Filter Design

052600 VU Signal and Image Processing

Filter Design

Torsten Möller + Jana Kemnitz + Raphael Sahann

[email protected] [email protected] [email protected] vda.cs.univie.ac.at/Teaching/SIP/18w/

1 Sources

• Oppenheim&Shafer — Chapter 7

• Fast Fourier transform (FFT) is an algorithm to implement DFT

• Most commonly used FFT is the Cooley–Tukey algorithm

2 • Number of multiplications are reduced from O ( N ) to O ( N l o g 2 (N) with N sampling points (data size)

• Basic idea is recursive -> divide the transform into two pieces of size N/2 at each step

• Widely used for many applications in engineering, science, and mathematics ©© TorstenJana Kemnitz Möller 3 FFT - Key Points

• Basic idea is recursive - > divide the transform into two pieces of size N/2 at each step

©© TorstenJana Kemnitz Möller 4 Overview

• Motivation

• What is a ﬁlter?

• Filter in image processing

• Anti-Aliasing Filter

• Realistic ﬁlters

– John von Neumann, 1979, The Dancing Wu Li Masters: An Overview of the New Physics (1979) by Gary Zukav, Bantam Books, p. 208, footnote. (https://en.wikiquote.org/wiki/John_von_Neumann)

6 What is a ﬁlter?

7 Filter - some Examples

In signal processing, a ﬁlter is a device or process which can be described by a mathematical function that modiﬁes an input signal x(t) or x[n] into an output signal y[t] or y[n]

x[n] Filter y[n]

y(t) x(t)

Optical ﬁlter (zone plate) x[n] Y[n]

^ f(x,y) f(x,y)

©© TorstenJana Kemnitz Möller 8 Filter - Image Processing

• Filtering is a technique for modifying or enhancing an image.

• For example, an image can be ﬁltered to emphasize certain features or remove other features.

• Image processing operations implemented with ﬁltering include smoothing, sharpening, and edge enhancement.

• In image processing, a kernel, convolution matrix, or mask is a small matrix. It is used for blurring, sharpening, embossing, edge detection, and more. This is accomplished by doing a convolution between a kernel and an image.

©© TorstenJana Kemnitz Möller 9 Filter - Image Processing

©© TorstenJana Kemnitz Möller 10 Filter - Image Processing

• Identity

• Edge detection

• Sharpen

• Gaussian blur

©© TorstenJana Kemnitz Möller 11 Linear Spatial Filtering

Gonzalez & Woods - Digital Image Processing (3rd Edition)

©© Raphael Torsten MöllerSahann 12 Smoothing Filter

Gonzalez & Woods - Digital Image Processing (3rd Edition)

©© Raphael Torsten MöllerSahann 13 Smoothing Filter

Gonzalez & Woods - Digital Image Processing (3rd Edition)

©© Raphael Torsten MöllerSahann 14 Smoothing Filter - Application

Gonzalez & Woods - Digital Image Processing (3rd Edition)

©© Raphael Torsten MöllerSahann 15 Instagram Filters

17 Use of ﬁlters

• histogram equalization • denoising • getting rid of artifacts (notch ﬁltering) • deconvolution • re-sampling / anti-aliasing

Gonzalez & Woods - Digital Image Processing (3rd Edition) ©© Raphael Torsten MöllerSahann 19 Noise Filtering

Gonzalez & Woods - Digital Image Processing (3rd Edition)

g(x, y)=h(x, y) ? f(x, y)+⌘(x, y)

G(u, v)=H(u, v)F (u, v)+N(u, v)

©© Raphael Torsten Möller Sahann 22 Estimating the Degradation Function

Gonzalez & Woods - Digital Image Processing (3rd Edition)

g(x, y)=h(x, y) ? f(x, y)+⌘(x, y) (Time domain)

G(u, v)=H(u, v)F (u, v)+N(u, v) (Frequency domain)

©© Raphael Torsten Möller Sahann 23 Inverse Filtering (Deconvolution)

G(u, v) Fˆ(u, v)= H(u, v)

• simple estimate of the transform by dividing the transform by the degradation function • cannot recover undegraded image fully, because N(u,v) is not known — very small values of H(u,v) will dominate the estimate N(u, v) Fˆ(u, v)=F (u, v)+ H(u, v)

©© Raphael Torsten Möller Sahann 24 Estimation by Experimentation

Gonzalez & Woods - Digital Image Processing (3rd Edition)

©© Raphael Torsten Möller Sahann 25 Gonzalez & Woods - Digital Image Processing (3rd Edition)

27 General Process of Sampling and Reconstruction Original function Sampled function Acquisition

e.g., supersampling Reconstructed Reconstruction Function Re-sampled function Resampling

e.g., resample at screen resolution © Torsten Möller 28 Sampling in frequency space F

y ω F

ω x F

ω © Torsten Möller 29 How so? – by Convolution Theorem

Spatial Domain: Frequency Domain:

Mathematically: • Convolution: f(x)*h(x) • Multiplication: ∞ ∫ f (t)⋅ h(x − t)dt −∞ Evaluated at discrete

points© Torsten(sum) Möller 30

€ Reconstruction? — Convolution Theorem Spatial Domain: Frequency Domain:

ω Convolution: Multiplication:

• A signal can be reconstructed from its samples without loss of information if the original signal has no frequencies above 1/2 of the sampling frequency • For a given bandlimited function, the rate at which it must be sampled (to have perfect reconstruction) is called the Nyquist frequency • Due to Claude Shannon (1949)

2D 1D Given Given

Needed Needed

Pre-filter

Post-aliasing

Pre-aliasingsampling Reconstruction filter

Original function Sampled function Acquisition

Reconstructed Reconstruction Function Re-sampled function Resampling

Spatial domain Frequency domain

sampling x *

x smoothing *

Spatial domain Frequency domain

smoothing * x

x Re-sampling *

Spatial domain Frequency domain

* x

• Non-bandlimited signal – prealiasing sampling

• Low sampling rate (<= Nyquist) – prealiasing

sampling

• Non perfect reconstruction – post-aliasing

• Zone plate: sin(x2 + y2)

• Sampled at 128 x 128 and reconstructed to 512 x 512 using windowed sinc

• Left rings: part of the signal • Right rings: aliasing due to undersampling

Original function Band-limited function Pre-Filtering

Sampled Acquisition Function Reconstructed function Reconstruction

• Increasing the sampling rate moves each copy of the spectra further apart, potentially reducing the overlap and thus aliasing • Low-pass ﬁlter and then the resulting signal is re-sampled at image resolution

45 Ideal Reconstruction

• Box ﬁlter in frequency domain = Pass-band stop-band • Sinc Filter in spatial domain • Sinc has inﬁnite extent – not practical Ideal filter Smoothing Practical filter Post-aliasing

• Use the sinc function – to bandlimit the sampled signal and remove all copies of the spectra introduced by sampling • But how? • ﬁlter in the frequency domain! • But how expensive? • Frequency domain: O(n.logn) • Spatial domain: O(n.k)

• Use the sinc function – to bandlimit the sampled signal and remove all copies of the spectra introduced by sampling • But: • The sinc has infinite extent (i.e. IIR = infinite impulse response) and we must use simpler filters with finite extents. • The windowed versions of sinc may introduce ringing artifacts which are perceptually objectionable.

• Nearest Neighbor (Box) • Linear

• Sinc

• Gaussian • Many others

Nearest neighbor Linear Interpolation

• Also have ringing in pass/stop bands • Realizable ﬁlters do not have sharp transitions

π π π Low-pass filter band-pass filter high-pass filter

53 Filter Speciﬁcations

0.99 ≤ Heff (jΩ) = 1.01 0 ≤ Ω ≤ 2π(2000)

Heff (jΩ) ≤ 0.001 2π(3000)≤ Ω

δ1 = 0.01

δ2 = 0.001

Ωp = 2π(2000)

Ωs = 2π(3000)

Adapted ©from Torsten 2005 Möller Güner Arslan 54 Butterworth Lowpass Filters •The Butterworth filter designed to have as flat frequency response as possible (no ripples) in the pass-band and zero roll off response in the stop-band •One of the most commonly used digital filters •Fast and simple to use •Since they are frequency-based, the effect of filtering can be easily understood and predicted

Main disadvantage: expense of a wide transition band as the ﬁlter changes from the pass band to the stop band

• Passband is designed to be maximally ﬂat • The magnitude-squared function is of the form

2 1 Hc (jΩ) = 2N 1 + (jΩ / jΩc )

2 1 Hc (s) = 2N 1 + (s / jΩc )

1 / 2N (jπ / 2N)(2k +N−1) sk = (− 1) (jΩc )= Ωce for k = 0,1,...,2N - 1

• You should know typical image ﬁlter kernels (edge detection, sharpen, smoothing)

• You should understanding the process: estimating the Degradation Function

• You should understand how ﬁltering is used to avoid aliasing

• You should understand the differences between an “ideal” and a “realistic” ﬁlter