11 Electron Tomography
265
It has been proved that CS presents in general higher quality reconstructions
than classic SIRT and WBP given the same number of iterations, and is capable
to retrieve highly accurate reconstructed images in cases of severe undersampling
(i.e. low number of projections available for the reconstruction). Examples of CSET reconstruction of magnetic nanoparticles are available in the literature [32–34],
illustrating the power of this method to reconstruct accurately complex 3D structures for quantitative analysis and, thus, allowing the comparative study of magnetic
properties and structural information.
11.2.1.1 Mathematical Principles
Understanding the principles behind the CS theoretical framework requires a fair
knowledge of the concepts of sparse representation, compressible signal and sensing
processes (measurement).
In a standard signal acquisition process (e.g. image acquisition in TEM), an initial
signal
x with n components will be measured against the so-called sensing waveform
(i.e. a functional basis Φ) giving a recorded signal
p with m components. This is:
p = Φ
x
In general, the process will suffer from undersampling (i.e. m n) and the
equation system is undetermined. CS theory shows that a unique solution can be
calculated for this problem, with two major restrictions: (i)
x is sparse in a certain
basis Ψ . (ii) The basis for the sparse representation Ψ and for the sensing waveform
Φ must be incoherent.
A signal
x is considered sparse in a certain domain (i.e. basis Ψ ), when all
the information can be expressed through a small set s of c s = 0 coefficients ( c).
Mathematically, the sparse transform is expressed as:
c = Ψ
x
The signal is said to be sparse only if s n, being
c the sparse representation
of
x. In practical applications, CS allows the relaxation of the strict constraint of
sparsity to compressibility. In a compressible
x,
c would contain k = 0 elements
such as k > s. Thus, the transform Ψ is allowed to retrieve a certain small number
of coefficients (k − s) with lower significance to the information recovery than the
s remaining, but still k n, whereas
x is sufficiently represented by the s < k n
coefficients with higher significance. Therefore, the small coefficients (k − s) can
be filtered away (or set to 0), providing that the inverse transform will be able to
effectively recover the initial image with minimal information loss. Then,
x is then
said to be a compressible signal.
Ψ and Φ incoherence means that the sensing basis cannot be sparsely represented.
Hence, it ensures that each p i contains information about many of the coefficients
265
It has been proved that CS presents in general higher quality reconstructions
than classic SIRT and WBP given the same number of iterations, and is capable
to retrieve highly accurate reconstructed images in cases of severe undersampling
(i.e. low number of projections available for the reconstruction). Examples of CSET reconstruction of magnetic nanoparticles are available in the literature [32–34],
illustrating the power of this method to reconstruct accurately complex 3D structures for quantitative analysis and, thus, allowing the comparative study of magnetic
properties and structural information.
11.2.1.1 Mathematical Principles
Understanding the principles behind the CS theoretical framework requires a fair
knowledge of the concepts of sparse representation, compressible signal and sensing
processes (measurement).
In a standard signal acquisition process (e.g. image acquisition in TEM), an initial
signal
x with n components will be measured against the so-called sensing waveform
(i.e. a functional basis Φ) giving a recorded signal
p with m components. This is:
p = Φ
x
In general, the process will suffer from undersampling (i.e. m n) and the
equation system is undetermined. CS theory shows that a unique solution can be
calculated for this problem, with two major restrictions: (i)
x is sparse in a certain
basis Ψ . (ii) The basis for the sparse representation Ψ and for the sensing waveform
Φ must be incoherent.
A signal
x is considered sparse in a certain domain (i.e. basis Ψ ), when all
the information can be expressed through a small set s of c s = 0 coefficients ( c).
Mathematically, the sparse transform is expressed as:
c = Ψ
x
The signal is said to be sparse only if s n, being
c the sparse representation
of
x. In practical applications, CS allows the relaxation of the strict constraint of
sparsity to compressibility. In a compressible
x,
c would contain k = 0 elements
such as k > s. Thus, the transform Ψ is allowed to retrieve a certain small number
of coefficients (k − s) with lower significance to the information recovery than the
s remaining, but still k n, whereas
x is sufficiently represented by the s < k n
coefficients with higher significance. Therefore, the small coefficients (k − s) can
be filtered away (or set to 0), providing that the inverse transform will be able to
effectively recover the initial image with minimal information loss. Then,
x is then
said to be a compressible signal.
Ψ and Φ incoherence means that the sensing basis cannot be sparsely represented.
Hence, it ensures that each p i contains information about many of the coefficients