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 MG 1ˆ Nx1NxM NF1My

Sequential architecture:

Parallel architecture: LCD

LCD

LCD

1f

2f

Mf

( 1)Nx

1T

1 fxy

2T

2 fxy

MMy fxT

G (NxM)

Reconstruction matrix G (NxM)

x̂

objectobject reconstruction

( 1)NxDMD

Imaging optics

light collection

single detector

( 1)i Nf

feature T

iiy fx

),,2,1( Mi projection vector

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.

Using Principal Component Analysis (PCA) projection as example

Testing sample

Training samples

Projection axis 2

Static PCA

Projection axis 1 Reconstruction

Adaptive PCA

Projection axis 2

Projection value

Training samples

for 2nd projection vector

Projection axis 1

Reconstruction

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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

K(i): # of samples for Ai+1

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

Ri: autocorrelation matrix of Ai

fi: dominate eigenvector of Ai

yi: feature value measured by fi

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

K(i) decreases

iteration index i

iteration index i

Reconstruction from static FSI (i = 100)

Reconstruction from AFSI (i = 100)

Projection vector’s implementation order is adapted.

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AFSI – Hadamard:

Hadamard-Based AFSI

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 <A1>

First 5 Hadamard basis←Static FSI AFSI→

according to 1st feature

according to 2nd feature

sample mean <A2>

projection axis 1

Sample mean for training set Ai is <Ai>

yj = fiT <Ai> j = 1,…,M

max{yj} corresponds to the dominant Hadamard projection vector

L : # of features in each adaptive step

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<Ai>: sample mean of Ai

fi: ith Hadamard vector for Ai

AFSI – Hadamard:

Hadamard-Based AFSI

K(1) nearest samples

testing sample

projection axis 1

sample mean

projection axis 2

Selected samplesaccording to 1st 2 features 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

<Ai+1>

<Ai>

Sort

Sort Hadamard basis vectors

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:

K(i) decreases

number of features M = LiL

de

cre

as

es

L in

cre

as

es

number of features M = Li

Reconstruction from adaptive FSI

Reconstruction from static FSI

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

Wiener operator is used for object reconstruction

Auto-correlation matrix is updated in each adaptation step

T : integration time

σ0

2 = 1

detector noise variance:

σ22 = σ0

2 /T

nFxnyy 0

1( )y y y n G R R R

{ }

{( )( ) }

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

<Ai+1>

<A1>

from de-noising yiL+j

Calculate Ri for Ai

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

K(i) decreases

L d

ec

rea

se

s

L in

cre

as

es

High diversity training data; σ02 = 1

AFSI – fixed Rx

AFSI – adapted Rx

Static FSI

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

Total feature collection time = T × M0

Reducing Measurement error

Losing adaptation advantage

Hadamard-Based AFSI – Noise

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

2 = 1

Minimum total feature collection time

Increasing T

Trade-off

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