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EECS 373Design of Microprocessor-Based Systems

Farhan HormasjiMatthew DiffenderferUniversity of Michigan

Efficient AlgorithmsNovember 27, 2012

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Outline

• Overview

• DFT algorithms– FFT– Cooley–Tukey FFT– Bluestein's FFT– Goertzel

• Edge Detection Algorithms– Canny– Sobel– Prewitt– Roberts Cross

• Questions

Algorithms

What to look for:• Usefulness• Methodology• Implementation

Tradeoffs:• Cost• Code Size• Runtime• Availability

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Outline

• Overview

• DFT algorithms– FFT– Cooley–Tukey FFT– Bluestein's FFT– Goertzel

• Edge Detection Algorithms– Canny– Sobel– Prewitt– Roberts Cross

• Questions

Discrete Fourier Transform (DFT): Overview

• What?– Converts a sampled function from time domain to frequency domain

• Use?– DFTs reveal periodicities in input data as well as the relative strengths of any periodic components

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Discrete Fourier Transform (DFT): Mathematical Interpretation

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Discrete Fourier Transform (DFT): Applications

• Spectral Analysis– Most signals are sinusoidal– DFT tells you frequency, phase, and amplitude components

• Frequency Response of Systems– All input and output signals can be represented as cosine waves

– By observing change in magnitude and phase, any linear system can be described

• Convolution in the Frequency Domain– Convolution in time domain is multiplication in frequency domain

– Speed of computation greatly reduced

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Fast Fourier Transform (FFT): Overview

• What?– An efficient algorithm to compute the DFT and its inverse

• Why use it?– O(Nlog2N)

• Tradeoff– Input data must be a power of 2– If it’s not, data is either truncated or padded with zeros

• Many different types

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FFT: Cooley-Tukey algorithm

• Most common FFT algorithm– Divide and conquer algorithm

• Methodology– Breaks up DFT of N samples into N=N1N2

• Benefit– Can be combined with any other DFT algorithm

– What Matlab fft function does for optimization

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FFT: Cooley-Tukey algorithmMethodology

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Fast Fourier Transform (FFT): Bluestein

• Also called chirp z-transform algorithm

• Methodology– Expresses DFT as a convolution

• Benefit– Computes DFT of arbitrary sizes– Can be used to compute more general transforms

• Tradeoff– Only O(Nlog2N) complexity for prime-sized DFTs

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Fast Fourier Transform (FFT): BluesteinMethodology

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Goertzel Algorithm

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Goertzel Algorithm

• Tradeoff– O(NM)– N is number of DFT terms, M is the set of DFT terms to calculate

• Benefit– Simple structure of algorithm makes it well suited to small processors

– More efficient than FFT for small number of frequencies ( if M < log2N)

• Applications– Used to recognize DTMF tones produced by buttons on telephone keypad

– Call progress (dial tone, busy)14

Runtime Performance Comparison

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Runtime Performance Comparison:Matlab

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Outline

• Overview

• DFT algorithms– FFT– Cooley–Tukey FFT– Bluestein's FFT– Goertzel

• Edge Detection Algorithms– Canny– Sobel– Prewitt– Roberts Cross

• Questions

Edge DetectionOverview

• Use– Tool used in image processing– Goal is to identify points in a digital image where image brightness changes sharply or has discontinuities

– Reduce amount of data in image so that further image processing may occur

– Reduce images to shapes

• Applications– Most image processing

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Edge Detection: Canny

• Use– Optimal edge detection

• the algorithm should mark as many real edges in the image as possible

• edges marked should be as close as possible to the edge in the real image

• a given edge in the image should only be marked once

• image noise should not create false edges

• Benefits– Adaptable to most images– Generally has short runtime, well-suited for real time implementations in FPGAs

• Constraints– Size of Gaussian filter– Calibration of thresholds

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Edge Detection: CannyMethodology

• Convolves image with Gaussian filter– To eliminate noise

• Uses other types of edge detection to determine intensity gradient vertically, horizontally, and diagonally– 4 filters

• Goes through matrix to determine if gradient magnitude is a local maximum based on gradient direction– Determine high and low threshold

• Go through image and mark pixel as edge or non-edge based on thresholds

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Edge Detection: Sobel

• Use– Discrete differentiation operator– Computes the gradient of the image intensity

• Benefits– Few number of computations– Can be implemented simply in both hardware and software

• Tradeoffs– Poor gradient approximation for high frequency variations in image

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Edge Detection: SobelMethodology

• 2 kernels convolved with original image to calculate gradient in vertical and horizontal direction

– Where A is the original image

• Compare gradient value to threshold similar to Canny algorithm

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Edge Detection: Prewitt

• Same as Sobel algorithm but with two key differences

• The two kernels used have slightly different values

• Magnitude and Direction of gradient in Sobel:

• Magnitude and Direction of gradient in Prewitt:

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Edge Detection: Roberts Cross

• Use– One of the first edge detectors– Intensity of edges should correspond as close as possible to what a human would perceive

• Benefits– Only finds gradient in diagonal direction due to simple kernels, so algorithm has high simplicity

• Tradeoffs– Speed of modern computers make this simplicity negligable

– Edge detection suffers from sensitivity to noise

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Edge Detection: Roberts CrossMethodology

• 2 kernels convolved with original image to calculate gradient in diagonal direction

• Magnitude and direction of gradient found similar to Sobal

• Compare gradient value to threshold similar to Canny algorithm

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Block M

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Canny

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Sobel

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Prewitt

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Roberts Cross

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Edge Pixels Detected: Block M

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Prabal

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Canny

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Sobel

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Prewitt

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Roberts Cross

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Edge Pixels Detected: Prabal

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Blur

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Canny

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Sobel

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Prewitt

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Roberts Cross

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Edge Pixels Detected: Blur

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Gray

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Canny

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Sobel

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Prewitt

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Roberts Cross

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Edge Pixels Detected: Grayscale

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Final Comparison

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Questions?

Comments?

Discussion?

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References

• Everything– Wikipedia– Mathworks

• DFT/FFT– The Scientist and Engineer's Guide to Digital Signal Processing By

Steven W. Smith, Ph.D.– http://www.robots.ox.ac.uk/~sjrob/Teaching/SP/l7.pdf– http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/OWENS/

LECT4/node3.html

• Goertzel– http://www.numerix-dsp.com/goertzel.html

• Edge Detection– http://dasl.mem.drexel.edu/alumni/bGreen/www.pages.drexel.edu/

_weg22/can_tut.html– http://www.cse.unr.edu/~bebis/CS791E/Notes/EdgeDetection.pdf

• Edge Detection Images– Prabal: web.eecs.umich.edu/~prabal/– Block M: www.umich.edu– Grayscale: www.colorsimulator.com– Blur: www.dreamstime.com (Royalty Free Stock Photos)