image “padding” in limited-memory fpga systems

1Turri MAPLD 2005/P160

Image “Padding” In Limited-MemoryFPGA Systems

William Turri ([email protected])Ken Simone ([email protected])Systran Federal Corp.4027 Colonel Glenn Highway, Suite 210Dayton, OH 45431-1672937-429-9008 x104

MAPLD 2005 Conference Presentation


Research Requirements

• Develop an efficient means of managing large (512 MB to 1 GB) raw SAR images in memory, for wavelet-based compression on an FPGA

• Minimize the number of read/write operations required to perform a wavelet transform

• Preserve quality and avoid creating artifacts


SAR Compression Background• What are we compressing?

– Collections of SAR data in a raw, unprocessed format, collected by a sensor and stored in contiguous memory

• Why are we compressing it?– Raw SAR images can be up to 1 GB in size, and must be

compressed to facillitate transmission to airborne platforms to ground-based processing stations

• How are we compressing it?– Wavelet transforms have proven effective for compressing

many kinds of images, including SAR. Research has identified the 5/3 Wavelet Transform as well-suited to raw SAR data


Wavelet Transform BackgroundOriginal Image Row Transformed Image

0 n-1… 0 n-1(n/2)

k

k+(n/2)


5/3 Wavelet Filters

Low pass filter uses 5 taps

High pass filter uses 3 taps


Filter Equations• The preceding transform may be represented by the

following algorithms, and we may identify the data points (k indices) required to perform a computation (assume N = 32 and k ranges from 0 to 15)

)12(2

)22()2(0

001

krkrkr

Intkd

High-Pass

4

)()1()2()( 11

01

kdkdIntkrkr

d1(k) needs [2k, 2k+2, 2k+1]k = 0 => [0, 1, 2]k = 15 => [30, 31, 32]

r1(k) needs [2k-2, 2k-1, 2k, 2k+1, 2k+2]k = 0 => [-2, -1, 0, 1, 2] k = 15 => [28, 29, 30, 31, 32]

Indices lie beyond the lowerand upper bounds of N

Low-Pass

Index lies beyond the upperbound of N


Need for “Extension”

These values must be supplied throughsome form of extension…

…and so mustthis value.

Extension minimizesthe degradationalong the image borders, resulting from the wavelettransform beingused.


Extension Options

a b c … x y zbc y x

Odd-Symmetric

a b c … x y zab z y

Even-Symmetric

a b c … x y zyx a b

Periodic

For the integer 5/3 transform, Odd-Symmetric extension gives the best numerical results


Values Needed for Odd-Symmetric Extension

These values must be supplied throughsome form of extension…

…and so mustthis value.

a b cbc w x y z y


Possible Odd-Symmetric Approaches• Extended pixels could be added to the image array in memory

and fetched normally– Fetching extended values through additional memory read operations will

result in unnecessary delays in processing– Storing additional values will unnecessarily consume available memory

• The extended values could be “fetched” by reading the same value twice when processing boundary coefficients– This presents the same problem of creating unnecessary delays by

introducing additional memory read operations• Additional logic dedicated to processing the boundary

coefficients could be added to the hardware– Must consume minimal resources and not slow the system when

processing non-boundary coefficients– Should account for the extended coefficients based on mathematical

operations, if possible, rather than having to create additional stored values


Left-Side Extension for r1

)12(2

)22()2(0

001

krkrkr

Intkd

4

)()1()2()( 11

01

kdkdIntkrkr

4

)12(2

)22()2()12(

2)2()22(

)2(0

000

00

0

krkrkr

krkrkr

Intkr

For k = 0, these values must be generated through extension, giving…

4

)12(2

)22()2()12(

2

)2()22(

)2(0

000

00

0

krkrkr

krkrkr

Intkr

4

)()()2( 11

0

kdkdIntkr

4

)(2)2( 1

0

kdIntkr

2

)()2( 1

0

kdIntkr


Right-Side Extension for d1

For we get: 12

N

k

1)12

(22

1)12

(2)12

(2

)12

( 0

00

1

Nr

Nr

Nr

IntN

d

)1(

2

)()2(0

00 NrNrNr

Int

This value must be generated through extension, giving…

)1(

2

)2()2(0

00 NrNrNr

Int

)1(

2

)2(20

0 NrNr

Int

)1()2( 00 NrNrInt

)12(2

)22()2(0

001

krkrkr

Intkd

)12(

2

)2(20

0 krkr

Intgeneralized


Final EquationsFor k = 0:

)12(2

)22()2(0

001

krkrkr

Intkd

4

)(2)2()( 1

01

kdIntkrkr

For k = 1 to (N/2 – 2): )12(

2

)22()2(0

001

krkrkr

Intkd

4

)()1()2()( 11

01

kdkdIntkrkr

For k = (N/2 – 1):

4

)()1()2()( 11

01

kdkdIntkrkr

These rational forms of the equations were chosen because

they best suit our hardware design

)12(

2

)2(2)( 0

01 kr

krIntkd


Top-Level Hardwared1(k)

d1(k-1)(registered)

Hard-wired right shift by 1(divide by 2)

Hard-wired right shift by 2(divide by 4)


r1 Hardware

d1(k)

d1(k-1)

When k = 0, MUX passes d1(k)to the ADD8, givingand performing extension.

Otherwise, MUX passes d1(k-1)to the ADD8, giving

4

)(2 1 kd

4

)()1( 11 kdkd


d1 Hardwarer0(2k)

r0(2k+2)

When k = (N/2-1), lower MUX passes r0(2k) to the ADD8, givingand performing extension.

Otherwise, lower MUX passes r0(2k+2)to the ADD8, giving

2

)2(2 0 kr

2

)22()2( 00 krkr


Results of Extension

• Incorporating odd-periodic extension into the hardware involved only two minor changes from a system that does not incorporate extension– Increase from a 3-to-1 MUX to a 4-to-1 MUX to

accommodate d1 extension

– Addition of a 2-to-1 MUX to accommodate r1 extension

– Impact on resource consumption and timing characteristics is minimal


Summary and Suggestions

• Adding odd-symmetric extension through dedicated hardware effectively met our research goal with minimal impact on performance

• Similar approach could be used with other wavelet transforms, with similar anticipated results

• Similar approach could be used with the other forms of extension (even-symmetric and periodic)

image “padding” in limited-memory fpga systems

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