Signal Processing - September 2016 - 53

Object

0.1

Amplitude
0.2
0.3

0.4

0.5

0.5

0.6

Phase
0.7
0.8

0.9

1

m

8.0
7

Transparency

K=3

MIT
m

Transparency

0

0.3
0m

3.6
8

Γ2

ToF

0

Amplitude
0.4
0.6

0.8

1

0.5

0.6

Phase
0.7
0.8

0.9

1

K=2

Experimental Setup

0.2

t2

15ω0
25ω0

Γ1

35ω0

0

1

Amplitude
2
3

4

t1
5

0.5

0.6

Phase
0.7
0.8

0.9

1

45ω0

65ω0

K=1

55ω0

Γ0

t0

Multiple Depth Imaging

Erroneous 3-D Measurements

Figure 5. Multidepth imaging with ToF sensors. The experimental setup describes a scene with up to K = 3 optical paths arising from three different

depths. With f0 = ~ 0 /2r = 0.8 MHz, the ToF sensor captures raw 3-D images using the four-bucket method described in the section "Frequency-Domain
3-D Imaging (FD-ToF)." The resulting 3-D maps are erroneous. We show 3-D images for modulation frequencies 15~ 0 -65~ 0 in steps of 10~ 0 . We
estimate " C k , t k ,2k = 0 using the spectrum estimation method implemented via the matrix pencil algorithm. The results are consistent with the experimental
setup. For example, the letters "MI" and "T" associated with C 1 and C 2 correspond to their respective depths.

sparse signal that has been low-pass filtered with a calibrated
sampling kernel z ^ t h. Furthermore, the bandlimited approximation property allows us to write,
, = + M0 t t
.,~ nT
m
z h (,~ 0) e 0 ,
n = (z ) h K ) (t) ; t = nT . /
, = - M0 , K

n = 0, f, N - 1,
where zt , = pt , 2 , since the sampling kernel z is obtained by
autocorrelating the probing function, and ht K ^,~ 0 h is the sampled Fourier transform of the SRF. Note that the discrete FouN-1
rier transform of the sampled measurements " m n ,n = 0 results
in the 2M 0 + 1 point-wise samples of the weighted Fourier
transform of the SRF, " zt , ht K ^,~ 0 h, , # M 0 . Hence, given

sampled measurements we can deconvolve the SRF in Fourier
domain, which is a sum of K complex exponential functions.
The only requirement being that zt , should not vanish in the
interval - M 0 # , # M 0 . In a noiseless setting, upon deconvolving the SRF " ht K ^,~ 0 h, , # M 0 , we can use spectral estimation [cf. "Spectral Estimation (Prony's Method)"] to estimate
the parameters " C k, t k ,Kk =-01 , provided that M 0 $ K . Several
approaches have been proposed in literature to stabilize the
spectral estimation method in presence of noise [26].
As before, we use the matrix pencil method for our experimental demonstrations.
Figure 6 illustrates the reinterpretation of the transient imaging
problem as a sparse sampling problem. Time-domain measurements are acquired for the case K = 2 and K = 3, respectively,

IEEE SIgnal ProcESSIng MagazInE

|

September 2016

|

53



Table of Contents for the Digital Edition of Signal Processing - September 2016

Signal Processing - September 2016 - Cover1
Signal Processing - September 2016 - Cover2
Signal Processing - September 2016 - 1
Signal Processing - September 2016 - 2
Signal Processing - September 2016 - 3
Signal Processing - September 2016 - 4
Signal Processing - September 2016 - 5
Signal Processing - September 2016 - 6
Signal Processing - September 2016 - 7
Signal Processing - September 2016 - 8
Signal Processing - September 2016 - 9
Signal Processing - September 2016 - 10
Signal Processing - September 2016 - 11
Signal Processing - September 2016 - 12
Signal Processing - September 2016 - 13
Signal Processing - September 2016 - 14
Signal Processing - September 2016 - 15
Signal Processing - September 2016 - 16
Signal Processing - September 2016 - 17
Signal Processing - September 2016 - 18
Signal Processing - September 2016 - 19
Signal Processing - September 2016 - 20
Signal Processing - September 2016 - 21
Signal Processing - September 2016 - 22
Signal Processing - September 2016 - 23
Signal Processing - September 2016 - 24
Signal Processing - September 2016 - 25
Signal Processing - September 2016 - 26
Signal Processing - September 2016 - 27
Signal Processing - September 2016 - 28
Signal Processing - September 2016 - 29
Signal Processing - September 2016 - 30
Signal Processing - September 2016 - 31
Signal Processing - September 2016 - 32
Signal Processing - September 2016 - 33
Signal Processing - September 2016 - 34
Signal Processing - September 2016 - 35
Signal Processing - September 2016 - 36
Signal Processing - September 2016 - 37
Signal Processing - September 2016 - 38
Signal Processing - September 2016 - 39
Signal Processing - September 2016 - 40
Signal Processing - September 2016 - 41
Signal Processing - September 2016 - 42
Signal Processing - September 2016 - 43
Signal Processing - September 2016 - 44
Signal Processing - September 2016 - 45
Signal Processing - September 2016 - 46
Signal Processing - September 2016 - 47
Signal Processing - September 2016 - 48
Signal Processing - September 2016 - 49
Signal Processing - September 2016 - 50
Signal Processing - September 2016 - 51
Signal Processing - September 2016 - 52
Signal Processing - September 2016 - 53
Signal Processing - September 2016 - 54
Signal Processing - September 2016 - 55
Signal Processing - September 2016 - 56
Signal Processing - September 2016 - 57
Signal Processing - September 2016 - 58
Signal Processing - September 2016 - 59
Signal Processing - September 2016 - 60
Signal Processing - September 2016 - 61
Signal Processing - September 2016 - 62
Signal Processing - September 2016 - 63
Signal Processing - September 2016 - 64
Signal Processing - September 2016 - 65
Signal Processing - September 2016 - 66
Signal Processing - September 2016 - 67
Signal Processing - September 2016 - 68
Signal Processing - September 2016 - 69
Signal Processing - September 2016 - 70
Signal Processing - September 2016 - 71
Signal Processing - September 2016 - 72
Signal Processing - September 2016 - 73
Signal Processing - September 2016 - 74
Signal Processing - September 2016 - 75
Signal Processing - September 2016 - 76
Signal Processing - September 2016 - 77
Signal Processing - September 2016 - 78
Signal Processing - September 2016 - 79
Signal Processing - September 2016 - 80
Signal Processing - September 2016 - 81
Signal Processing - September 2016 - 82
Signal Processing - September 2016 - 83
Signal Processing - September 2016 - 84
Signal Processing - September 2016 - 85
Signal Processing - September 2016 - 86
Signal Processing - September 2016 - 87
Signal Processing - September 2016 - 88
Signal Processing - September 2016 - 89
Signal Processing - September 2016 - 90
Signal Processing - September 2016 - 91
Signal Processing - September 2016 - 92
Signal Processing - September 2016 - 93
Signal Processing - September 2016 - 94
Signal Processing - September 2016 - 95
Signal Processing - September 2016 - 96
Signal Processing - September 2016 - 97
Signal Processing - September 2016 - 98
Signal Processing - September 2016 - 99
Signal Processing - September 2016 - 100
Signal Processing - September 2016 - 101
Signal Processing - September 2016 - 102
Signal Processing - September 2016 - 103
Signal Processing - September 2016 - 104
Signal Processing - September 2016 - 105
Signal Processing - September 2016 - 106
Signal Processing - September 2016 - 107
Signal Processing - September 2016 - 108
Signal Processing - September 2016 - 109
Signal Processing - September 2016 - 110
Signal Processing - September 2016 - 111
Signal Processing - September 2016 - 112
Signal Processing - September 2016 - 113
Signal Processing - September 2016 - 114
Signal Processing - September 2016 - 115
Signal Processing - September 2016 - 116
Signal Processing - September 2016 - 117
Signal Processing - September 2016 - 118
Signal Processing - September 2016 - 119
Signal Processing - September 2016 - 120
Signal Processing - September 2016 - 121
Signal Processing - September 2016 - 122
Signal Processing - September 2016 - 123
Signal Processing - September 2016 - 124
Signal Processing - September 2016 - 125
Signal Processing - September 2016 - 126
Signal Processing - September 2016 - 127
Signal Processing - September 2016 - 128
Signal Processing - September 2016 - 129
Signal Processing - September 2016 - 130
Signal Processing - September 2016 - 131
Signal Processing - September 2016 - 132
Signal Processing - September 2016 - 133
Signal Processing - September 2016 - 134
Signal Processing - September 2016 - 135
Signal Processing - September 2016 - 136
Signal Processing - September 2016 - 137
Signal Processing - September 2016 - 138
Signal Processing - September 2016 - 139
Signal Processing - September 2016 - 140
Signal Processing - September 2016 - 141
Signal Processing - September 2016 - 142
Signal Processing - September 2016 - 143
Signal Processing - September 2016 - 144
Signal Processing - September 2016 - 145
Signal Processing - September 2016 - 146
Signal Processing - September 2016 - 147
Signal Processing - September 2016 - 148
Signal Processing - September 2016 - 149
Signal Processing - September 2016 - 150
Signal Processing - September 2016 - 151
Signal Processing - September 2016 - 152
Signal Processing - September 2016 - 153
Signal Processing - September 2016 - 154
Signal Processing - September 2016 - 155
Signal Processing - September 2016 - 156
Signal Processing - September 2016 - 157
Signal Processing - September 2016 - 158
Signal Processing - September 2016 - 159
Signal Processing - September 2016 - 160
Signal Processing - September 2016 - 161
Signal Processing - September 2016 - 162
Signal Processing - September 2016 - 163
Signal Processing - September 2016 - 164
Signal Processing - September 2016 - 165
Signal Processing - September 2016 - 166
Signal Processing - September 2016 - 167
Signal Processing - September 2016 - 168
Signal Processing - September 2016 - 169
Signal Processing - September 2016 - 170
Signal Processing - September 2016 - 171
Signal Processing - September 2016 - 172
Signal Processing - September 2016 - 173
Signal Processing - September 2016 - 174
Signal Processing - September 2016 - 175
Signal Processing - September 2016 - 176
Signal Processing - September 2016 - Cover3
Signal Processing - September 2016 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201809
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201807
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201805
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201803
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201801
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1117
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0917
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0717
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0517
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0317
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0117
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1116
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0916
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0716
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0516
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0316
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0116
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1115
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0915
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0715
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0515
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0315
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0115
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1114
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0914
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0714
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0514
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0314
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0114
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1113
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0913
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0713
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0513
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0313
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0113
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1112
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0912
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0712
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0512
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0312
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0112
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1111
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0911
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0711
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0511
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0311
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0111
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1110
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0910
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0710
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0510
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0310
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0110
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1109
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0909
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0709
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0509
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0309
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0109
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1108
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0908
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0708
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0508
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0308
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0108
https://www.nxtbookmedia.com