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Measuring Depth and Velocity withDefocus and Differential Motion
Emma Alexander1, Qi Guo1, Sanjeev Koppal2, Steven Gortler1, Todd Zickler1 1 Harvard SEAS2 University of Florida
Focal Flow
Motivation Low power (mW) depth sensing
[Rubenstein et al. 14] [Ma et al. 13]
~200 mW ~20 mW
Contribution
Optical Flow
Depth & 3D Velocity
Focal Flow
Image Motion
[Photo: Tony Hisgett]
Idea Combine motion and defocus blur
Wid
e ap
ertu
reP
inho
le
[Photo: Lost and Taken]
Textured plane
Pinhole
Wide aperture(Thin-lens Model)
In-focus plane
Optical Flow
Residual
Derivation Gaussian blur reveals depth
=Z
Z − Zf(2k + rkr) ∗ P = v(Z, �X) m ∗ k ∗ P
Theorem The residual can be factored into scene information and an image convolution exactly when the blur is Gaussian and the operator is the Laplacian, i.e.
m ∗ k ∝ 2k + rkrm ∗ k ∝ 2k + rkrm ∗ k ∝ 2k + rkrm ∗ k ∝ 2k + rkrm ∗ k ∝ 2k + rkrm ∗ k ∝ 2k + rkrm ∗ k ∝ 2k + rkrm ∗ k ∝ 2k + rkr
m ∗ k ∝ 2k + rkrm ∗ k ∝ 2k + rkrm ∗ k ∝ 2k + rkr? ×
Filter
R(Z, �X, filter k, pinhole image P ) ∝ m ∗ I
m ∗ k ∝ 2k + rkr
wmk = −rkr
k ∝ e−w
∫r
0m(s)s
ds
m ∝ rn
n ∈ {2, 4, 6, ...}
n = 2
Texture independence
Fourier transform
Solve differential equation
Compact operator
Nonnegative transmittance
All kernels from same filter
Inverse Fourier Transform
True depth (mm)250 350 450 550 650 750
Est
imat
ed d
epth
(m
m)
250
350
450
550
650
750
Proof of Concept
x
yTrue
dep
th (
mm
)
x
y
Est
imat
ed d
epth
(m
m)
x
y
Est
imat
ed d
epth
(m
m)
x
y
10mm
10mm
x
y
x
y
Input Image True Depth Result Sample PSF
Experimental results
[Photo: Thorlabs]True depth (mm)
Est
imat
ed s
peed
(m
m/fr
ame)
250 350 450 550 650 7500
0.2
0.4
0.6
0.8
1
1.2
Code, equipment, results: https://vision.seas.harvard.edu/focalflow