Twodimensional Image Filter Design for Multiplierless Implementation Using Genetic Algorithms

1
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

2
Objective Statement

• To devise an improved methodology for the
  performance enhancement of hardware-based image
  filters through the use of genetic optimization
  techniques.

3
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

4
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.

5
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.

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Multiplierless Approach

• Power-of-two coefficients (2n, 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 2n, 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 00010000012
6510
13 x8 104
00001101 x 00001000 011010002 Shift
left 3 10410
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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.

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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.

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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.

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Multiplierless GA For IIR Filters
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Crossover Example
Current Population
Member 1
Member 2
A coefficients
B coefficients
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Crossover Example
Current Population
Member 1
Member 2
A coefficients
B coefficients
Two coefficient sets (members) selected to cross
over
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Crossover Example
Member 1
Member 2
A coefficients
B coefficients
Random crossover points selected for A and B
coefficients
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Crossover Example
Member 1
Member 2
A coefficients
B coefficients
Range of coefficients to be exchanged between
members
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Crossover Example
Member 1
Member 2
A coefficients
B coefficients
Coefficients exchanged
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Crossover Example
Member 1
Member 2
A coefficients
B coefficients
Offspring as a result of crossover
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Crossover Example
Member 1
Member 2
A coefficients
B coefficients
New Population
Offspring join new population
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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).

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GA Output - Generic

• Ideal High-pass filter
• 2000 Iterations
• 10th-Order
• Mutation Probability 0.001

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GA Output - Generic

• Ideal High-pass filter
• 15000 Iterations
• 50th-Order
• Mutation Probability 0.001

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GA Output- Blurring
Magnitude Frequency Response Analysis
Magnitude Frequency Response
Relative Error of Fittest Member vs. Iteration

• Blurring (Low-Pass) Filter Parameters
• 2000 Iterations
• 15th-Order
• Mutation Probability 0.001
• Cutoff frequency 0.66radians

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Image Analysis- Blurring
Input bitmap image
Filtered bitmap image

• Blurring (Low-Pass) Filter Parameters
• 15th-Order
• Cutoff frequency 0.66radians

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GA Output- Canny Edge Detection
Canny An established method for detecting edges
in images. Described by a high-pass filter,
Gaussian impulse response.
Magnitude Frequency Response Analysis
Magnitude Frequency Response, Derivative Gaussian
Magnitude Frequency Response, Gaussian
Relative Error of Fittest Member vs. Iteration

• Canny Edge Detection (Gaussian) Filter
  Parameters
• 1000 Iterations
• 10th-Order
• s 1
• Mutation Probability 0.001

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GA Output- Canny Edge Detection
Magnitude Frequency Response Analysis
Magnitude Frequency Response, Gaussian
Relative Error of Fittest Member vs. Iteration

• Canny Edge Detection (Gaussian) Filter
  Parameters
• 300 Iterations
• 10th-Order
• s 2
• Mutation Probability 0.001

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Image Analysis-Canny Edge Detection
Input bitmap image
Filtered bitmap image

• Canny Edge Detection Filter Characteristics
• 10th-Order
• s 1

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Image Analysis-Canny Edge Detection
Ideal Impulse Response, Gaussian
Input bitmap image
Filtered bitmap image

• Canny Edge Detection Filter Characteristics
• 10th-Order
• s 1

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Image Analysis- Canny Edge Detection
Ideal Impulse Response, Gaussian
Input bitmap image
Filtered bitmap image

• Canny Edge Detection Filter Characteristics
• 10th-Order
• s 1

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Proposed Multiplierless Model

• Block Diagram for functional block implementing
  an IIR difference equation

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Hardware Implications FPGA

• Macro-cell oriented
• Hardware cost is function of word length (n)

Hardware costs (in macro cell counts) for
different implementations of 10th-order IIR
system.
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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)).

10th Order ASIC Functional Block Cost
Costmultiplier adders (18
delay registers) Costmultiplierless 18 adders
18 delay registers
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Hardware Implications ASIC

• 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.

Wired shift in fixed coefficient ASIC
configuration.
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Conclusions

• Developed a genetic algorithm with several
  unique, application-oriented attributes capable
  of optimizing filter coefficients such that the
  corresponding filters 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.

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Acknowledgements

• Professor Michael Rudko
• Professor François Cabestaing
• Professor Cherrice Traver