sparse image representation via combined transforms - Convex ...

74 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES
is a ridge function, where u denotes a constant vector, b denotes a scalar constant, and
u · x is the inner product of vector u and vector x. This idea is originally from neural
networks. When a function f is a sigmoid function, the function f(u · x − b) isasingle
neuron in neural networks. For Candès’ ridgelets, a key idea is to cleverly choose some
conditions for f; for example, an admissibility condition, so that when a function f satisfies
these conditions, we not only are able to have a continuous representation based on ridge
functions that have the form f(a linear term), but also can construct a frame for a compactsupported
square-integrable functional space. The latter is based on carefully choosing a
spatial discretization on the (u, b) plane. Note that a compact-supported square-integrable
functional space should be a subspace of L 2 (R). Candès [21] proves that his ridgelet system
is optimal in approximating a class of functions that are merely superpositions of linear
singularities. A function that is a superposition of linear singularities is a function that is
smooth everywhere except on a few lines. A 2-D example of this kind of function is a half
dome: for x =(x 1 ,x 2 ) ∈ R 2 , f(x) =1 {x1 >0}e −x2 1 −x2 2 . More detailed discussion is in [21].
A ridge function is generally not in L 2 (R 2 ), so in L 2 (R 2 ), Candès’s system cannot be
a frame, or a basis. Donoho [51] constructs another system, called orthonormal ridgelets.
In Donoho’s ridgelet system, a basic element is an angularly-integrated ridge function. He
proves that his ridgelet system is an orthonormal basis in L 2 (R 2 ). The efficiency of using
Donoho’s ridgelet basis to approximate a ridge function is explored in [52]. In discrete
cases (this is for digital signal processing), a fast algorithm to implement a quasi-ridgelet
transform for digital images is proposed in [49]. This algorithm is based on a fast Cartesian
to polar coordinate transform. The computational complexity (for an N × N image) is
O(N 2 log N).
3.5 Discussion
In the design of different transforms, based on what we have seen, the following three
principles are usually followed:
• Feature capture. Suppose T is a transform operator. We can think of T as a linear
operator in a function space Ω, such that for a function f ∈ Ω, T(f) isasetof
coefficients, T(f) ={c α ,α∈I}. Furthermore, we impose that f = ∑ α∈I c αb α , where
b α ’s are the basis functions. If basis functions of T reflect features that we try to catch,
and if a signal f is just a superposition of a few features, then the transform T should