Digital Signal Processing Reference
In-Depth Information
the isotropic undecimated wavelet transform for isotropic features; this wavelet
transform is well adapted to the detection of isotropic features such as the
clumpy structures to which we referred earlier
the anisotropic biorthogonal wavelet transform; we expect the biorthogonal
wavelet transform to be optimal for detecting mildly anisotropic features
the ridgelet transform, developed to process images that include ridge elements
and so to provide a good representation of perfectly straight edges
the curvelet transform to approximate curved singularities with few coefficients
and then provide a good representation of curvilinear structures
Therefore, when we choose one transform rather than another, we introduce, in fact,
a prior on what is in the data. The analysis is optimal when the most appropriate
decomposition to our data is chosen.
1.3.2 Toward Morphological Diversity
The morphological diversity concept was introduced to model a signal as a sum of a
mixture, each component of the mixture being sparse in a given dictionary (Starck
et al. 2004b; Elad et al. 2005; Starck et al. 2005a). The idea is that a single trans-
formation may not always represent an image well, especially if the image contains
structures with different spatial morphologies. For instance, if an image is composed
of edges and texture, or alignments and Gaussians, we will show how we can ana-
lyze our data with a large dictionary and still have fast decomposition. We choose
the dictionary as a combination of several subdictionaries, and each subdictionary
has a fast transformation/reconstruction. Chapter 8 will describe the morphological
diversity concept in full detail.
1.3.3 Compressed Sensing: The Link between Sparsity and Sampling
Compressed sensing is based on a nonlinear sampling theorem, showing that an
N
-
sample signal
x
with exactly
k
nonzero components can be recovered perfectly from
order
k
log
N
incoherent measurements. Therefore the number of measurements
required for exact reconstruction is much smaller than the number of signal samples
and is directly related to the sparsity level of
x
. In addition to the sparsity of the sig-
nal, compressed sensing requires that the measurements be incoherent. Incoherent
measurements mean that the information contained in the signal is spread out in the
domain in which it is acquired, just as a Dirac in the time domain is spread out in
the frequency domain. Compressed sensing is a very active domain of research and
applications. We will describe it in more detail in Chapter 11.
1.3.4 Applications of Sparse Representations
We briefly motivate the varied applications that will be discussed in the following
chapters.
The human visual interpretation system does a good job at taking scales of a
phenomenon or scene into account simultaneously. A wavelet or other multiscale
transform may help us with visualizing image or other data. A decomposition into
different resolution scales may open up, or lay bare, faint phenomena that are part
of what is under investigation.
