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Peyman Milanfar

@docmilanfar

Oct 7, 2022 • 8 tweets • 8 min read • Read on X

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@GooglePhotos

Today we announced a new feature on Pixel 7/Pro and @GooglePhotos called "Unblur". It's the culmination of a year of intense work by our amazing teams. Here's a short thread about it

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#MadeByGoogle #fixedonpixel #Pixel7 #PhotoUnblur
@GooglePixel_US

Last yr we brought two new editor functions to Google Photos: Denoise & Sharpen. These could improve the quality of most images that are mildly degraded. With Photo Unblur we raise the stakes in 2 ways:

First, we address blur & noise together w/ a single touch of a button.

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Second, we're addressing much more challenging scenarios where degradations are not so mild. For any photo, new or old, captured on any camera, Photo Unblur identifies and removes significant motion blur, noise, compression artifacts, and mild out-of-focus blur.

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Photo Unblur works to improve the quality of the 𝘄𝗵𝗼𝗹𝗲 photo. And if faces are present in the photo, we make additional, more specific, improvements to faces on top of the whole-image enhancement.

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One of the fun things about Photo Unblur is that you can go back to your older pictures that may have been captured on legacy cameras or older mobile devices, or even scanned from film, and bring them back to life.

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It's also fun to go way back in time (like the 70s and 80s !) and enhance some iconic images like these photos of pioneering computer scientist Margaret Hamilton, and basketball legend Bill Russell.

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@2ptmvd

Recovery from blur & noise is a complex & long-standing problem in computational imaging. With Photo Unblur, we're bringing a practical, easy-to-use solution to a challenging technical problem; right to the palm of your hand
w/ @2ptmvd @navinsarmaphoto @sebarod & many others

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Bonus: Once you have a picture enhanced with #PhotoUnblur, applying other effects on top can have an even more dramatic effect. For instance, here I've also blurred the background and tweaked some color and contrast.

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More from @docmilanfar

Peyman Milanfar

@docmilanfar

Dec 20

Years ago when my wife and I we were planning to buy a home, my dad stunned me with a quick mental calculation of loan payments.

I asked him how - he said he'd learned the strange formula for compound interest from his father, who was a merchant in 19th century Iran.

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The origins of the formula my dad knew is a mystery, but I know it has been used in the bazaar's of Iran (and elsewhere) for as long as anyone can remember

It has an advantage: it's very easy to compute on an abacus. The exact compounding formula is much more complicated

2/4

I figured out how the two formulae relate: the historical formula is the Taylor series of the exact formula around r=0.

But the crazy thing is that the old Persian formula goes back 100s (maybe 1000s) of years before Taylor's, having been passed down for generations

3/4

Read 4 tweets

Peyman Milanfar

@docmilanfar

Dec 8

How are Kernel Smoothing in statistics, Data-Adaptive Filters in image processing, and Attention in Machine Learning related?

My goal is not to argue who should get credit for what, but to show a progression of closely related ideas over time and across neighboring fields.

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In the beginning there was Kernel Regression - a powerful and flexible way to fit an implicit function point-wise to samples. The classic KR is based on interpolation kernels that are a function of the position (x) of the samples and not on the values (y) of the samples.

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Instead of a fixed smoothing parameter h, we can adjusted it dynamically based on the local density of samples near the point of interest. This enables accounting for variations in the spatial distribution of samples, but doesn't take into account of the values of samples

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Read 11 tweets

Peyman Milanfar

@docmilanfar

Dec 3

“On a log-log plot, my grandmother fits on a straight line.”
-Physicist Fritz Houtermans

There's a lot of truth to this. log-log plots are often abused and can be very misleading

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A plot of empirical data can reveal hidden phenomena or scaling. An important and common model is to look for power laws like

p(x) ≃ L(x) xᵃ

where L(x) is slowly varying, so that xᵃ is dominant

Power laws appear all over physics, biology, math, econ. etc., however...

2/5

...just because on a log-log plot your data looks like a line, you can't conclude that you're looking at a power law

In fact, a roughly straight behavior on a log-log scale is like a necessary condition, but it is not sufficient for power-law behavior. Take this example:

3/5

Read 5 tweets

Peyman Milanfar

@docmilanfar

Nov 10

Integral geometry is a beautiful topic bridging geometry, probability & statistics

Say you have a curve with any shape, possibly even self-intersecting. How can you measure its length?

This has many applications - curve could be a strand of DNA or a twisted length of wire

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A curve is a collection of tiny segments. Measure each segment & sum. You can go further: make the segments so small they are essentially points, count the red points

A practical way to do this: drop many lines, or a dense grid, intersecting the shape & count intersections

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Curve's length is the sum of intersections n(ρ,θ) of all lines (in polar coords) with the curve (counting multiplicities). This is the beautiful Crofton formula:

Length = 1/2 ∫∫ n(ψ,p) dψ dp

The 1/2 is there because oriented lines are a double cover of un-oriented lines

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Read 6 tweets

Peyman Milanfar

@docmilanfar

Sep 24

Smoothing splines fit function to data as the sol'n of a regularized least-squares optimization problem.

But it’s also possible to do it in one shot with an unusually shaped kernel (see figure)

Is it possible to solve other optimization problems this way? Surprisingly yes

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This is just one instance of how one can “kernelize” an optimization problem. That is, approximate the solution of an optimization problem in just one-step by constructing and applying a kernel once to the input

Given some conditions you can it do much more generally

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If you specialize the regularization to be of the form
φ(x) = ρ( ||Ax|| ) where A= R(|i-j|) is a stationary & isotropic, this gives tidy conversions between φ(x) and the kernel K(x).

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Read 4 tweets

Peyman Milanfar

@docmilanfar

Sep 18

Mean-shift iteratively moves points towards regions of higher density. It does so by placing a kernel at each data point, calculating the mean of the data points within that window, shifting points towards this mean until convergence: Look familiar?

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(Animation @gabrielpeyre)

The first term on the right hand side of the ODE has the form of a pseudo-linear denoiser f(x) = W(x) x. A weighted average of the points where the weights depend on the data. The overall mean-shift process is a lot like a residual flow:

d/dt x(t) = f(x(t)) - x(t)

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Residual on the RHS is an approximation of the “score” -the gradient of the empirical density of x making it a gradient flow

d/dt x(t) ≈ ∇ log p̂(x(t))

So mean-shift a) estimates the empirical density & b) flows points to nearby peaks. Similarly to flow-matching & InDI

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Read 4 tweets

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