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POCS reconstruction of stereoscopic views
to signals defined over irregular sampling grids.

In such cases, Shannon’s theory is not applica-
This paper presents an application of POCS (pro-
ble and alternative methods must be found to
jection onto convex sets) methodology to the re-
reconstruct or approximate the original contin-
construction of intermediate stereoscopic views.

The basic problem in such a reconstruction, re-
Although some results on the reconstruction
sulting from disparity compensation, is that of
of band-limited functions from irregularly-spaced
the recovery of a regularly-sampled image from
samples are available (e.g., [1]), their practical
its irregularly-spaced samples. This problem also
usefulness is limited; theoretical constraints on
arises in other image processing and coding ap-
the maximum spacing of irregular samples under
plications. The results reported here improve
the perfect reconstruction condition cannot be
our previous POCS-based reconstruction method
satisfied by arbitrarily-distributed image sam-
by locally adapting the algorithm to the density
ples after disparity or motion compensation. By
of image samples. We also extend the method
relaxing the perfect reconstruction condition, al-
to color images by implementing the method in
ternative methods have been proposed such as
the luminance-chrominance (Y -U -V ) space.

the polynomial interpolation or iterative recon-struction [2].

In this paper, we extend a reconstruction
method based on projections onto convex sets
In order to compute intermediate views in a
(POCS) [3] proposed by us earlier [4]. We im-
stereoscopic or multiview representation of a 3-
prove the convergence of reconstruction by lo-
D scene, the usual problem is that of recovering
cally adapting the algorithm to the density of
regularly-spaced images samples (intensity and
image samples. We also extend the method to
color) based on irregularly-spaced samples. This
color images by implementing it in the luminance-
is due to disparity compensation and is similar
to motion compensation used for temporal in-terpolation of image sequences.

The interpolation of either regularly- or irre-
gularly-spaced samples based on the knowledgeof a regularly-sampled image has been exten-
Let g = {g(x), x = (x, y)T ∈ R2} be a con-
sively treated in the literature and has found
tinuous 2-D projection of the 3-D world onto
numerous practical applications in image pro-
an image plane and let gΛ = {g(x), x ∈ Λ} be
cessing and coding. The case of computation of
a discrete image obtained from g by sampling
regularly-spaced samples based on regular ones
over a lattice Λ [5]. Let’s assume that g is band-
has been explored to a lesser degree. The pri-
limited, i.e., G(f ) = F{g}=0 for f ∈ Ω where
mary reason for this are difficulties associated
F is the Fourier transform, f = (f1, f2)T is a
with the extension of Shannon’s sampling theory
frequency vector and Ω ⊂ R2 is the spectralsupport of g. If the lattice Λ satisfies the multi-
This work was carried out with partial support from
dimensional Nyquist criterion [5], the Shannon
the Natural Sciences and Engineering Research Councilof Canada under Strategic Grant STR224122 while the
sampling theory allows to perfectly reconstruct
g from gΛ. However, in the case of irregular sam-
pling the theory is not applicable. Therefore,
there is little flexibility in shaping the spectrum
the general goal is to develop a method for the
of gk ; any practical lowpass filtering on Λ must
reconstruction of g from an irregular set of sam-
suppress high frequencies since a slow roll-off
ples gΨ = {g(xi), xi ∈ Ψ ⊂ R2, i = 1, ., K},
transition band must be used to minimize ring-
where Ψ is an irregular sampling grid.

ing on sharp luminance/color transitions. Sec-ondly, the interpolation operator IΨ/Λ, espe-
cially the one based on triangulation (better per-formance), is involved computationally.

We use the POCS methodology [3] to recon-
In our earlier work [4] an alternative imple-
struct image g. This methodology involves a
mentation of the algorithm (1) has been pro-
set theoretic formulation, i.e., finding a solution
posed. Since our goal is the reconstruction of
as an intersection of property sets rather than
image samples obtained from motion or dispar-
by a minimization of a cost function. We use
ity compensation, a 1/2-, 1/4- or 1/8-pixel pre-
cision of motion or disparity vectors is usually
sufficient. Therefore, it has been proposed to
0 - set of all images g such that at xi ∈
Ψ, i = 1, ., K (irregular sampling grid)
implement (1) on an oversampled grid matching
• A1 - set of all band-limited images g, i.e.,
where B is implemented on ΛP , that is a P ×P -
If the membership in A0 can be assured by a
times denser (oversampled) lattice than Λ, and
sample replacement operator R (to enforce proper
P equals 2, 4, or 8 depending on motion/disparity
image values on Ψ), and the membership in A1
vector precision. Clearly, Λ is a sub-grid of ΛP ,
– by suitable bandwidth limitation (low-pass fil-
i.e., x ∈ Λ ⇒ x ∈ ΛP . gΨ/Λ is the nearest-
tering) B, then the iterative reconstruction al-
neighbor interpolation of gΨ on ΛP , defined at
where gk is the reconstructed image after k it-
the nearest-neighbor sampling, i.e., sampling on
tracts image values (luminance/color) on the ir-
regular grid Ψ. Note that equation (1), pro-
words, the implementation (3) is performed on a
posed in [6], results in an approximation rather
lar samples from Ψ are quantized to the nearest
low-pass filtering. In order to implement equa-
tion (1) on a computer, a suitable discretization
must to be applied. In [6], equation (1) was im-
tion that a suitable value of P is selected.

2.2. Adaptation of the relaxation coeffi-cient
where the lowpass filtering B is implementedover Λ and α is a parameter that allows control
The choice of the relaxation coefficient α in equa-
of convergence and stability of the algorithm.

tion (3) has a direct impact on the convergence
The symbol gk denotes a bilinearly-interpolated
properties of the algorithm; the greater the α,
image gk needed to recover image samples on Ψ.

the faster the convergence, but only up to some
Also, note that an interpolation function IΨ/Λ
αmax above which the algorithm becomes unsta-
replaces the sampling operator SΨ. This func-
ble. Experiments have shown that the value of
tion interpolates image samples (gΨ − ˜
max in (3) is closely related to the properties of
fined on Ψ in order to recover samples on Λ.

the irregular sampling grid. Namely, the algo-
Sauer and Allebach have studied three interpo-
rithm has been most prone to instability in im-
lators IΨ/Λ: one derived from bilinear interpola-
age regions where irregular sampling grid is the
tion and two based on triangulation with planar
densest. Clearly, when increasing α above αmax,
facets [6]. The implementation (2) of the re-
the algorithm starts to diverge in those image re-
construction algorithm (1) suffers from two de-
gions where the number of irregular samples per
ficiencies. First, by processing all images on Λ
area is the highest. That is why it is proposed
to introduce an additional α-correcting term in
Ω, the fewer the samples of Ek(f ) thatneed to be computed. This allows signif-
Ψ are samples of a function describing lo-
cal density of irregular grid. We expect that al-
gorithm implementations based on (4) will allow
vergence than those based on formulation (3).

tion by zeroing parts of the spectrum leads
in spatial domain to oscillations at sharp
only marginally dependent on the degree of vari-
luminance/color transitions. In [4], a de-
ation in the local densities of irregular grids.

tailed discussion of the design of lowpass
To be a good descriptor of local grid den-
filters that minimize these effects can be
sity, the function d should equal 1 where the
grid is regular, should be greater than 1 in areas
where there are more samples of irregular gridthan those of regular one, and less than 1 when
converse is true. Experiments show that the ac-tual definition of the function is not critical; var-
This operation can be very efficiently im-
ious functions d seem to work almost equally
well. It has been decided that the d function is
computed by counting occurrences of irregularsamples in a 1 × 1 square neighborhood of each
node of the regular grid, and then by filteringthe results by a 5 × 5 separable smoothing fil-
The proposed reconstruction algorithm has been
ter. We obtain in this way a regularly-sampled
tested experimentally on images with various ir-
function d; as samples on the irregular grid d
regular sampling grids Ψ. In our previous work
the nearest-neighbor samples of d (on the regu-
[4], we presented results for both synthetic and
natural disparity fields. Since the local sampledensity is less predictable in the case of natu-ral disparity fields here we are comparing the
new algorithm with the previous, non-adaptive
The implementation of equation (4) would, in
one on natural data only. The disparities were
general, require more memory and be less effi-
computed from an ITU-R 601 stereopair Flow-
cient than that of equation (2), however we opt
erpot using an optical flow-type algorithm [8],
for an implementation in the frequency domain
and subsequently used in disparity compensa-
in order to reduce the computational complex-
tion to obtain the irregular grid Ψ. Then, the
luminance and color of gΨ were computed us-ing bicubic interpolation [9]. Using gΨ, gΛ was
reconstructed and compared with the original
be the reconstruction error defined on Λ
image. We tested the algorithm for P =4, 8 and
by the inverse of the irregular grid density func-
16 and various α’s. We used lowpass filters pro-
tion. Then, each iteration of the reconstruction
posed earlier in [4] since they give a good com-
promise between detail loss and aliasing.

Fig. 1(a) shows the PSNR evolution for lumi-
1. Fourier transform of the error ek sampled
nance reconstruction error with fixed and adap-
tive α. Note that the experimentally optimizedfixed α (highest stable value) for the luminance
component was 0.4 for P =4, 0.3 for P =8 and
0.2 for P =16, while it was 0.7 for all P ’s in
the adaptive case. Clearly, the convergence in
the adaptive case is faster and the steady-statePSNR is higher; the benefits of α adaptation are
chrominance errors. Note that similarly to the
luminance case the higher the P , the better the
performance of the algorithm, although the higherthe computational complexity due to the higheroversampling rate. While the increase of over-
sampling from P =4 to P =8 shows up to a 1dB
PSNR gain, a similar increase from P =8 to P =16
Subjectively, the reconstructed images were
of very high quality. This suggests the viabil-
ity of the proposed algorithm for various high-
quality reconstructions in image processing and
ory and practice of irregular sampling,”in Wavelet: Mathematics and Applications,
J. Benedetto and M. Frazier, Eds., chapter 8,pp. 305–363. CRC Press, Boca Raton FL,
[2] A. Sharaf and F. Marvasti, “Motion com-
pensation using spatial transformations with
forward mapping,” Signal Process., ImageCommun., vol. 14, pp. 209–227, 1999.

[3] P. Combettes, “The foundations of set the-
oretic estimation,” Proc. IEEE, vol. 81, no.

based image reconstruction from irregularly-
spaced samples,” in Proc. IEEE Int. Conf.

Image Processing, Sept. 2000, vol. II, pp.

[5] E. Dubois, “The sampling and reconstruc-
tion of time-varying imagery with applica-
tion in video systems,” Proc. IEEE, vol. 73,no. 4, pp. 502–522, Apr. 1985.

reconstruction of band-limited images fromnonuniformly spaced samples,” IEEE Trans.

Circuits Syst., vol. 34, no. 12, pp. 1497–1506,Dec. 1987.

proach,” in Signal Process. III: Theories and
Applications (Proc. Third European Signal
Process. Conf.), 1986, pp. 267–270.

[8] R. March, “Computation of stereo dispar-
ity using regularization,” Pattern Recognit.

Lett., vol. 8, pp. 181–187, Oct. 1988.

[9] R.G. Keys, “Cubic convolution interpolation
for digital image processing,” IEEE Trans.

Acoust. Speech Signal Process., vol. 29, no.

Figure 1: Evolution of PSNR of the reconstruc-

Source: http://iss.bu.edu/data/jkonrad/cpapers/Stas01icav3d.pdf

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Psychological Treatment for AdolescentDepression: Perspectives on the Past,Present, and FutureLouise Hayes,1,2 Patricia A. Bach3 and Candice P. Boyd4 1 School of Behavioural and Social Sciences and Humanities, University of Ballarat, Australia2 Ballarat Health Services, Child and Adolescent Mental Health Service, Ballarat, Australia3 Illinois Institute of Technology, Chicago, United States of Ame