reconstruction with adaptive feature-specific imaging jun ke 1 and mark a. neifeld 1,2 1 department...
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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
Frontiers in Optics 2007
Motivation for FSI and adaptation.
Adaptive FSI using PCA/Hadamard features.
Adaptive FSI in noise.
Conclusion.
Motivation - FSI Reconstruction with Feature-specific Imaging (FSI) :
Frontiers in Optics 2007
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
Frontiers in Optics 2007
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
Frontiers in Optics 2007
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
Frontiers in Optics 2007
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.
Frontiers in Optics 2007
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.
Frontiers in Optics 2007
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
Frontiers in Optics 2007
<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
Frontiers in Optics 2007
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
Frontiers in Optics 2007
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
Frontiers in Optics 2007
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
Frontiers in Optics 2007
High diversity training data; σ02 = 1High diversity training data; σ0
2 = 1
Minimum total feature collection time
Increasing T
Trade-off
Conclusion
Frontiers in Optics 2007
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