Signal Processing - September 2016 - 134

use with photometric stereo algorithms. Ikehata et al. [23]
By taking a series of measurements I , with different, but
models the non-Lambertian, specularities and shadows as
known lighting direction L , the surface normal n, and
additive corruption E, so that the observed image intensity
albedo k can be estimated from a system of linear equations
is I = k (n : L) + E . Assuming the corruption E is spatially
using, for instance, a least squares method.
Classical photometric stereo assumes a distant-light model.
sparse, the problem can be solved by compressive sensing
This considerably simplifies the problem, as it produces conalgorithms by modeling the optimization similar to the Lastant lighting angle and incident radiance across the object surgrange form of (1) as:
face. However, distant-light sources are impractical due to finite
space and energy constraints. As a result, for a typical photomin I - k (n : L) - E 2 + m E 0 ,
(3)
k, n, E
metric stereo capture setup, lighting angle and incident radiance
vary across the object surface. Using the simplified far-light
where m is an nonnegative parameter controls the balance
model from (2) with such a setup produces a 3-D shape with
between data fit and sparsity.
large global error [25]. Recently, researchWhile photometric stereo can produce
ers have explored the near-light photometric
submillimeter precision surface measureWhile photometric stereo
stereo method to recover millimeter to subments with a large field of view, other
can produce submillimeter methods can be used to measure surface
millimeter scale markings on the surfaces of
precision surface
Paul Gauguin's paintings (Figure 2) [9]. The
detail on the microscopic scale. Optical codepth maps acquired achieve a depth preciherence tomography (OCT) has recently
measurements with a
sion of fewer than 100 nm for a field of view
been employed for examining the layer
large field of view, other
structure of paintings [13]. High-resoluas large as 300 mm. These depth maps have
methods can be used to
tion 3-D images at a micron scale can be
revealed new details of how Gauguin promeasure surface detail on reconstructed thus revealing the underlayduced his paintings using his unique drawthe microscopic scale.
ers of paintings and their corresponding
ing transfer techniques.
depth positions. Originally proposed for
Classical photometric stereo also asbiomedical imaging of structures such as the eye, OCT can
sumes Lambertian surfaces with perfect diffuse reflection.
produce high-resolution contrast depth maps. OCT presents
However, this assumption is invalid for a large class of real
challenges in that the instrumentation is expensive and can
materials such as metals, plastics, and glass, which exhibit
only scan centimeter-sized areas. The depth maps obtained
different combinations of diffuse and specular reflections.
are also not linked to material color information, so interThe most accurate way to model how light is reflected from
preting these data is not immediately intuitive.
an opaque surface uses the bidirectional reflectance distribution function (BRDF), which is a four-dimensional function
fr (~ i, ~ o) , which depends on the incoming light direction
Image relighting for cultural heritage
~ i and outgoing light direction ~ o . The BRDF is the most
In addition to the 3-D geometry, characterizing surface
general way to model surface reflection (not considering
appearance under different lighting conditions is also critical
subsurface scattering), but it also severely complicates the
for cultural heritage. The appearance of an artwork is the
photometric stereo problem. As a result, several researchers
sum result of how its material and microstructure interact
have investigated lower-dimensional reflectance models for
with all possible incoming light rays and all the possible
subsequent measured outgoing light
rays that may have been reflected,
absorbed, scattered, refracted, and
Calibration
transmitted from the artwork's surReflective
Target
face. This compressive light-transport
Sphere
Color
Nativity
function combines each possible inciChecker
dent light location, wavelength, direction, polarization with how this
incident electromagnetic radiation
scatters underneath the object's surface, and global illumination effects
such as self-shadowing and interreflection. It is an immense totality of meaFigure 2. (a) The setup for capturing photometric stereo of Gauguin's Nativity: a color checker for
surements that is only theoretically
color calibration, a 3-D calibration target for 3-D surface calibration, a reflective sphere for calibrating
possible to collect completely. Conselight direction, and the work of art. (b) Several frames from an animation visualizing the 3-D surface
quently, the light-transport function at
shape at the location of the lines drawn in Nativity. The 3-D reconstruction shows clear evidence of
a fixed viewpoint may be easier to
protrusions on the page where ink has been deposited. This is solid evidence for the ink being transgather by capturing images of artwork
ferred from a matrix such as that in a monotype transfer process. (Images and figure caption used
with permission from [9].)
lit from various light directions.
134

IEEE SIgnal ProcESSIng MagazInE

September 2016

Table of Contents for the Digital Edition of Signal Processing - September 2016