Signal Processing - May 2017 - 63

and the square binary patterns, a superfast 3-D measurement system has been developed with a speed of 667 fps at
480 × 480 pixels [34]. Under such a high frame rate, the motion effect could be largely alleviated. Nevertheless, the square binary pattern has two main drawbacks. First, the depth measurement range
needs to be small to guarantee a proper degree of projector defocusing, since the stripes will not become sinusoidal when the degree of
defocusing is not proper. Second, when the stripes are wide, it requires a large degree of defocusing to reshape the square stripes to
the sinusoidal fringes, which may exceed the capability of the
projector lens.
To overcome the shortcomings of the square binary patterns, in [35] Wang and Zhang propose another kind of binary pattern by borrowing the idea of dithering, also known as
half toning, which has been extensively used in digital image
processing. The dithered binary pattern they use is shown in
Figure 8(b). Specifically, the desired sinusoidal fringes are
produced using the Bayer-ordered dithering technique or the
error-diffusion dithering technique. Different from the square
binary pattern, the dithered pattern appears sinusoidal even
before processing. Therefore, even when the fringes are very
wide, or the projector is nearly focused, the dithered binary
pattern can produce ideal sinusoidal fringes. In other words,
the dithered binary pattern is less dependent on the degree of
projector defocusing and the width of the fringes. The former
robustness extends the depth measurement range, while the latter robustness enables multifrequency phase shifting for absolute depth measurement. Using the dithered binary patterns,
a superfast two-frequency phase-shifting method is developed
for absolute 3-D measurement of live rabbit hearts at 800 fps
with a resolution of 576 × 576 pixels [36].
The aforementioned defocusing techniques generally
require much calibration effort, since most of the existing calibration techniques assume the projector to be in focus. Alternatively, Yang et al. propose density-modulated binary patterns

(a)

to carry the phase information [37]. Different from defocusing, the phase is implicitly represented by the 1-D density
variation of bright dots in the pattern. Figure 8(c) shows an
exemplar pattern with density variation in the vertical direction. Specifically, when denoting the pattern as P B (x, y), the
number of bright dots in different rows of P B (x, y) is defined
as a sinusoidal function
k (y) = Round $8sin ` 2r

y
+ i j + 1B # c + 1 .,
T

(12)

where Round () is a function to round a floating number to an
integer, T is the number of rows in a sinusoidal period, and c is a
scaling factor controlling k (y) as an integer from 1 to K that
determines the number of different densities in a period. The three
patterns for three-step phase shifting are generated by setting i to
-2r/3, 0, and 2r/3, respectively. A low-pass filter is then applied
to the captured images to extract the phase, which is accomplished
by calculating the average energy in a small sliding window. It is
verified that the energy images well approximate the phase-shifted
sinusoidal fringes and thus support high-accuracy depth measurement. Furthermore, an error correction method is proposed to
reduce the quantization errors introduced by approximating sinusoidal fringes with a limited number of densities.
Figure 9 shows the depth reconstruction results of some
indoor scenes in comparison with original phase shifting and
the Kinect. It can be seen that depth from the density-modulated
patterns is consistently better than that from the Kinect, in terms
of surface details and object boundaries. Compared with original phase shifting, some details are lost as binary patterns
cannot represent phase as perfectly as grayscale patterns.
However, a distinct property of the density-modulated binary
pattern is that it still preserves the locally unique distribution
of bright dots in the vertical direction. Therefore, using three
density-modulated binary patterns, the advantages of scalable
depth sensing in [32] and unambiguous 3-D measurement in

(b)

(c)

(d)

FIGURE 9. Depth reconstruction results for two indoor scenes: (a) the color image, (b) the Kinect results, (c) the results of density-modulated binary patterns, and (d) the results of original phase shifting.

IEEE SIgnal ProcESSIng MagazInE

May 2017

Table of Contents for the Digital Edition of Signal Processing - May 2017