3D-样条重建：一种新的形状从阴影方法

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"这篇论文提出了一种新的shape from shading (SFS)方法，通过结合3D-spline重建和‘有用领域’的概念来解决SFS问题。这种方法无需边界条件即可使问题有唯一解，并且适用于正交和透视投影的两种SFS形式。论文包括对合成图像和真实图像的验证。关键词：shape from shading; 3D-spline" 在计算机视觉领域，shape from shading 是一种从二维图像中恢复三维形状的技术。这篇论文引入了一种原创的方法，将3D-spline（三维样条）用于重建，从而解决了SFS问题。3D-spline 是一种参数化表面模型，它能够灵活地表示复杂的形状，且无需传统SFS方法中的边界条件来确保问题的稳定性。这为解决光照阴影带来的形状反求问题提供了新的思路。传统的SFS问题通常会遇到未定义的边界条件，导致反求结果的不确定性。论文中提出的3D-spline 方法通过构建参数化表面，消除了对这些条件的需求，使得整个问题变得有界且可解。这种方法的核心是将场景建模为一个连续、平滑的3D-spline 表面，允许对整个重建区域进行形状反求。同时，论文还引入了“有用领域”的概念。这是指图像中的一部分像素集合，其灰度值符合SFS假设，可以用来解算整个重建区域的SFS问题。通过选择这些像素，算法能够在保持准确性的同时，减少噪声和光照变化的影响，提高重建质量。论文详细描述了该方法在正交投影和透视投影下的实现，这两种投影方式在实际应用中非常常见。正交投影常用于简单的几何形状，而透视投影则更接近人眼观察的实际效果。通过在这两种情况下验证方法的有效性，论文证明了该方法的普适性和可靠性。最后，作者们使用合成图像和真实图像进行了实验，以验证所提出方法的性能。合成图像可以提供已知的地面真相，用于评估算法的准确性；而真实图像则能检验算法在实际复杂环境中的表现。这些实验结果为新方法的实际应用提供了有力支持。这篇论文为shape from shading 的研究提供了一个创新的框架，结合了3D-spline 和“有用领域”这两个关键概念，有望改善从二维图像中恢复三维形状的精度和鲁棒性。这种方法对于机器人导航、遥感图像分析以及虚拟现实等领域具有重要的理论和实践价值。

work of deformable models: the SFS equation is expressed

as a constraint, all along the evolution. This last method,

which provides rather good results, has been extended to

perspective images in [28,29]. Albeit we consider this latter

work clearly as that yielding the be st results, the problem of

slowness (e.g., 10,000 iterations of 2 s each, at the most pre-

cise level with 10,000 triangles) makes this method techni-

cally impracticable to some extent. Another kind of

deformable models, namely, 2D-snakes, has been used in

[30], but once again, a special eﬀort has to be done after-

wards, in order to connect the diﬀerent snakes, which cor-

respond to parallel sections of the shape. The results are

not so good, to say at least.

A few papers have dealt with more complex models than

linear or quadratic ones. In [31], a superquadrics is used,

whose 10 parameters are estimated via genetic algorithms,

but no convincing result is shown. On the other hand, the

results reported in [32] are of good quality, even if the

method is intrinsically dedicated to face reconstruction.

Dealing with other types of scenes would require to refor-

mulate the main part of the problem.

Finally, let us mention some papers [33,34] in which the

use of a 3D (quadratic) model is not aimed at modeling the

surface, but at enforcing some ‘‘coherence criteria’’ on the

normals at each step of an iterative process.

All of existing works, as discussed below, aim at avoid-

ing the necessity of boundary conditions, thus solvi ng Pb2.

The methods described in [3,26,27,29], respectively, in

[3,23,24,31,32], more or less successfully grapple with

Pb1, respectively, with Pb3. On the other hand, Pb4 was

not addressed in these works and only [31] dealt with

Pb5, to some extent.

2.1.2. Our contribution

Using a 3D parametric model for the scene representa-

tion in SFS clearly prevents from the necessity of boundary

conditions, which are implicitly replaced by constraints on

the scene surface: Pb2 is solved at once. Nevertheless, as

this comes out again from the previous state of the art,

the choice of the model has great consequences upon the

other aforementioned problems. In this paper, we aim at

simultaneously circumscribe problems Pb1, Pb2, Pb3 but

also Pb4 (as explained in Section 2.2.2). We will consider

a 3D-sp line to model the surface shape which surprisingly

enough, has never been done before. It will be shown that

this model allows us to reach this aim, contrary to the pre-

viously cited works.

2.2. Selecting the reliable pixels

2.2.1. State of the art

Aiming at applying SFS methods not only to synthetic,

but also to real images, one must take into account the fact

that a number of pixels could be ‘‘unreliable’’ i.e., not in

accordance with the SFS hypotheses (Pb4). The classical

methods of resolution which consider one unknown per

pixel (the height of the conjugate point in the scene) cannot

straightforwardly discard the unreliable pixels, except

either by imposing smoothness on the height, or by making

holes in the reconstruction domain and by using a posteri-

ori interpolation. To cope with problem Pb4, interesting

alternatives have been proposed in some papers, where

the unreliable pixels, whose greylevels are yet biased, are

kept in the reconstruction domain, at the expense of great

eﬀorts to make the methods more robust. We list now what

we think to be the most signiﬁcant contributions dealing

with Pb4 in classical SFS. In [35], the Lambertian and spec-

ular reﬂection components are separated, using probabilis-

tic tools, in order to consider more realistic photometric

modelings than the purely Lambertian one. In [36], the

assumption of uniform albedo is questioned: the scene is

segmented into regions having diﬀerent albedos. In [37],

shadow areas, which could be confused with areas of lower

albedos, are identiﬁed through the so-called ‘‘shading ﬂow

ﬁeld’’ procedure. The simpler situation of ‘‘black shadows’’

is astutely taken into account in [10], where it is considered

that the limit between light an d shadow belongs to the

shape to be reconstructed. An iterative algorithm is

reported in [38], which takes secondary reﬂections into

account. Finally, in [39,40], the standard regularization

theory is extended, in order to take visual discontinuities

such as occlusions into account.

2.2.2. Our contribution

It is then judicious to consider the parametric approach

to SFS versus the classical approach, since the former

intrinsically solves the problem of interpolation, and it also

prevents from the use of a smoothness term, which usually

moves the solution away from a minimizer of the intensity

error term. In parallel with the concept of reconstruction

domain X

, we introduce that of ‘‘us eful domain’’ X

.We

deﬁne the ‘‘maximal useful domain’’

as the set contain-

ing all the reliable pixels. To illustrate this notion, consider

the following example. Let us imagine that the rectangles in

Fig. 1-a represent unreliable areas in the image. We aim at

reconstructing the surface on the reconstruction domain X

represented in black in Fig. 1-b, which does not contain

holes. The maximal useful domain

, which contains

holes, is represented in black in Fig. 1-c. In order to recon-

struct the scene on the whole, while decreasing the comput-

ing time, we choose as useful domain X

only a subset of

, made up for example by the pixels represented in black

in Fig. 1-d. Only the pixel s of X

will be taken into account

to check the concordance of the 3D parametric model with

the image. This simple idea makes it possible to give a sat-

isfactory response to problem Pb4, while reducing the com-

puting time, as this will be seen in the experiments.

3. Spline from shading

3.1. Two SFS modelings

We attach a 3D frame (Cxyz ) to the camera, whose ori-

gin C is the optical center and such that axis C

coincides

468 F. Courteille et al. / Image and Vision Computing 26 (2008) 466–479

朗伯体反射和镜面发射成分相互独立，使用概率工具进行融

合，从而得到一个更加比纯朗伯体更加真实的情景。

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