Reconstruction with Adaptive Feature-specific Imaging

Jun Ke1 and Mark A. Neifeld1,2

1Department of Electrical and Computer Engineering,

2College of Optical Sciences

University of Arizona

Frontiers in Optics 2007

Outline

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Motivation for FSI and adaptation.

Adaptive FSI using PCA/Hadamard features.

Adaptive FSI in noise.

Conclusion.

Motivation - FSI Reconstruction with Feature-specific Imaging (FSI) :

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FSI benefits:

Lower hardware complexity

Smaller equipment size/weight

Higher measurement SNR

High data acquisition rate

Lower operation bandwidth

Less power consumption

1myn mM 1ˆ nx1nxnmF 1my

Reconstruction matrix M (nxm)

x̂

object

object estimate

)1( nx

DMD

Imaging optics Imaging

optics

single detector

)1( nif

feature T

iiy fx

),,2,1( Mi

Sequential architecture:

Parallel architecture:

LCD

LCD

LCD

1f

2f

Mf

)1( nx

1T

1 fxy

2T

2 fxy

MMy fxT

M (nxm)

Motivation - Adaptation

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Acquire feature measurements sequentially

Use acquired feature measurements and training data to adapt the next projection vector

The design of projection vector effects reconstruction quality.

Estimation

Poorly designed projection vectors

Testing sample

Training samples

Projection axis 2

Static PCA

Projection axis 1

Using PCA projection as example Well designed projection vectorsAdaptive PCA

Estimation

Projection axis 2

Projection value

Training samples

for 2nd projection vector

Projection axis 1

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1...

ˆ i m mm i

y

x fObject estimate

yi = fiTx

Calculate fi+1

Reconstruction

Object x

Update Ai to Ai+1

according to yi

Computational Optics

Calculate f1

Ri+1

Calculate R1 from A1

Adaptive FSI (AFSI) – PCA:

i: adaptive step index

Ai: ith training set

Ri: autocorrelation matrix of Ai

fi: dominate eigenvector of Ai

yi: feature value measured by fi High diversity of training data helps adaptation

PCA-Based AFSI

Testing sample

K(1) nearest samplesProjection axis

Testing sample

K(1) nearest samples

Selected samples

According to 1st feature

According to 2nd feature

K (2) nearest samples

Projection axis 2

Projection axis 1

Object examples (32x32):

Tx̂ = F y Reconstructed object:

2ˆ{|| || }/E N x - x RMSE:

y = Fx Feature measurements: where, : 1 : N M N x F

is the total # of featuresM

PCA-Based AFSI

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Number of training objects: 100,000

Number of testing objects: 60

RMSE reduces using more features

RMSE reduces using AFSI compare to static FSI

Improvement is larger for high diversity data

RMSE improvement is 33% and 16% for high and low diversity training data, when M = 250.

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AFSI – PCA:

PCA-Based AFSI

Reconstruction from static FSI (M = 100)

Reconstruction from AFSI (M = 100)

K increases

Projection vector’s implementation order is adapted.

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xi(mean): average vector of Ai

fi: dominant Hadamard vector for Ai

AFSI – Hadamard:

Hadamard-Based AFSI

Object estimate

yiL+j = f iL+jTx

(j=1,…,L)

Choose fiL+1 ~ f(i+1)L

ReconstructionObject x

Update Ai to Ai+1

according to yiL+j

Computational Optics

Choose f1~fL

xi+1(mean)

x1(mean)

Sort

SortHadamard

bases

Selected samples

K(1) nearest samples

testing sample

projection axis 1

K(1) nearest samples

testing sample

projection axis 2

K(2) nearest samples

sample mean

First 5 Hadamard basis←Static FSI AFSI→

according to 1st feature

according to 2nd feature

sample mean

projection axis 1

RMSE reduces in AFSI compared with static FSI

RMSE improvement is 32% and 18% for high and low diversity training data, when M = 500 and L = 10.

AFSI has smaller RMSE using small L when M is also small

AFSI has smaller RMSE using large L when M is also large

Hadamard-Based AFSI

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AFSI – Hadamard:

Reconstruction from adaptive FSI

Reconstruction from static FSI

K increases

L d

ec

rea

se

s

L in

cre

as

es

Hadamard-Based AFSI – Noise

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AFSI – Hadamard:

Hadmard projection is used because of its good reconstruction performance

Feature measurements are de-noised before used in adaptation

Auto-correlation matrix is updated in each adaptation step

Wiener operator is used for object reconstruction

nFxnyy 0

1)( nyyy RRRM

{ }

{( )( ) }

Ty

T

Tx

E

E

R yy

Fx Fx

FR F

1...ˆ i m mm i

y

x fObject estimate

yiL+j = fiL+jTx+niL+j

(j = 1,2,…L)

Choose fiL+1~f(i+1)L

Reconstruction

Object x

Update Ai to Ai+1

according to

Computational OpticsChoose f1~fL

xi+1(mean)

Calculate x1(mean)

from de-noising yiL+j

Calculate Ri for Ai

T : integration time

σ0

2 = 1

ˆiL jy

Sort Hadamard bases

Sort

ˆiL jy

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RMSE in AFSI is smaller than in static FSI

RMSE is reduced further by modifying Rx in each adaptation step

RMSE improvement is larger using small L when M is also small

RMSE is small using large L when M is also large

Hadamard-Based AFSI – Noise

High diversity training data; σ02 = 1

T : integration time/per feature

σ0

2 = 1

detector noise variance σ2 = σ02 /T

High diversity training data; σ02 = 1

K increases

L d

ec

rea

se

s

L in

cre

as

es

T : integration time/per feature; M0: the number of features

Total feature collection time = T × M0

RMSE reduces as T increases

High reconstruction quality requirement needs longer total feature collection time

To achieve each RMSE requirement, there is a minimum total feature collection time.

Hadamard-Based AFSI – Noise

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High diversity training data; σ02 = 1High diversity training data; σ0

2 = 1

Conclusion

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Noise free measurements:

PCA-based and Hadmard-based AFSI system are presented

AFSI system presents lower RMSE than static FSI system

Noisy measurements:

Hadamard-based AFSI system in noise is presented

AFSI system presents smaller RMSE than static FSI system

There is a minimum total feature collection time to achieve a reconstruction quality requirement