Images are composed of geometric structure and texture. Large scale structures
are considered to be the geometric structure, while small scale details
are considered to be the texture. In this dissertation, we will argue that the
most important difference between geometric structure and texture is not
the scale - instead, it is the requirement on representation or reconstruction.
Geometric structure must be reconstructed exactly and can be represented
sparsely. Texture does not need to be reconstructed exactly, a random sample
from the distribution being sufficient. Furthermore, texture can not be
represented sparsely.
In image inpainting, the image content is missing in a region and should
be reconstructed using information from the rest of the image. The main
challenges in inpainting are: prolonging and connecting geometric structure
and reproducing the variation found in texture. The Filter, Random fields
and Maximum Entropy (FRAME) model [213, 214] is used for inpaining texture.
We argue that many ’textures’ contain details that must be inpainted
exactly. Simultaneous reconstruction of geometric structure and texture is
a difficult problem, therefore, a two-phase reconstruction procedure is proposed.
An inverse temperature is added to the FRAME model. In the first
phase, the geometric structure is reconstructed by cooling the distribution,
and in the second phase, the texture is added by heating the distribution.
Empirically, we show that the long range geometric structure is inpainted in
a visually appealing way during the first phase, and texture is added in the
second phase by heating the distribution.
A method for measuring and quantifying the image content in terms of
geometric structure and texture is proposed. It is assumed that geometric
structures can be represented sparsely, while texture can not. Reversing
the argumentation, we argue that if the image can be represented sparsely
then it contains mainly geometric structure, and if it cannot be represented
sparsely then it contains texture. The degree of geometric structure is determined
by the sparseness of the representation. A Truncated Singular Value
Decomposition complexity measure is proposed, where the rank of a good
approximation is defining the image complexity.
Image regularization can be viewed as approximating an observed image
with a simpler image. The property of the simpler image depends on the regularization
method, a regularization parameter and the image content. Here
we analyze the norm of the regularized solution and the norm of the residual
as a function of the regularization parameter (using different regularization
methods). The aim is to characterize the image content by the content in the
residual. Buades et al. [27] used the content in the residual - called ’Method Noise’ - for evaluating denoising methods. Our aim is complementary, as we
want to characterize the image content in terms of geometric structure and
texture, using different regularization methods.
The image content does not depend solely on the objects in the scene, but
also on the viewing distance. Increasing the viewing distance influences the
image content in two different ways. As the viewing distance increases, details
are suppressed because the inner scale also increases. By increasing the
viewing distance, the spatial lay-out of the captured scene will also change.
At large viewing distances, the sky occupies a large region in the image
and buildings, trees and lawns appear as uniformly colored regions. The
following questions are addressed: How much of the visual appearance in
terms of geometry and texture of an image can be explained by the classical
results from natural image statistics? and how does the visual appearance
of an image and the classical statistics relate to the viewing distance?