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10.3 Optical Microangiography (OMAG) 225
iFFT
(
H
−
(
z, n
))
=
A(z)e
−i
(
k
0
z+2πu
0
t
)
2
(10.33)
Figure 10.16d shows the inverse Fourier transformed signal. The absolute ampli-
tude of the flow signal obtained by taking the FT can be expressed as
A(z) = 2
I fft[H
(
z, n
)
]
(10.34)
From Eq. (10.34), it is evident that the imaging contrast is a result that represents
the optical reflectance scattered from the flow particles and that it does not relate
to the flow velocity. The amplitude of the flow signal A(z) can be used to reconstruct
the flow, just like that in the conventional OCT.
10.3.3.1.5 Directional Flow Imaging
Based on the above signal processing, the real flow signal is transformed to
an analytical signal by Hilbert transform, which enables the bidirectional flow
configuration, that is, the positive flow and the negative flow to be separated in
different imaging plane. Figure 10.17 shows the Hilbert-transformed analytical
flow signal, which is obtained by filtering out the negative frequency components
and the structural signals and then taking the inverse FT as shown below [62]
iFFT[H
(
z, n
)
] =
A(z)e
ik
0
z
2
(
cos
(
2π
|
u
0
|
t
)
± i sin
(
2π
|
u
0
|
t
))
(10.35)
where u
0
is the modulation frequency, which relates to the flow direction. If u
0
>
0,
the Fourier transformation along the depth direction resolves the positive flow
(a) Strctural image (b) Doppler image
(c) Positive flow (d) Ne
g
ative flow
2.5
2
1.5
1
−0.8
−1.2
−1.6
−0.8
−1.2
−1.6
1
−1
3
−3
Figure 10.17 Bidirectional image with OMAG: (a) the struc-
ture image; (b) the Doppler image, left v
0
= 7.5mms
−1
(red color), middle v
1
=−7.5mms
−1
(blue color), right
v
2
= 4.5mms
−1
(orange color); (c) the positive flow image;
and (d) the negative flow image.
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226 10 Optical Microangiography: Theory and Application
components in the positive image plane, and if u
0
< 0, it resolves the negative flow
components in the negative imaging plane.
10.3.3.1.6 Motion Compensation in OMAG
The sample movement during in vivo imaging conditions is an inevitable artifact
in OMAG and which is particularly very dominant for imaging of the human eye
or skin. The relative sample motion represents as a Doppler shift in the light
frequency between the light it emits and that it receives. This shift is imparted
when the light strikes an object that is moving relative to the probe. For the
OMAG system, the Doppler frequency due to the sample motion is depicted as an
additional modulation frequency, f
O
, in the interferogram. Therefore, the decisive
frequency mixing in the interferogram, as outlined in the previous section, will
now become f
M
− f
D
+ f
O
, which determines whether the Hilbert transform of
Eq. (10.18) is turned into Eq. (10.19) or (10.20). In addition, the sample movement
is directional when the OMAG system sees it and it is not predictable. As such,
if f
O
is larger than f
M
, the structural signals will appear in the flow image plane,
giving severe artifacts on the blood flow measurement. As an example to illustrate
the effect of subject motion on the OMAG results, Figure 10.18 shows a typical
in vivo B-scan of the posterior part of the human eye from a volunteer. Figure 10.18a
is the conventional OCT image in which the important physiological layers, such
as the retina and choroids, can be delineated. Figure 10.18b is the corresponding
OMAG flow image in which it can be seen that although the blood flow signals
from retinal and choroidal layers are evident, the subject movement deteriorates
the blood flow image, particularly in the right part of the image. Further analysis
from Figure 10.18b implies that the motion is not uniform, even during one single
(a) (b)
(c) (d)
Retina
Choroids
Sclera
Figure 10.18 Typical in vivo B-scan of
posterior part of the human eye. (a) Conven-
tional OCT/OMAG structural image, where
the retina and choroidal layers are demar-
cated, and the corresponding (b) OMAG
flow image, (c) PRDOCT image with gray
levels coded from −π (black) to π (white),
and (d) power Doppler OCT image. Scale
bar = 500 µm. (Source: Images are repro-
duced from Ref. [67].)
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10.3 Optical Microangiography (OMAG) 227
B-scan (within 50 ms). This nonuniform subject movement can also be seen in the
PRDOCT image as shown in Figure 10.18c,d.
A casual inspection of Figure 10.18 shows that the blood flow imaging is better
delivered by OMAG than that by PRDOCT, because the blood flow signals as
seen in the choroidal layer in Figure 10.18b are not seen in PRDOCT images
(Figure 10.18c,d), as illustrated by the arrows. However, the subject movement
artifacts are still present. Thus, for in vivo OMAG imaging of the human eye,
a solution to the bulk motion artifacts is required so that the quality of OMAG
blood flow image is improved. Because OMAG imaging has its origin in FDOCT,
the OCT signals can be directly calculated from the captured interferograms for
sequential A-scans using conventional FDOCT algorithms. It is known that the
OCT signal so computed is a complex function,
˜
B
OCT
(t, z), which can be written as
˜
B
OCT
(t, z) = A(t,z)exp[iφ(t,z)] (10.36)
where A(t, z) and φ(t, z) are the amplitude and phase of
˜
B
OCT
(t, z), respectively. t
is the time when the probe beam proceeds in the B-scan, and z represents the
coordinate along the imaging depth. In our OMAG system, the phase term φ(t, z)
can be expressed by the following equation:
φ(t, z) = φ
P
(t, z) + φ
B
(t, z) + φ
M
(t, z) (10.37)
where φ
P
(t, z) is the phase term caused by the moving red blood cells, φ
B
(t, z)is
the phase term caused by the subject movement, and φ
M
(t, z) is the phase term
caused by the modulation frequency, f
M
. Following PRDOCT [49, 50], the phase
differences between the ith and (i – 1)th A-lines can be written as
φ
i
(t, z) = φ
i
P
(t, z) + φ
i
B
(t, z) + φ
i
M
(t, z) (10.38)
where φ
i
P
(t, z) and φ
i
B
(t, z) are the phase changes caused by the moving particle
and the subject movement between the ith and (i – 1)th A-lines, respectively.
Because in OMAG system, the introduced modulation frequency, f
M
, is constant
during imaging, φ
i
M
(t, z) is a constant value that is known a priori,thatis
φ
i
M
(t, z) = 2πf
M
T (10.39)
where T is the time interval between adjacent A-lines. This phase term can be
directly subtracted from Eq. (10.38). Thus, the phase term directly related to the
movement, φ
i
m
(t, z)is
φ
i
m
(t, z) = φ
i
P
(t, z) + φ
i
B
(t, z) (10.40)
In Eq. (10.40), the variable t is the timing of the probe beam proceeding in a B-scan,
which corresponds to the position of the probe beam, x. Thus Eq. (10.8) can be
replaced by
φ
i
m
(x, z) = φ
i
P
(x, z) + φ
i
B
(x, z) (10.41)
Because the OMAG system used in this study has an imaging speed of 20 000
A-scans per second, it is reasonable to assume that the subject is static within
a single A-scan. Under this condition, Eq. (10.41) represents the bulk motion of
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228 10 Optical Microangiography: Theory and Application
subject between adjacent A-scans, plus the phase changes due to the blood flow,
if presented. Usually, within a single B-scan, the regions occupied by the blood
vessels are much smaller than the areas occupied by the surrounding tissue. Thus,
the phase change between the ith and (i – 1)th A-scans caused by the bulk motion
could be approximated by averaging all the phase differences along the A-scan
Df
i
m
(x) =
1
z
f
i
m
(x, z)dz (10.42)
to minimize the errors brought by the system noise and flow signals [13, 31]. Here
φ
i
m
(x) is the bulk motion corresponding to the phase difference between ith and
(i – 1)th A-scans. Therefore, with reference to the first A-scan in the B-scan, the
phases at the ith A-line caused by the bulk subject motion, φ
i
m
(x), can be obtained
by a recursive function
φ
1
m
(0) = 0,φ
i
m
(x) = φ
i−1
m
(x) + φ
i
m
(x), i = 2, 3, ..., N (10.43)
where N is the number of A-scans that composes the B-scan. Thus far, the phase
changes along the B-scan due to the bulk tissue motion, φ
i
B
(x, z), have been
computed. Subtracting Eq. (10.43) from Eq. (10.37) and then inserting into Eq.
(10.36), we have
˜
B
OCT
(x, z) = A(x, z)exp[i(φ
P
(x, z) + φ
M
(x, z))] (10.44)
where the frequency modulation term due to the bulk tissue motion is eliminated.
Subsequently, the spectral interferograms without the tissue motion effect can be
obtained via an inverse FFT to Eq. (10.44)
b(x, λ) = FFT
−1
˜
B
(x, z)
(10.45)
Finally, the conventional OMAG algorithm as detailed in the previous section is
applied to Eq. (10.45) to obtain the OMAG flow image that would be ideally free of
motion artifacts. In Eq. (10.42), the phase due to the bulk tissue motion is estimated
by the averaged value of phase changes along the A-scan. However, the phases are
random for the OCT signals that fall below the system noise floor. This will have a
dramatic effect on the estimation of bulk tissue motion. To reduce this effect, we
used the OCT structural image to segment the regions of interest, R(x,z), within
which the OCT signals are 15 dB above the system noise level. Only the phases
within the region of interest are calculated. Below, we detail the steps to perform
the segmentation to find R(x,z).
First, the SD-OCT structural image was used to segment the regions of interest
that represent retina and choroids. Two boundaries were determined. One cor-
responded to the vitreoretinal interface, that is, anterior boundary, and the other
to the deepest penetration points, posterior boundary. The anterior boundary was
obtained by using an algorithm that is similar to that described by Mujat et al. [68].
First, we used a Gaussian filter with a standard deviation of 4 × 4pixelstoblurthe
cross-sectional image (B-scan) obtained previously. Then the edge of the image was
obtained by calculating the gradient of the blurred structural image. It was then con-
verted to a binary image by setting a threshold. The first nonzero points of the binary
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10.3 Optical Microangiography (OMAG) 229
image, represented by a curve, were detected, and the speckle noises were elimi-
nated. The anterior boundary was then obtained by fitting and smoothingthis curve.
The posterior boundary was obtained by setting a threshold to the blurred retina
structural image, which was equal to the noise floor of the image obtained. The
intensity was treated as the signal if it was larger than the threshold and was set to
unity; on the other hand, it was treated as noise if it was smaller than the threshold
and was set to zero. Using this method, the structure image was converted to a
binary image. The posterior boundary was obtained by the last nonzero points
after the isolated speckle noises were eliminated. The final posterior boundary
was then obtained by a fitting and smoothing procedure. Finally, the region of
region R(x,z) was obtained by bounding the anterior and posterior boundaries as
described above. The phase estimation in Eq. (10.42) is therefore performed within
this region of interest
φ
i
m
(x) ≈
1
z
φ
i
m
(x, z)dz
|
z ∈ R
(
x, z
)
(10.46)
To further improve the accuracy of the phase estimation, the phases in Eq. (10.46)
are only calculated for the OCT signals that are stronger than a preset threshold,
which is 15 dB above the system noise floor. Figure 10.19 illustrates the flow chart
Spectral
interferograms
b(t,l)
FDOCT
Structural
OCT image
Complex OCT
signals, B(t,z )
Phases f(t,z)
Structral region
of interests, R(x,z)
PRODT
Segmentation
Eq.(11)
FFT
−1
OMAG
algorithm
Phase
differences
∆f(t, z)
Bulk motion
estimates
∆f
i
(x)
Corrected
interferograms
Corrected
OCT
signals
Recursive
function, f
i
(x)
OMAG Flow
image
Figure 10.19 Flow chart detailing the steps to computation-
ally compensate the bulk tissue motion artifacts in OMAG
imaging of blood perfusion in retina and choroids. (Source:
Image is reproduced from Ref. [67].)
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230 10 Optical Microangiography: Theory and Application
300
Phase (rad)
X-position (mm)
250
200
150
100
50
0
0 0.5 1.51 2 2.5
(a) (b)
(c) (d)
Figure 10.20 Final results after applying
the motion compensation method to dig-
itally stabilize the B-scan as detailed in
Figure 10.3. (a) Accumulated changes along
the B-scan from Eq. (10.42), (b) the result-
ing OMAG blood flow image after motion
compensation. Also shown are (c) PRDOCT
blood flow image and (d) Power Doppler
blood flow image after motion compensa-
tion for comparison. Scale bar = 500 µm.
(Source: Images are reproduced from
Ref. [67].)
of the operations that are used to calculate the final OMAG flow images with the
subject motion artifacts minimized.
To test the algorithm, we used the raw interferogram data for Figure 10.18.
Figure 10.20 show the results after the bulk tissue motion was compensated to
stabilize the B-scan image. The phase accumulation due to the bulk tissue motion
amounted to ∼250 rad within the single B-scan (Figure 10.20a). This phase change
along the B-scan was used to compensate the spectral interferograms to obtain
the final OMAG blood flow image as shown in Figure 10.20b. Compared with
Figure 10.18b, the OMAG flow image after phase compensation is almost free of
motion artifacts as shown in Figure 10.20b. Such results show that the algorithm
described above works well. For comparison, we also applied the compensation
method to compensate the phases in the OCT signals to obtain the final PRDOCT
flow images. The results are given in Figure 10.20c,d for PRDOCT power DOCT
images, respectively. Comparing OMAG and PRDOCT blood flow images, the
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10.3 Optical Microangiography (OMAG) 231
former delivers superior imaging performance, as evidenced by the fact that (i) the
blood signals circled in Figure 10.20b cannot be easily identified in Figure 10.20c,d
and (ii) the flow signals within choroidal layer are abundant in Figure 10.20b which
are, however, almost absent in PRDOCT images.
10.3.4
Doppler OMAG
The key advantage of OMAG is that only the signals backscattered by the moving
blood appear in the OMAG flow output plane; this makes the blood flow imaging
almost free of artifact-induced noise. However, the OMAG is incapable of providing
flow velocity information like PRDOCT. This is because in the OMAG flow image,
the regions that are occupied by the microstructural signals are rejected by OMAG;
thus the required correlation between the adjacent A-scans within the static tissue
regions are totally lost, leading to a rather noisy appearance in the output plane
of flow velocity. To overcome this problem, we digitally reconstruct an ideal static
background tissue that is totally optically homogeneous to replace the real hetero-
geneous tissue sample in OMAG. This ideal background tissue provides a constant
background signal that makes the adjacent A-scans totally correlated, leading to a
dramatic increase of the phase SNR for the phase-resolved signals that represent
flow velocities. This method inherits the advantages of both OMAG and PRDOCT.
The spectral interference signal captured by each pixel of the CCD camera in
OMAG/SD-OCT is essentially the same, except the wavelength, λ. We assume the
wave numbers of the broadband light source are from k
0
to k
0
+, where k
0
= 2π/λ
0
,
and these wave numbers cover 1024 pixels of the line scan camera. Consequently,
the camera records the spectral interference fringe signal formed between the
reference light and the light backscattered from within sample, which can be
written as a function of k
j
I
k
j
= S
k
j
E
R
exp
i2k
j
r
+
∞
−∞
a(z)exp
i2k
j
[r + nz]
dz
2
j = 1, 2, 3, ..., 1024 (10.47)
where i = – 1, <> is the time average, k
j
is the wave number of the light captured
by the jth detector (pixel) of the CCD camera, I(k
j
) is the light intensity captured by
the jth detector, S(k
j
) is the spectral density of the light source at k
j
, r is the optical
path length for the light traveled in the reference arm, n is the refractive index of
the sample, and a(z) is the magnitude of the light backscattered at depth z.Inour
system, each B-scan contains 1500 A-lines and covers about 2.5 mm in the lateral
direction. Thus the signal captured by the jth pixel in each B-scan can be written as
a function of time variable t that relates to the position of focus beam spot on the
sample, steered by the x-scanning mirror. Thus
I
k
j
, t
= S
k
j
E
R
exp
i2k
j
r
+
∞
−∞
a(z)exp{i2k
j
[r + nz]}dz
2
j = 1, 2, 3, ..., 1024 (10.48)
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232 10 Optical Microangiography: Theory and Application
Because the light backscattered from the sample is quite weak compared to the
light reflected from the RM, we do not consider the self-cross-correlation between
the light backscattered from different positions within the sample. We also do not
consider the DC signals because they do not contribute to useful OMAG signals.
Therefore, Eq. (10.48) can be written as
I
k
j
, t
= 2S
k
j
E
R
∞
−∞
a(z, t)cos
2k
j
nz
dz (10.49)
It is clear that Eq. (10.48) is constant if the sample is totally optically homogeneous,
which means that a(z, t) and n do not vary within the entire sample. If this
is the case, then the spatial frequency components of the sample in the lateral
direction presented by Eq. (10.49) will be a delta function, shown as a red arrow in
Figure 10.21a. However, in real situations, our imaging sample is often optically
heterogeneous, which means that a(z, t) and n are functions of time variable t.
Thus, Eq. (10.49) can be expressed as
I
k
j
, t
= 2S
k
j
E
R
+∞
−∞
a
(
z
1
)
cos
2k
j
n(z, t)zdz
(10.50)
As a consequence, Eq. (10.50) is no longer constant. The intensity captured by the
CCD camera will be modulated by the heterogeneous properties of the sample
along each B-scan. The spatial frequency components of a static tissue sample,
which we call the heterogeneous frequencies, will exhibit as a randomly distributed
function around zero frequency with a bandwidth of BW, as shown in red curve in
Figure 10.21b [70].
When there is a patent blood vessel buried within motionless tissue at position
(z
1
,t
1
), we assume that the blood cells (scattering particles) within the vessel move
toward the incident beam at a velocity v. The frequency of the light backscattered
from these blood cells will be modulated by its velocity. Then, Eq. (10.50) can be
expressed as
I
k
j
, t
= 2S
k
j
E
R
∞
−∞
a(z, t)cos(2k
j
n(t)z)dz
+a(z
1
, t
1
) cos[2k
j
n(t)(z
1
− vt)]
(10.51)
Magnitude
Frequency of
static tissue
Magnitude Magnitude
Heterogeneous
frequency
Heterogeneous
frequency
Doppler beatin
g
frequency
f
c
−f
c
(a) (b) (c)
−BW/2 −BW/2
BW/2 BW/2
ƒ(Hz)
ƒ(Hz)
ƒ(Hz)
Figure 10.21 Diagram of frequency com-
ponents for different tissue samples: (a)
an ideal tissue sample (optically homoge-
neous sample) with no moving particles, (b)
a real tissue sample (optically heterogeneous
sample) with no moving particles, and (c) a
real tissue sample (optically heterogeneous
sample) with moving particles. (Source:
Images are reproduced from Ref. [69].)
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10.3 Optical Microangiography (OMAG) 233
Here, we also do not consider the self cross-correlation signal from within the
sample. The first term on the right side of Eq. (10.51) represents backscattering
signals from a static sample with reflectivity of a(z,t), while the second term
represents backscattering from the moving particles with reflectivity of a(z
1
,t
1
) with
a velocity of v at position a(z
1
,t
1
). Moving particles produce a frequency shift caused
by the Doppler effect of the moving particles. This is illustrated in Figure 10.21c;
the blue curve is the Doppler beat frequency part. Recall that the phase-resolved
technique requires a correlation between adjacent A-scans to determine v of the
moving particles [49], and the correlation, in conventional PRDOCT, provided
by the first term on the right side of Eq. (10.51) is also required to suppress
the noise signals in the nonflow regions in order to increase the flow imaging
contrast. Because of the optical heterogeneity n(z,t) of a tissue, n(z,t) imposes a
noise background onto the blood flow signals, making it difficult for PRDOCT to
measure precisely the blood flow velocity, particularly in capillaries [60, 65, 67]. In
contrast, OMAG eliminates the first term on the right side of Eq. (10.51) in order
to image blood flow. This elimination renders the noise production due to n(z,t)to
a minimum, but unfortunately, results in OMAG losing its correlation condition
between adjacent A-scans for the heterogeneous tissue regions. Consequently, the
phase-resolved technique cannot be directly applied to OMAG blood flow signals.
To solve this problem, our proposed strategy is to digitally re-construct an ideal
sample background with a constant backscattering coefficient a
0
and a refractive
index n
0
throughout the sample – thus creating a totally homogeneous sample that
reinforces a complete correlation among OMAG A-scan signals.
I
0
k
j
, t
= 2S
k
j
E
R
∞
−∞
a
0
(z, t)cos
2k
j
n
0
(z, t)z
dz
(10.52)
where a
0
(z, t) ≡ a
0
and n
0
(z, t) ≡ n
0
throughout the scanned tissue sample. In
condition, a
0
= 10
−6
, and n
0
= 1.35 when we construct the homogeneous tissue
background using Eq. (10.52). These values were taken according to the typical
optical properties of biologicaltissues, that is, theaverage reflectivity is between10
−4
and 10
−7
and the average refractive index is 1.35 [71]. The digitally reconstructed
homogeneous tissue sample subsequently replaces the first term on the right side
of Eq. (10.51). In doing so, OMAG blood flow signal now becomes
I
k
j
, t
= 2S
k
j
E
R
∞
−∞
a
0
(z, t)cos
2k
j
n
0
z
dz
+ E
R
a
(
z
1
, t
1
)
cos
2k
j
n
(
z
1
, t
1
)(
z
1
− vt
)
(10.53)
The construction of the ideal tissue sample does not affect OMAG signals of blood
flow because it only replaces the tissue background signals with a homogeneous
background without affecting the blood flow signals within the B-scan. If we treat
time variable t as a constant and apply a FT on wavelength k, we can obtain
˜
I(z, t) = FT
−1
I
k
j
, t
k = A(z, t)exp[iφ(z, t)] (10.54)
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234 10 Optical Microangiography: Theory and Application
where ϕ(z, t) is the phase of the analytic signal. The phase difference between
adjacent A-scans, n and n– 1, is then evaluated
φ(z, t) = tan
−1
Im
˜
I
(
z, t
n
)
·
˜
I
(
z, t
n−1
)
Re
˜
I
(
z, t
n
)
·
˜
I
(
z, t
n−1
)
(10.55)
On the basis of the linear relationship between phase difference between adjacent
A-lines and velocity [4, 5], the velocity of flow signal imaged by OMAG can be
directly written as
ν(z, t) =
λφ(z, t)
4πt
(10.56)
where v(z, t) is the flow velocity at depth z, t is the time interval between adjacent
A-lines. Note that there is a small constant offset induced by a
0
in Eq. (10.53), which
may perturb the evaluated ϕ(z, t). However, the small offset is usually at least 2
orders of magnitude smaller than the OMAG flow signals, leading to a negligible
effect on the final evaluated ϕ(z, t).
10.3.4.1 Signal Processing
The flow chart of the processing procedure is illustrated in Figure 10.22. It is
because the image data was achieved by scanning, the variable t present in the
theory section could be directly replaced by the position of the x-direction. In this
case, we use x to replace t for better illustration of the cross-sectional image. The
raw fringe signal, I(k,x), captured by our system was first separated into two parts
based on the heterodyne technique by a carefully selected frequency, f
c
, along lateral
direction, which depends on the characteristic of the sample and system setup.
High pass
filtering
at each
wavelength
Low pass
filter at each
wavelength
Back ground
reconstruction
algorithm
Frequency
synthesizing
Phase-resolved
optical doppler
tomography
i = 1
j = 1
N
k
i
x
j
M
l(k,x)
i = 1
j = 1
N
k
i
x
j
M
l
S
(k,x)
i = 1
j = 1
N
k
i
x
j
M
l
ideal
(k,x)
i = 1
j = 1
N
k
i
x
j
M
l
syn
(k,x)
z
i
0
j = 1
N/2
x
j
M
Dj(z,x)
i = 1
j = 1
N
k
i
x
j
M
l
f
(k,x)
Figure 10.22 Flow chart showing the steps
for DOMAG to evaluate the velocities of
blood flow from a B-scan data set, I(k,t). The
data coordinates are indicated in the lower
right corner of each data block, where t is
the time variable of probe beam scanning
over a sample, k is the wave number, f is
the spatial frequency, and z is the imaging
depth. FT|t represents the Fourier trans-
form (FT) against the time variable t in the
B-scan, FT −1|f indicates the inverse FT
against the spatial frequency, f,andFT|k is
FT against the wave number k.
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