Two

dimensional Image Filter
Design for Multiplierless
Implementation Using Genetic
Algorithms
Senior Capstone Design Project
Spring, 2003:
Douglas Lockett & Christopher Roblee
Advisor: Professor Michael Rudko
Objective Statement
To devise an improved methodology for
the performance enhancement of
hardware

based image filters through the
use of genetic optimization techniques.
Agenda
Image Filtering, IIR Systems
Multiplierless approach for performance
enhancement
Genetic optimization for multiplierless filter
design
Genetic Algorithm (GA) design process
GA and filter output results
Proposed hardware model
Conclusion
Image Filtering
Inherently two

dimensional.
Performed in spatial domain.
Many real

world applications.
Filters are either finite impulse response
(FIR) or infinite impulse response (IIR).
Defined by a series of filter coefficients.
IIR Filters
Motivations:
Require fewer coefficients (lower order) than FIR
filters for a given response.
Allow for more precise and sharper approximations of
the ideal frequency response.
Tradeoffs:
Issues in stability.
Issues with phase response linearity.
Multiplierless Approach
Power

of

two coefficients (
2
n
, where n is whole number
)
A single arithmetic shift left (ASL) corresponds to a
multiplication by two. Likewise, an n

bit shift left or right
(ASR) corresponds to a multiplication or division by 2
n
,
respectively.
Computationally intensive and logically complex
multiplication operations within a filtering algorithm can be
reduced to fast and efficient binary shifts.
Highly beneficial for real

time image processing systems.
Much more difficult to obtain suitable coefficients, however.
13
x5
65
00001101
x 00000101
00001101
Add
00000000
Add
00001101
Add
0001000001
2
=
65
10
13
x8
104
00001101 x 00001000
01101000
2
Shift left 3
=
104
10
Example: Real multiplication
Example: Power

of

two multiplication
Genetic Algorithms (GAs)
Used to optimize systems through evolutionary
breeding (crossover).
Based on Darwinian genetic theory of fittest
member (system) survival.
Random search, as opposed to exhaustive search
approach.
Many generations (iterations) to improve overall
population fitness.
Decided on GA to realize multiplierless enhancement:
Classical filter design techniques (Chebyshev,
Butterworth, Elliptical, Bilinear Transform) are
incapable of satisfying criteria imposed by
multiplierless systems.
Innovative application.
Execution Flow of a Basic Genetic
Algorithm
Begin with random initial
population of possible outcomes.
Evaluate fitness of each member in
the population.
Probability of crossover assigned
to each member based upon
fitness.
Members selected to crossover
pseudo

randomly, with fitter
members having higher
probabilities of being chosen to
crossover. Corresponds to an
exchange of “genetic material”
between parents such that
subsequent generations become
fitter over time.
Offspring from the crossover
process are subject to a random
mutation.
Above steps are repeated until
population converges to a single
(ideal) outcome representing the
fittest solution.
n
y
Fitness Evaluation
Initial
Population
of Members
Crossover Selection
Crossover of selected
members
Mutation inserted into
population
Random
Mutation
Converge
d
?
Fittest
Member
New
Population
Desired
Output
Characteristics of Designed
Genetic Algorithm
Population comprises possible sets of multiplierless (power

of

two) filter coefficients.
Filter (member) stability is initial criterion.
Member fitness inversely proportional to the squares of the
differences between the member magnitude frequency response
and that of a specified ideal filter.
Crossover procedure defined by randomly selected crossover
points, corresponding to ranges of coefficients exchanged
between crossover pairs (breeding).
A specific number of fittest parent members are set aside and
inserted into the subsequent generation to prevent regression.
Coefficients within each member have a random chance of
mutation based upon a pre

defined mutation probability.
Process repeated for a specified number of iterations.
Fittest member after evolution cycle determined to be the optimal
multiplierless filter representation.
Y
N
More
Iterations?
Fittest
Member
New Population
Initial Population of
power

of

2
coefficients.
Desired
Response
Remove Unstable
Coefficient Sets
Parameters:

Number of Iterations

Filter Order

Mutation probability

Word length

Sampling frequency
Fitness Evaluation

Crossover Member Selection

Crossover Point Selection
Crossover of selected
coefficients
Mutation inserted into
population
Random
Mutation
Selected
Fittest
Members
Remove Unstable
Coefficient Sets
Newly

generated
power

of

2
coefficients.
Population replenished with
new members to maintain
original size
Multiplierless GA
For IIR Filters
Crossover Example
A coefficients
B coefficients
Member 1
Member 2
Current Population
Crossover Example
Two coefficient sets (members) selected to cross over
A coefficients
B coefficients
Member 1
Member 2
Current Population
Crossover Example
Random crossover points selected for A and B coefficients
A coefficients
B coefficients
Member 1
Member 2
Crossover Example
Range of coefficients to be exchanged between members
A coefficients
B coefficients
Member 1
Member 2
Crossover Example
Coefficients exchanged
A coefficients
B coefficients
Member 1
Member 2
Crossover Example
Offspring as a result of crossover
A coefficients
B coefficients
Member 1’
Member 2’
Crossover Example
Offspring join new population
New Population
A coefficients
B coefficients
Member 1’
Member 2’
Major Differences in Designed GA
Much more specific in order to accommodate multiplierless criteria.
Members represented as power

of

two decimal numbers as
opposed to binary strings.
Pre

emptive removal of unstable members before fitness evaluation
& pervasive stability checking throughout.
Crossover of entire coefficient values as opposed to fractions of
binary strings.
Specified number of fittest members held over to subsequent
generation to prevent fitness regression.
Subdivision of members into numerator and denominator coefficient
subsets (B and A coefficients, respectively).
GA Output

Generic
Ideal High

pass filter:
•
2000 Iterations
•
10
th

Order
•
Mutation Probability = 0.001
Magnitude Frequency Response
Relative Error of Fittest
Member vs. Iteration
GA Output

Generic
Ideal High

pass filter:
•
15000 Iterations
•
50
th

Order
•
Mutation Probability = 0.001
GA Output

Blurring
Blurring (Low

Pass) Filter Parameters:
•
2000 Iterations
•
15
th

Order
•
Mutation Probability = 0.001
•
Cutoff frequency = 0.66radians
Magnitude Frequency Response
Relative Error of Fittest
Member vs. Iteration
Magnitude Frequency Response Analysis
Image Analysis

Blurring
Blurring (Low

Pass) Filter Parameters:
•
15
th

Order
•
Cutoff frequency = 0.66radians
Input bitmap image
Filtered bitmap image
GA Output

Canny Edge Detection
Canny Edge Detection (Gaussian) Filter Parameters:
•
1000 Iterations
•
10
th

Order
•
σ = 1
•
Mutation Probability = 0.001
Magnitude Frequency Response,
Gaussian
Relative Error of Fittest
Member vs. Iteration
Magnitude Frequency Response,
Derivative Gaussian
Canny:
An established method for detecting edges in images. Described
by a high

pass filter, Gaussian impulse response.
Magnitude Frequency Response Analysis
GA Output

Canny Edge Detection
Canny Edge Detection (Gaussian) Filter Parameters:
•
300 Iterations
•
10
th

Order
•
σ = 2
•
Mutation Probability = 0.001
Magnitude Frequency Response, Gaussian
Relative Error of Fittest
Member vs. Iteration
Magnitude Frequency Response Analysis
Image Analysis

Canny Edge
Detection
Canny Edge Detection Filter Characteristics:
•
10
th

Order
•
σ = 1
Input bitmap image
Filtered bitmap image
Image Analysis

Canny Edge
Detection
Canny Edge Detection Filter Characteristics:
•
10
th

Order
•
σ = 1
Ideal Impulse Response, Gaussian
Input bitmap image
Filtered bitmap image
Image Analysis

Canny Edge
Detection
Canny Edge Detection Filter Characteristics:
•
10
th

Order
•
σ = 1
Ideal Impulse Response, Gaussian
Input bitmap image
Filtered bitmap image
Proposed Multiplierless Model
Block Diagram for functional block implementing an IIR difference equation
:
Hardware Implications:
FPGA
Macro

cell oriented
Hardware cost is function of word length (n)
Hardware costs (in macro cell counts) for different implementations of 10
th

order IIR system.
Word
Length
Multiplier

based
Multiplierless
Total Multiplier Cost
Total Cost
Total Shifter Cost
Total Cost
3
rd
Order
10
th
Order
3
rd
Order
10
th
Order
10
th
Order
n = 8
640
2,432
708
2,738
152
458
n = 16
2,560
9,728
2,689
10,322
304
898
n = 24
5,760
21,888
5,892
22,770
456
1,338
Operator
Required Number of Macro Cells
Shifter
n
Multiplier
2n
2
Adder
2n
Unit Delay
n
Hardware Implications: ASIC
Application specific design.
Can optimize logic at the gate or transistor
level, as opposed to the macrocell level
Can hard

code the coefficients into circuitry
to completely eliminate the need for any
gate logic to perform multiplications.
Multiplier

based, fixed ASIC implementation
require a series of shifts and additions to
perform multiplications (equal to # of 1s in
binary coefficient,
J(i)
).
Cost
multiplier
=
adders
+ (18
delay registers
)
Cost
multiplierless
= 18
adders
+ 18
delay registers
10
th
Order ASIC Functional Block Cost:
Hardware Implications: ASIC
Power of 2 coefficient = 0.125
Shift Right 3
.
0
0
1
0
0
0
0
0
1
1
1
0
1
1
0
0
Multiplicand = 234
x(n), 8 bits
.
0
1
0
0
0
0
0
0
0
0
0
1
1
1
0
1
Product/output register: product = 29.25
y(n),16 bits
Wired shift

by

3 logic.
Wired shift in fixed coefficient ASIC configuration.
•
Multiplications are constant shifts, so the bit signals of the multiplicand
(input) can be rerouted to effectively realize a wired shift by the amount
specified in the static coefficient.
•
Rerouted (shifted) input can be sent directly to the output register without
any shift logic or additional clock cycles.
Conclusions
Developed a genetic algorithm with several unique,
application

oriented attributes capable of optimizing filter
coefficients such that the corresponding filter
’
s frequency
response matches that of an ideal system with the constraint
that all coefficients are powers

of

two and the resulting filter
is stable.
Genetically optimized multiplierless filters consistently yield
comparable image results as their ideal counterparts.
In many cases the multiplierless systems have a definite
advantage in terms of their efficiency while maintaining a
desired response.
In all cases, the multiplierless approach allows for substantial
reductions in hardware cost and computational intensity.
Multiplierless

based systems are a viable alternative for
implementing image filters.
Acknowledgements
Professor Michael Rudko
Professor Fran
ç
ois Cabestaing
Professor Cherrice Traver
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