compressive ghost imaging - uteero/sc/compressive_ghost_imaging.pdf · multicolor ghost imaging ....
Post on 06-Oct-2020
7 Views
Preview:
TRANSCRIPT
COMPRESSIVE GHOST IMAGING Andreas Valdmann
1
Traditional imaging
Light source
Matrix sensor (Camera)
Object with reflectivity 𝑂 𝑥,𝑦
Signal 𝑆𝑐𝑐𝑐(𝑥,𝑦) ~ 𝑂(𝑥,𝑦)
Uniform illumination
2
Ghost imaging
Light source
Matrix sensor (Camera) measures 𝐼(𝑥, 𝑦)
Object 𝑂(𝑥,𝑦)
Beam splitter
𝐼(𝑥,𝑦)
𝐼(𝑥,𝑦) 𝑂(𝑥,𝑦)
Single pixel detector measures 𝑆 = ∫ 𝐼 𝑥,𝑦 𝑂 𝑥, 𝑦 𝑑𝑥𝑑𝑦
𝑂(𝑥, 𝑦) = 𝑆 − 𝑆 𝐼(𝑥, 𝑦) − 𝐼(𝑥, 𝑦) 𝑂 𝑥,𝑦 =
1𝑀�𝑆𝑖𝐼𝑖(𝑥,𝑦)
𝑀
𝐼=1
M – number of measurements
Random pattern
3
Image reconstruction
Computational ghost imaging
Object 𝑂(𝑥,𝑦)
𝐼(𝑥,𝑦)
𝐼(𝑥,𝑦) 𝑂(𝑥,𝑦)
Single pixel detector measures 𝑆 = ∫ 𝐼 𝑥,𝑦 𝑂 𝑥, 𝑦 𝑑𝑥𝑑𝑦
𝑂(𝑥, 𝑦) = 𝑆 − 𝑆 𝐼(𝑥, 𝑦) − 𝐼(𝑥, 𝑦) 𝑂 𝑥,𝑦 =
1𝑀�𝑆𝑖𝐼𝑖(𝑥,𝑦)
𝑀
𝐼=1
M – number of measurements
𝐼(𝑥, 𝑦) is determined by user. No camera is needed.
Digital projector
4
Image reconstruction
5
6
7
Multicolor ghost imaging
Object 𝑂𝜇(𝑥,𝑦)
𝐼(𝑥,𝑦)
𝐼(𝑥,𝑦)𝑂𝜇(𝑥,𝑦)
Single pixel detector measures 𝑆𝜇 = ∫ 𝐼 𝑥,𝑦 𝑂𝜇 𝑥,𝑦 𝑑𝑥𝑑𝑦
𝑂𝜇(𝑥, 𝑦) = 𝑆𝜇 − 𝑆𝜇 𝐼(𝑥, 𝑦) − 𝐼(𝑥, 𝑦) 𝑂𝜇 𝑥,𝑦 =
1𝑀�𝑂𝜇𝑖𝐼𝑖(𝑥,𝑦)
𝑀
𝐼=1
M – number of measurements
𝐼(𝑥, 𝑦) is determined by user. No camera is needed.
Digital projector
8
Experimental results
S. Welsh, M. Edgar, R. Bowman, P. Jonathan, B. Sun, and M. Padgett, "Fast full-color computational imaging with single-pixel detectors," Opt. Express 21, 23068-23074 (2013).
1 million measurements!
9
Inversion image reconstruction
𝐈 = [𝐼1, 𝐼2, … , 𝐼𝑀] matrix of all light patterns
𝑆 = [𝑠1, 𝑠2, … , 𝑠𝑀] array of single pixel detector signals
𝐼1, 𝐼2, … , 𝐼𝑀 are flattened light pattern arrays
Pixels (N)
Mea
sure
men
ts (M
) =
𝑆 𝑂 𝐈
M measurements; N pixels
1 2 3 4 1 2 3 4
x
y
𝐈𝑂 = 𝑆
𝑂 flattened image of the object (unknown)
System of linear equations
10
Inversion image reconstruction
M measurements; N pixels
𝐈𝑂 = 𝑆
M = N: perfect image recovery if no noise is present (only in theory). M > N: can be solved with least squares methods. Larger M gives better signal to noise ratio, but measurements and calculations take more time. M < N: Theoretically infinite number of solutions. Additional constraints must be given. Compressive sensing
Pixels (N)
Mea
sure
men
ts (M
)
=
𝑆 𝑂 𝐈
11
Discrete cosine transform
12
JPEG compression DCT frequencies
DCT of a natural image gives a sparse matrix.
13
Compressive sensing Is it necessary to make many measurements if most of the data is omitted during compression anyway? No, if the measured image is natural, i.e. it is sparse in some basis.
Idea of compressive sensing algorithm
• Solve the underdetermined system in some sparse basis.
• Additional constraint: result vector must contain as many zeros as possible: L0 norm minimization.
14
Lp norm
𝑥 𝑝 = �𝑥𝑖𝑝𝑛
𝑖=0
1𝑝
𝑥 0 counts non-zero elements in vector
𝑥 1 sum of absolute values of vector components (Manhattan geometry)
𝑥 2 Euclidian norm
Minimizing L0 norm is NP-hard. Minimizing L1 norm also gives a small L0 norm. Examples of implementation: • Matlab L1-MAGIC toolbox • Python CVXOPT package
15
Comressive ghost imaging 𝐈𝑂 = 𝑆 Initial problem
Discrete cosine transform 𝐈 → 𝐈𝑫𝑫𝑫
𝐈𝑫𝑫𝑫𝑂𝐷𝐷𝐷 = 𝑆 𝑂𝐷𝐷𝐷 is in sparse basis
Minimize 12
𝑆 − 𝐈𝑫𝑫𝑫𝑂𝐷𝐷𝐷 2 + λ� 𝑂𝐷𝐷𝐷
λ is a small regularization parameter
Inverse discrete cosine transform 𝑂𝐷𝐷𝐷 → 𝑂
16
Experimental results
S. Welsh, M. Edgar, R. Bowman, P. Jonathan, B. Sun, and M. Padgett, "Fast full-color computational imaging with single-pixel detectors," Opt. Express 21, 23068-23074 (2013).
17
Potential applications of compressive sensing • Enegry efficient miniature cameras [David Schneider
(March 2013). "New Camera Chip Captures Only What It Needs". IEEE Spectrum.]
• Magnetic resonance imaging (MRI) [Michael Lustig. Compressed sensing MRI resources. http://www.eecs.berkeley.edu/~mlustig/CS.html]
• Hyperspectral imaging [Y. August, C. Vachman, Y. Rivenson,
and A. Stern, "Compressive hyperspectral imaging by random separable projections in both the spatial and the spectral domains," Appl. Opt. 52, D46-D54 (2013).]
18
Summary Ghost imaging • Single pixel detector • Many measurements • Replaces matrix sensor
Compressive sensing • Number of measurements is
less than number of pixels • Uses sparseness of natural
images
Thank you
19
top related