Quadratic Programming Solver for Image Deblurring Engine

Rahul Rithe, Michael Price

Massachusetts Institute of Technology

Image Deblurring

Blur Kernel

• For image deblurring, the solution is constrained to be non-negative l = 0, u = +∞

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Cauchy Point Computation:

First local minima along the gradient projected on to the search space

Algorithm

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Gradient (Ax – b)

OptimizationsDimension Reduction• Ignore the dimensions that

have active constraints by holding their solution to zero till the next outer iteration

• If all but 100 constraints are active: 100×100 matrix/vector operations instead of 1000×1000

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Gradient (Ax – b)

OptimizationsIncremental Update• Incrementally update

matrix/vector product in CP• Incrementally update

gradient throughout both CP and CG steps, based on incremental changes to x

• At the end of each CG refinement, recalculate cost using updated gradients

• Avoids explicit computation of Ax product every outer iteration

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Gradient (Ax – b)

OptimizationsPerformance Improvement• N outer iterations with M1

breakpoints checked for CP and M2 CG iterations per outer iteration

• Direct implementation: N(3+M1+M2) matrix/vector multiplications

• Optimized implementation:1+N(2+M2

) matrix/vector multiplications

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Gradient (Ax – b)

Optimized implementation typically achieves ~ 50% performance improvement

Architecture

• Control logic determines resource access • Memory controller connects the design to external

DDR2 memory

• A, b, x stored in DRAM• On-chip SRAMs used for

temporary variables• Single-precision floating

point arithmetic• Iterative execution of CP

and CG• Use non-concurrency of

CP and CG to share SRAMs

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Matrix Multiplier

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Multiplication in chunks of m:• m elements of A are fetched per clock cycle from DRAM• One element of x, b can be accessed per clock cycle from

SRAM

Matrix Multiplier

Active Columns• Check if any columns in

a group of m columns are active

• Skip over the group if no active columns

Active Rows• Check if any rows in a

group of m rows are active

• Skip over the group if no active rows

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Matrix Multiplier

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Sort

• Cauchy Point Computation requires sorting an array of breakpoints

• Sort implemented using merge sort

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Main Modules• The control logic in both CP and CG modules are FSMs

that sequence the external operators • Each state corresponds to a discrete step of the

algorithm• Each step evaluates as many operations as possible

concurrently

Conjugate Gradient Architecture

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FPGA ImplementationVitrex-5 LX110T• QP Solver design integrated with DDR2 memory using a

Request/Response interface• Integrated with Sce-Mi to communicate between a

processor and the FPGA• Verified in simulation• Performance after

synthesis: 51.3 MHz

Total LUTs 78743/69120 113%

LUTs as Logic

76975/51200 150%

LUTs as Memory

1768/17920 9%

FF 69485/69120 100%

Resource utilization during placement13

FPGA ImplementationKintex-7 K325T• QP Solver design integrated with DDR3 memory using a

Request/Response interface• Integrated with USB interface to communicate between a

processor and the FPGA• Performance after synthesis: 67.2 MHz

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FPGA ImplementationKintex-7 K325T• QP Solver design integrated with DDR3 memory using a

Request/Response interface• Integrated with USB interface to communicate between a

processor and the FPGA• Performance after synthesis: 67.2 MHz

Dual Port RAMs 33

Simple Dual Port RAMs 610

Block RAMs 114/148 77%

DSP48s 58/840 6%

Total LUTs 69073 33%

Resource utilization after synthesis

Slice LUTs 64,522/203,800 31%

Slice Registers 55,406/407,600 13%

Occupied Slices 23,206/50,950 45%

DSP48E1s 58/840 6%

RAMB36E1/FIFO36E1s

113/445 25%

Resource utilization after placement

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Results

Synthetic problem of size 256

Real problem of size 361 from image deblurring16

Results

FPGA implementation is faster for larger problem sizes

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Conclusions• QP Solver module designed and implemented on

Kintex-7 FPGA• Optimized the implementation to reduce

matrix/vector multiplications• Maximized concurrent execution of processing steps• FPGA implementation verified to be functional for

problem sizes ranging from 16 to 361

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Acknowledgements

Priyanka Raina

Richard Uhler, Myron King, Prof. Arvind