3D Reconstruction of Human Face Based on Single or Several Images
Предложен новый подход к трехмерной реконструкции человеческого лица по одному или нескольким изображениям. Подход основан на генеративной модели человеческого лица. Также предложен новый способ построения генеративной модели по набору нетекстурированных поверхностей. A new approach to a human face...
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Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України
2011
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| Cite this: | 3D Reconstruction of Human Face Based on Single or Several Images / М. Тyshchenko // Управляющие системы и машины. — 2011. — № 2. — С. 79-85. — Бібліогр.: 8 назв. — англ. |
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| citation_txt | 3D Reconstruction of Human Face Based on Single or Several Images / М. Тyshchenko // Управляющие системы и машины. — 2011. — № 2. — С. 79-85. — Бібліогр.: 8 назв. — англ. |
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| description | Предложен новый подход к трехмерной реконструкции человеческого лица по одному или нескольким изображениям. Подход основан на генеративной модели человеческого лица. Также предложен новый способ построения генеративной модели по набору нетекстурированных поверхностей.
A new approach to a human face 3D reconstruction by single or several images is suggested. The approach is based on a morphable model of a human face. A new way to design the morphable model by a set of untextured 3D shapes is suggested as well.
Запропоновано новий підхід до тривимірної реконструкції людського обличчя за одним або кількома зображеннями. Підхід базується на генеративній моделі людського обличчя. Також запропоновано новий спосіб побудови генеративної моделі за набором нетекстурованих поверхонь.
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УСиМ, 2011, № 2 79
UDC 004.93’1
М. Тyshchenko
3D Reconstruction of Human Face Based on Single or Several Images
Предложен новый подход к трехмерной реконструкции человеческого лица по одному или нескольким изображениям. Подход
основан на генеративной модели человеческого лица. Также предложен новый способ построения генеративной модели по
набору нетекстурированных поверхностей.
A new approach to a human face 3D reconstruction by single or several images is suggested. The approach is based on a morphable
model of a human face. A new way to design the morphable model by a set of untextured 3D shapes is suggested as well.
Запропоновано новий підхід до тривимірної реконструкції людського обличчя за одним або кількома зображеннями. Підхід
базується на генеративній моделі людського обличчя. Також запропоновано новий спосіб побудови генеративної моделі за на-
бором нетекстурованих поверхонь.
Introduction1
A three-dimensional (3D) reconstruction of a sce-
ne based on its photographs forms a vast stratum
in contemporary computer vision. State of the art
provides no common technology capable of arbi-
trary object’s automatic 3D reconstruction by a set of
photographs taken from unknown viewpoints. A
sparse reconstruction of rigid scenes is well studied
so far. A software reconstructing camera trajecto-
ries and 3D model of a scene as a point cloud is
available [1], but the subsequent triangulation of
such models cannot be accurately done without
human interaction [2]. Moreover, it is not always
possible to achieve automatic reconstruction even
significantly reducing a class of objects to be
processed [3].
A 3D reconstruction of a human face occupies
a separate, rather vast, niche in the indicated area.
The research on this topic can be divided into two
large classes: geometry oriented and morphable
model oriented.
Geometry oriented approaches mostly rely on
establishing the correspondence between points in
different photographs of an object; surface recon-
struction is then performed by determining such
location of points in 3D space and such camera
parameters which fit in a best way to the point cor-
respondence. Such approaches require either sub-
stantial organizational development: special cam-
era positioning and tuning, special illumination
conditions, or significant amount of manual work:
Keywords: 3D reconstruction, morphable model, human
face, motion field, labeling problem.
indication of point correspondence, segmentation
of scene objects etc.
Morphable model based techniques utilize the
analysis-through-synthesis approach. A morphable
model of a human face can be depicted as a box
with tunable handles. Depending on positions of
those handles the box generates one or another pic-
ture of one or another human face. The problem is to
set handles of those box to such positions that it
would generate exactly the input image, or at least
the closest possible. A 3D shape of a face is uni-
quely determined by positions of those handles or,
speaking more precisely, by parameters of the mor-
phable model. An ability to reliably reconstruct a
3D shape of a human face even by a single photo-
graph is a considerable advantage of such ap-
proaches.
1. Prior works
V. Blanz and T. Vetter’s research [3] on design
and use of a morphable model of human face is
one of the most well known works in the area.
They have managed to design an accurate morpha-
ble model and implement an efficient scheme of
particular individual’s face 3D reconstruction ba-
sed on one or several images. Their approach is ba-
sed on a conjecture that a set of all possible 3D
shapes of a human face is convex in a sense. Loose-
ly speaking, by averaging two shapes of different
faces one gets a shape of some human face as well.
Under this assumption, one can design a basis out
of 3D shapes of several typical faces, and appro-
ximate a shape of an arbitrary face by convex com-
bination of those basis shapes. Then a problem of
human face 3D reconstruction based on its image
80 УСиМ, 2011, № 2
consists in search for such a convex combination
of basis shapes and such camera and illumination
parameters, under which a generated image is le-
ast different from an input. A problem statement
of such sort is a classical instance of the analysis
through a synthesis approach.
Nevertheless, a method suggested by V. Blanz
and T. Vetter contains a number of considerable
drawbacks. The first one consists in the fact that
for construction of a morphable model and for its
further service in 3D reconstruction technology,
the basis models have to be textured. The paper
does not contain any reference for the case of un-
textured models. Meanwhile, bases of untextured
models are much more available as compared to
textured ones. Thus, it is reasonable to design a
3D reconstruction technology which makes use of
just untextured models.
One more remarkable shortcoming of the tech-
nology is that for an accurate surface reconstruc-
tion one needs to specify initial approximation of
orientation and illumination rather precisely. This
drawback is vital for technology usage in fully auto-
matic systems where any human interaction is im-
possible in principle.
A number of research efforts, carried out mainly
by commercial companies, addressed to overcome
the aforementioned drawbacks of V. Blanz and
T. Vetter’s technology.
A method, suggested in [4], is based on the de-
tection of silhouettes on images of a face, and the
search for such a convex combination of basis
shapes which fits best to those silhouettes. The
invariance to illumination is an advantage of this
approach. But it is achieved by loss of consider-
able amount of information, so that an accurate
surface reconstruction becomes possible only with
very huge amount of images. Not to speak about
3D reconstruction of a face by a single image.
A method [5] is suggested for the reconstruc-
tion of a facial shape by series of slightly different
photographs. It is founded on feature point track-
ing, based on which a camera position for each ima-
ge is reconstructed and a rough approximation of
3D shape is built. Further, images are pairwise rec-
tified with respect to camera positions in order to
use a stereo reconstruction algorithm to refine a
model. The fact that the method works only with
those series of images which can be arranged in
such a way that each subsequent picture does not
differ much from the previous one is a substantial
drawback of the method, as well as its inapplica-
bility in the case of a single input image.
A method [6] is designed for 3D reconstruction
of a human face by two slightly different photo-
graphs and two video sequences. The method per-
forms a search for such a convex combination of
basis shapes, which fits best to location of some
feature points in a pair of input images. Video se-
quences are used for accurate model texturing.
The facts that the method works only for a pair of
slightly different images, is not applicable in the
case of single input image, and requires manual
indication of large number of feature points are
considerable shortcomings of the technique.
In the present paper we suggest a method for
3D reconstruction of a human face which does not
have the aforementioned drawbacks.
2. Morphable model of a human face
A morphable model of a human face can be de-
picted as a box with tunable handles. Depending
on positions of those handles the box generates one
or another picture of one or another human face.
This box contains handles of three types (fig. 1).
Some of them are charged with 3D shape of the
surface (α), another are charged with camera posi-
tion (β), the other are charged with illumination (γ).
Let us admit, that by depicting a facial shape, a ca-
mera position and an illumination by handles of that
box, we virtually claim that all these quantities
can be described by a finite number of parameters.
Fig. 1. Three different types of morphable model parameters
Camera position is indeed specified by finite,
moreover, small number of parameters. It is not that
easy in case of illumination and 3D shape. In fur-
УСиМ, 2011, № 2 81
ther subsections we will consider means to param-
eterize a set of 3D shapes of a human face. At pre-
sent, let us consider the illumination parameteri-
zation.
2.1. Illumination model
Fig. 2. Directional illumination model
Let us suppose that a surface is lit by a single
directional light source, i.e. a source which shines
from infinity with parallel beams of light, and is
specified by a single vector d; direction of the vec-
tor indicates light direction, and its length defines
light intensity. Brightness of a surface in some
point t is defined as a scalar product of normal to
the surface in this point and vector d:
3
1
( , , , ) ( , , )i i
i
L d t d n t
.
Let us suppose that there exists so called ambi-
ent light as well. It is just a constant added to the
brightness of all surface points:
3
1
( , , , , ) ( , , )i i
i
L d c t d n t c
. (1)
Let us take a gander what happens if we have
two directional light sources rather than one. Then
the brightness of some surface point (not taking
into account ambient light) equals to:
3 3
1 2
1 1
( , , ) ( , , )i i i i
i i
d n t d n t
, (2)
where d1 is a vector, describing the first light
source, d2 describes the second one. The equation
(2) can be obviously rewritten as
3
1 2
1
( ) ( , , )i i i
i
d d n t
. (3)
Thus, in this illumination model no matter we
have a single light source, two, or thousand of
them. All these sources are equal to some single
light source.
Thus, we have parameterized illumination by a
vector d and a scalar c:
γ = {d, c}. (4)
2.2. Convex combination of faces
We assume that a set of 3D shapes of a human
face is convex in a sense. Loosely speaking, by
averaging two surfaces which are faces one gets a
face as well.
Suppose we have a collection of surfaces repre-
senting some typical human faces 1 2, ,..., ns s s .
We call them basis shapes. Let us assume that
these basis shapes can be represented as points in
some linear space. The means of doing so will be
described in section 2.4. We approximate a shape
of an arbitrary human face by a convex combina-
tion of the basis shapes:
1 1
, 1
n n
i i i
i i
s s
. (5)
Convex combination coefficients {i:i =1,2,…,n}
are just the aforementioned parameters of a facial
shape α.
But what exactly should be meant by averaging
of two surfaces? Let us clarify it on a very simple
example.
2.3. Morphable model of rectangle
Fig. 3. Two basis rectangles Fig. 4. Wrong averaging of basis
rectangles
Fig. 5. Labeling of basis rectan-
gles
Fig. 6. Correct averaging of basis
rectangles
Suppose that we would like to design a mor-
phable model of rectangle. Let us do so in the fol-
lowing way: represent any rectangle as a convex
combination of those, shown in fig. 3. If we con-
sider each of introduced rectangles as a function
of a single variable and average these functions,
we come up with a function depicted by thick line
in fig. 4, which, obviously, is not a rectangle. It
has happened because we had averaged semanti-
cally different points. If we specify a proper map-
ping between points of the two basis rectangles,
that is, map a point A of the darker rectangle to a
point A' of the brighter rectangle, map a point B to
82 УСиМ, 2011, № 2
a point B', and uniformly map all points of a seg-
ment AB to a segment A'B', then we get a correct
averaging (fig. 6). Indeed, a point A'' is just half
way along the points A and A', and point B'' is ex-
actly in the middle between the points B and B'.
Thus, in order to get a rectangle by averaging
two rectangles, one needs to average semantically
identical, i.e. correspondent, points. This reason-
ing is also applicable for the case of much more
complicated curves in 2D plane and surfaces in
3D space. In particular, it is applicable for the sur-
face of a human face.
2.4. A morphable model of a human face
Let us suppose that basis shapes nsss ,...,, 21
are represented as vectors containing 3D coordi-
nates of surface points:
1 1 1 2 2 2, , , , , ,..., , ,
i i i
Ti i i i i i i i i
i m m ms x y z x y z x y z , (6)
where mi – is a number of points in i-th surface.
In order to make the operation (5) formally ap-
plicable to such surfaces one has to make sure that
number of points in all surfaces is the same:
mi = m, i = 1,2, …, n.
In order to make the operation (5) provide a hu-
man face as a result, not an arbitrary surface, one
has to guarantee that the following informally de-
scribed property is also fulfilled (a formal descrip-
tion of the property will be given in the next sub-
section): components of the vectors have to be ar-
ranged in such a way that semantically identical po-
ints are placed in the same positions in their vectors.
For instance, if a coordinate of a tip of the nose of
the first basis shape is located in the first position in
vector s1, then coordinates of a tip of the nose of
all other basis shapes have to be located in the first
positions in their vectors s2,s3,…,sn as well (fig. 7).
Fulfilling these requirements comes easy if the
correspondences between the points of different
basis shapes are specified. A search for such cor-
respondences is the major problem of fulfilling the
requirements.
2.4.1. Establishing the correspondence between
points of basis shapes
Let us choose one of the basis shapes, and call
it the principal shape. If we were able to establish
a correspondence between points of the principal
shape and each of the other basis shapes, then we
could easily fulfill requirements, specified in p. 7.
Thus we have reduced our task to a problem of
establishing correspondence between points of
two surfaces.
Fig. 7. Corresponding points of different 3D models have to be
placed in the same positions in their vectors
On this stage we suppose that we have some
fixed triangulation of surface points. Let us define
a normal vector nt as a vector of a unit length
which is orthogonal to a triangle t.
We get a normal to each of the triangulation
vertices by averaging normals of triangles com-
prising the vertex:
( )
( )
1
( )
,
i
k p
t p
i
p
n
n
k p
(8)
where k(p) is a number of triangles comprising the
point p, ( )
it
n p is a normal to i-th triangle com-
prising the point p. The inner part of the triangle is
filled with a linear function of its vertex normals.
Normals can be visualized by ascribing to each
surface point a color, which RGB components are
proportional to (x, y, z) components of a normal.
Frontal projections of surfaces, colored in such a
way, are depicted on fig. 8. We refer to such im-
ages as to normal maps.
Let us remark that semantically identical points
of different models have similar colors on these
images. For instance, a tip of the nose is colored
almost identical in the two models. In compliance
with the remark, we define a difference between a
point (x, y) in a normal map of the first surface
and a point ( , ), ( , )x yx k x y y k x y in a normal
map of the second surface as follows:
УСиМ, 2011, № 2 83
,
3
1
1
2
( , ), ( , )
( , , )
( ( , ), ( , ), ) ,
x y x y
c
x y
q k x y k x y
I x y c
I x k x y y k x y c
(9)
where c is a number of RGB channel, 1I and 2I
are normal maps of the first and the second model.
Fig. 8. Visualization of normals to a surface of a human face
Let us introduce some natural constraints on mu-
tual alignment of the corresponding pixels in a pair
of normal maps, namely forbid them to get «entan-
gled»:
, , 1, ( , ), ( , ), ( 1, ), ( 1, )
0, 1, , 1,
, 1,
, ,
i j i j x y x y
x x
y y
g k i j k i j k i j k i j
if k i j k i j
k i j k i j
otherwise
(10)
, , , 1 ( , ), ( , ), ( , 1), ( , 1)
0, , 1 , 1,
, , 1
, .
i j i j x y x y
y y
x x
g k i j k i j k i j k i j
if k i j k i j
k i j k i j
otherwise
(11)
Functions (10) and (11) are equal to zero for al-
lowed correspondence pairs, and infinity for for-
bidden ones. Fig. 9 visualizes these constraints: if
the black pixel of the left image corresponds to the
black pixel of the right image, then the grey pixel
of the left image can correspond only to one of the
grey pixels of the right image.
Fig. 9. Constraints on mutual location of corresponding pixels in
pair of normal maps
Let us define a penalty for a collection of cor-
respondences k:
,
1 1
1
, , 1,
1 1
1
, , , 1
1 1
( ) ( , ), ( , )
( , ), ( , ),
( 1, ), ( 1, )
( , ), ( , ),
( , 1), ( , 1) ,
X Y
x y x y
x y
X Y
x y x y x y
x y
x y
X Y
x y x y x y
x y
x y
Q k q k x y k x y
g k x y k x y
k x y k x y
g k x y k x y
k x y k x y
(12)
where k is a vector comprising a pair of numbers
(kx(x, y), ky(x, y)) for each pixel of first model’s
normal map, X and Y are vertical and horizontal
dimensions of the normal maps respectively.
A problem of optimal mapping between two
normal maps consists in search for a collection of
correspondences k with minimal penalty:
k* = argminkQ(k) . (13)
A problem (1) is NP-hard in general case. For the
particular case, when functions gi,j,i+1,j and gi,j,i,j+1
are given by (10) and (11), a polynomial algorithm is
not known as well. To get an approximate solution
of the problem we use a method, suggested in [7].
By solving a problem (13) we establish a corre-
spondence between pixels of two normal maps.
On the ground of these correspondences it is easy
to retrieve correspondence between points of 3D
models. And based on this data it is easy to fulfill
the requirements indicated at p. 8. Thus by choos-
ing different weights {i : i = 1, 2, , n} one can
generate surfaces of different human faces.
2.4.2. Expanding the possibilities of the morpha-
ble model
In order to increase a diversity of faces which
our morphable model is able to generate let us
split a set of facial points into four segments: eyes,
nose, mouth and cheeks, as it is depicted in fig. 10.
Let us introduce a set of convex combination co-
efficients : 1, 2,...,S
i i n for each of the seg-
ments S S . It is obvious that the arbitrary cho-
sen coefficients produce gaps on borders of the
segments i.e. we get for disconnected segments of
a face rather than a single face. We use the follow-
ing technique in order to eliminate those gaps.
84 УСиМ, 2011, № 2
Fig. 10. Face segmentation
For each segment S S and for each point p
of the surface calculate a distance from this point
to the nearest point of the segment:
( ) minS v Sd p p v . (14)
Normalize this distance over all segments and
over all points of the surface:
'
' '
( )ˆ
( ')maxmax
S
S
S
S S p
d p
d p
d p
. (15)
Calculate a coefficient of point p belonging to a
segment S:
4ˆ( ) 1 ( )S Sb p d p . (16)
The fourth power is chosen to guarantee a suf-
ficient attenuation speed for the function bS(p)
while moving away from the segment border. Nor-
malize this coefficient:
'
'
( )ˆ
( ')
S
S
S
S S
b p
b p
b p
. (17)
Calculate a convex combination coefficient for
point p:
ˆ( ) S
i i S
S S
p b p
. (18)
Convex combination coefficients of the basis
shapes are now different for different points of the
model. Such a modification substantially increases
a diversity of faces produced by the morphable
model.
3. 3D reconstruction of a human face based
on a photograph
In the previous section we have in depth rese-
arched the interior of the morphable model. It is
now convenient for us to return to such an abstrac-
tion level where a morphable model is just a box
with three types of parameters: those charged with
3D shape of the surface (α), those charged with ca-
mera position (β), and those charged with illumi-
nation (γ). Under particular values of these parame-
ters a morphable model generates a picture of a hu-
man face with a 3D shape given by parameters α,
taken under a viewpoint, given by parameters β,
and illumination conditions, given by parameters γ.
This is the time to ask a question: what exactly
this generated image looks like? Let us recall that
our morphable model generates a non-textured
surface. For certainty let us suppose that all points
of a surface are colored in white. Then in the gen-
erated image we get something similar to a plaster
statue photograph. Let us first describe a method
of 3D reconstruction for the case when we have a
photograph of a plaster statue of a human face as
an input image. Afterwards we will consider a way
to modify the suggested algorithm to work with
photographs of real human faces.
3.1. 3D reconstruction of a plaster statue
Suppose we have an input image
I: T→RGB, (19)
where T is a set of pixels, RGB = {0,1,…,255}×
× {0,1,…,255}×{0,1,…,255}.
Suppose we have an image, generated by the
morphable model with respect to parameters α, β
and γ under illumination model (1):
, , :L T RGB . (20)
Let us define a difference between the input
and the generated image:
2
, , , ,( , ) ( ( ) ( ))
t T
F I L I t L t
, (21)
where T is a subset of pixels, into which at least
one of the surface points is projected.
The problem of 3D reconstruction of a human fa-
ce based on its image consists in looking for such
parameters α, β and γ, under which a difference bet-
ween the input image and the generated one is mi-
nimal:
, ,
, ,
( , ) minF I L
. (22)
Given fixed parameters α and β a function (22)
under illumination model (1) is minimized with
respect to γ by least squares. The minimization
with respect to α and β is performed by Nelder–
Mead method [8]. To make the paper self-conta-
ined we give a brief description of the method.
УСиМ, 2011, № 2 85
Suppose a continuous function f : R
n R is given. The
Nelder–Mead algorithm maintains a simplex {x1, x2, , xn+1} in
Rn. The algorithm requires a termination threshold ε as well as
parameters α, γ, and σ which are explained later on. The
algorithm performs the following sequence of operations:
1. Order all simplex vertices according to the values of
function f: f(x1) f(x2) … f(xn+1) .
2. Calculate x0 , the center of gravity of all simplex points,
except xn+1 .
3. Reflection. Compute a reflected point:
xr = x0 + (x0 – xn+1).
If: f (x1) f (xr) f (xn),
then substitute a point xn+1 by xr , and go to step 1.
4. Expansion. If the reflection point is the best so far:
1( ) ( )rf fx x
then compute an expansion point:
xe = x0 + (x0 – xn+1).
If the expansion point is better than the reflection point:
( ) ( )e rf fx x
then substitute the worst point xn+1 by expansion point xe
and go to step 1. Otherwise, substitute the point xn+1 with
point xe and go to step 1.
5. Contraction. On this stage it is certain that:
f(xr) f(xn).
Compute a contraction point:
xc = xn+1 + (x0 – xn+1).
If the contraction point is better than the worst point:
1( ) ( )c nf f x x
then substitute a point xn+1 with xc, and go to step 1.
6. Reduction. For all point, except the best one, assign:
1 1: ( ), 2,3,..., 1i i i n x x x x .
Calculate distances between the best point and all other
simplex vertices: 1 , 2,3,..., 1i i i n x x .
If max i
i
, go to step 1. Otherwise, end up with a point x1.
Standard values for parameters α, γ, and σ are the fol-
lowing: 1, 2, 1/ 2, 1/ 2 .
3.2. 3D reconstruction of a human face
In order to process photographs of real human fa-
ces, rather than images of single-colored plaster stat-
ues, we remove all non-skin segments from the im-
age, i.e. eyebrows, eyes and lips, as it is shown in
fig. 11. As semantically identical points of all basis
shapes are located in the same positions in their vec-
tors, it is sufficient to remove the indicated segments
only from one of the basis shapes. This operation is
then mechanically transferred to the other models.
Fig. 11. Non-skin segments removal
Fig. 12 and 13 show examples of human face
3D reconstruction made by the suggested method.
Fig. 12. An example of human face 3D reconstruction by single
imageFig. 12 and 13 show examples of human face 3D re-
construction made by the suggested method.
Fig. 13. An example of human face 3D reconstruction by several
images
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// Proc. of SIGGRAPH 2008..
3. Blanz V., Vetter T. Face identification across different
poses and illumination with a 3D morphable model
// IEEE Trans. Pattern Analysis and Machine Intelli-
gence. – 2003. – 25, – P. 1063–1074.
4. US Patent N 7212664. Mitsubishi Electric Research
Laboratories, Inc. Constructing heads from 3D models
and 2D silhouettes.
5. US Patent N 7103211. Geometrix, Inc. Method and appa-
ratus for generating a 3D face models from one camera.
6. US Patent N 7212656. Microsoft Corporation. Rapid
computer modeling of faces for animation.
7. Kolmogorov V. Convergent Tree-Reweighted Message
Passing for Energy Minimization // IEEE Transactions
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© М. Тyshchenko, researcher, the International Research and Trai-
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| id | nasplib_isofts_kiev_ua-123456789-82926 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0130-5395 |
| language | English |
| last_indexed | 2025-12-07T18:28:50Z |
| publishDate | 2011 |
| publisher | Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України |
| record_format | dspace |
| spelling | Tyshchenko, M. 2015-06-11T20:12:14Z 2015-06-11T20:12:14Z 2011 3D Reconstruction of Human Face Based on Single or Several Images / М. Тyshchenko // Управляющие системы и машины. — 2011. — № 2. — С. 79-85. — Бібліогр.: 8 назв. — англ. 0130-5395 https://nasplib.isofts.kiev.ua/handle/123456789/82926 004.93’1 Предложен новый подход к трехмерной реконструкции человеческого лица по одному или нескольким изображениям. Подход основан на генеративной модели человеческого лица. Также предложен новый способ построения генеративной модели по набору нетекстурированных поверхностей. A new approach to a human face 3D reconstruction by single or several images is suggested. The approach is based on a morphable model of a human face. A new way to design the morphable model by a set of untextured 3D shapes is suggested as well. Запропоновано новий підхід до тривимірної реконструкції людського обличчя за одним або кількома зображеннями. Підхід базується на генеративній моделі людського обличчя. Також запропоновано новий спосіб побудови генеративної моделі за набором нетекстурованих поверхонь. en Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України Управляющие системы и машины Оптимизационные задачи структурного распознавания образов 3D Reconstruction of Human Face Based on Single or Several Images Трехмерная реконструкция человеческого лица на основе одного или нескольких изображений Тривимірна реконструкція людського обличчя на основі одного або декількох зображень Article published earlier |
| spellingShingle | 3D Reconstruction of Human Face Based on Single or Several Images Tyshchenko, M. Оптимизационные задачи структурного распознавания образов |
| title | 3D Reconstruction of Human Face Based on Single or Several Images |
| title_alt | Трехмерная реконструкция человеческого лица на основе одного или нескольких изображений Тривимірна реконструкція людського обличчя на основі одного або декількох зображень |
| title_full | 3D Reconstruction of Human Face Based on Single or Several Images |
| title_fullStr | 3D Reconstruction of Human Face Based on Single or Several Images |
| title_full_unstemmed | 3D Reconstruction of Human Face Based on Single or Several Images |
| title_short | 3D Reconstruction of Human Face Based on Single or Several Images |
| title_sort | 3d reconstruction of human face based on single or several images |
| topic | Оптимизационные задачи структурного распознавания образов |
| topic_facet | Оптимизационные задачи структурного распознавания образов |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82926 |
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