The approximation to the HR image after n
iterations.• : The LR image obtained by applying the
simulated imaging process to .• : The point spread function of the imaging blur.• : a HR pixel• : a LR pixel influenced by x• : The center of the receptive field of y.
nkg
nf
kg
PSFh
x
y
nf
yz
Problem Formulation
• Find a HR image , that gives .
–
nf gg n
Algorithm Overview
• Register the LR images.
• Guess the HR image .
• Iteration n:– Simulate the imaging process to create
from .– Compare and . – Correct in the direction of the error.
• output
nf0
nkg
nf
nkg kg
nf
nf
Registration
HR HR HR HRHR HR HR HR HR HR HR
LRLR
LRLR
LRLR
LRLR
LRLR
LRLR
LRLR
LRLR
LRLR
LRLR
LRLR
LRLR
Iteration• Take the current guess.• Decrease its resolution to get • 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.
– is a back-projection kernel applied on that represents the way the HR pixel x should be updated from y.
BPxyh yzx
nkg
Wasach
One of three input images
Initial guess (average of input images)
Output
Debluring
Original Image Blurred Image Restored Image
Wasach
Initial GuessBlurred Image Restored Image
Limits on Reconstruction Based Methods
fromLimits on Super-Resolution and How to
Break ThemSimon Baker and Takeo Kanade
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.
HR HR HR
LR
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
HR HR HR
LR
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
fromLimits on Super-Resolution and How to Break Them
Simon Baker and Takeo Kanade
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
- Lo: the LR images
Reconstruction Constraints
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– PSF: Point Spread Function– : The pixel in Lo that corresponds to pixel z in Su.– m: a LR pixel index zri
Recognition: LR features
• We use “Parent Structures” to describe LR features.
Recognition: Choosing the Pixels from the DB
PS = Parent StructureF = 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 derivativeV0 – 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.
• Do gradient descent.
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
Wasach
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
Unified single-image SR (x3)Bicubic interpolation
Wasach
Wasach
Wasach
Wasach
Wasach
Wasach
Wasach
Wasach
Wasach
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?