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.

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Super-Resolution

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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

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.

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

LR HR HR HR Deresolution • Given HR image • Project to LR image • Each LR pixel is a linear combination of HR pixels

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

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.

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

LR LR LR LR LR LR LR LR LR LR LR LR LR LR LR LR LR LR LR LR LR LR LR LR HR HR HR HR HR HR HR HR HR HR HR Registration

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.

Wasach One of three input images Initial guess (average of input images) Output

Debluring Original Image Blurred Image Restored Image

Large Magnification Factor is Problematic • Large magnification factor causes: • Overly smooth HR image • Fine details are not recovered • An explanation is needed.

LR HR HR HR 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.

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!

Limits of Super-Resolution • Size of LR images: N pixels. • Size of HR image: NM 2pixels. • Each HR pixel can be added noise of amplitude smaller than M 2which 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

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.

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

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. • 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.

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