Introduction - inst.eecs.berkeley.edu

Introduction - inst.eecs.berkeley.edu

The Frequency Domain, without tears Somewhere in Cinque Terre, May 2005 CS194: Image Manipulation & Computational Photography Many slides borrowed Alexei Efros, UC Berkeley, Fall 2017 from Steve Seitz

Salvador Dali Gala Contemplating the Mediterranean Sea, which at 30 meters becomes the portrait of Abraham Lincoln, 1976 A nice set of basis Teases away fast vs. slow changes in the image.

This change of basis has a special name Jean Baptiste Joseph Fourier (1768-1830) ...the manner in which the author arrives at had crazy idea (1807): these equations is not exempt of difficulties Any univariate function can and...his analysis to integrate them still leaves something to be desired on the score of

be rewritten as a weighted generality and even rigour. sum of sines and cosines of different frequencies. Dont believe it? Neither did Lagrange,

Laplace, Poisson and other big wigs Not translated into English until 1878! Laplace But its (mostly) true!

called Fourier Series Lagrange Legendre A sum of sines Our building block:

Asin(x Add enough of them to get any signal f(x) you want! How many degrees of freedom? What does each control? Which one encodes the coarse vs. fine structure of

the signal? Fourier Transform We want to understand the frequency of our signal. So, lets reparametrize the signal by instead of x: f(x)

Fourier Transform F() For every from 0 to inf, F() holds the amplitude A and phase of the corresponding sine Asin(x

How can F hold both? F ( ) R( ) iI ( ) 2 2 1 I ( ) A R( ) I ( ) tan

R( ) We can always go back: F() Inverse Fourier Transform

f(x) Time and Frequency example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t) Time and Frequency example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)

= + Frequency Spectra example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t) =

+ Frequency Spectra Usually, frequency is more interesting than the phase Frequency Spectra

= = + Frequency Spectra

= = + Frequency Spectra

= = + Frequency Spectra

= = + Frequency Spectra

= = + Frequency Spectra

1 = A sin(2 kt ) k 1 k Frequency Spectra FT: Just a change of basis

M * f(x) = F() * . . .

= IFT: Just a change of basis M-1 * F() = f(x) * .

. . = Finally: Scary Math Finally: Scary Math

ix not really scary: e cos(x ) i sin(x ) is hiding our old friend: sin(x phase can be encoded by sin/cos pair

P cos( x ) Q sin( x ) A sin( x P 2 Q 2 P tan 1 Q So its just our signal f(x) times sine at frequency

Extension to 2D = Image as a sum of basis images Extension to 2D

in Matlab, check out: imagesc(log(abs(fftshift(fft2(im))))); Fourier analysis in images Intensity Image Fourier Image

http://sharp.bu.edu/~slehar/fourier/fourier.html#filtering Signals can be composed + =

http://sharp.bu.edu/~slehar/fourier/fourier.html#filtering More: http://www.cs.unm.edu/~brayer/vision/fourier.html Man-made Scene Can change spectrum, then reconstruct

Local change in one domain, courses global change in the other Low and High Pass filtering The Convolution Theorem The greatest thing since sliced (banana) bread! The Fourier transform of the convolution of two functions is the product of their Fourier transforms

F[ g h] F[ g ] F[h] The inverse Fourier transform of the product of two Fourier transforms is the convolution of the two inverse Fourier transforms 1 1

1 F [ gh] F [ g ] F [h] Convolution in spatial domain is equivalent to multiplication in frequency domain! 2D convolution theorem example

|F(sx,sy)| f(x,y) * h(x,y) |H(sx,sy)|

g(x,y) |G(sx,sy)| Filtering Why does the Gaussian give a nice smooth image, but the square filter give edgy artifacts?

Gaussian Box filter Fourier Transform pairs Gaussian

Box Filter

Recently Viewed Presentations

  • Introduction to The Giver - Valerie Nafso

    Introduction to The Giver - Valerie Nafso

    My utopia is called Awesomeville. It is very awesome here! It is everything you would ever want. There are big buildings everywhere! Everybody has a huge awesome house with 14 rooms, 15 bathrooms, and in every room of every house...
  • Non native teachers of English - Home - BALEAP

    Non native teachers of English - Home - BALEAP

    Non-native teachers in EAP "Academic language is nobody's mother tongue" (Bourdieu, 1994) Native speaker also 2nd language learner. Easier for NS to learn academic language? (Jenkins, 2016) EAP: teaching language AND academic culture Non-natives as model learners on several levels
  • Muscle Contractions, Fatigue & Names

    Muscle Contractions, Fatigue & Names

    Isotonic contractions "same tone" Myofilaments are able to slide past each other during contractions. The muscle shortens, tension rises. Walking, running, lifting an object. 9/17/2018. SAP2b. Constant tension on muscles as muscle shortens. Muscle changes length. 2 types: Concentric-muscle shortens....
  • Evolving the Web Platform - download.microsoft.com

    Evolving the Web Platform - download.microsoft.com

    Evolving the Web Platform. Jeff Burtoft. Senior Program Manager, Developer Evangelism. @boyofgreen. Click to edit title. Click to edit subtitle
  • 'Who's Afraid Of Virginia Wolf - Yola

    'Who's Afraid Of Virginia Wolf - Yola

    Who's Afraid of Virginia Woolf ? is a play by Edward Albee that opened on Broadway at the Billy Rose Theater on October 13, 1962. Who's Afraid of Virginia Woolf ? won both the 1963 Tony Award for Best Play...
  • Chapter 4 Structuring the Interview McGraw-Hill/Irwin Copyright  2011

    Chapter 4 Structuring the Interview McGraw-Hill/Irwin Copyright 2011

    The Body of the Interview Question Sequences Combination Sequences: begins with open questions, proceeds to one or more closed questions, and ends with open questions The Body of the Interview Question Sequences The Diamond Sequence: enables interviewers to begin with...
  • CACFP Pre-K Meal Pattern - Child Nutrition

    CACFP Pre-K Meal Pattern - Child Nutrition

    CACFP Pre-K Meal Patterns. The recently updated CACFP meal patterns ensure children have access to healthy, balanced meals throughout the day. While the new meal patterns are a significant change for day cares and other child-care institutions, as we said,...
  • MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY

    MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY

    MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT'S RESISTANCE TO ROTATION IT IS RESISTANCE OF BENDING MOMENT OF A BEAM IT IS THE...