We will first provide a brief introduction to framelets, followed by details on how framelets can be incorporated into an minimization framework for DWI denoising.

Tight Framelets

A system X с L_{2}(R) is called a tight wavelet frame of Ь_{2}(Ш) if

where (•, •) is the inner product of L_{2}(R). It is clear that an orthonormal basis is a tight frame, since the identity hold for arbitrary orthonormal bases in L_{2}(R). Tight frames are generalization of orthonormal bases with greater redundancy—a property central to sparse representation and often desirable in applications such as denoising [10].

Given a set of generators Ф := ff_{R}) с L_{2}(R^{d}), which are desirably (anti)symmetric and compact functions, the corresponding quasi-afflne system Х(Ф) generated by Ф is the collection of dilations and shifts of Ф: Х(Ф) = f fi,_{r},_{k} : 1 < r < R; l, k e Z), where f_{ln},_{k} is defined by

When X(Ф) forms an orthonormal basis of L_{2}(R), Х(Ф) is called an orthonormal wavelet basis. When Х(Ф) forms a (tight) frame of L_{2}(R), each function f_{r}, r = 1,R, is called a (tight) framelet and the whole system Х(Ф) is called a (tight) wavelet frame system. A tight wavelet frame is also called a Parseval frame. Note that in the literature, the affine (or wavelet) system, which corresponds to the decimated wavelet (frame) transforms, is commonly used. The quasi-affine system, introduced and analyzed in [11], corresponds to the undecimated wavelet (frame) transforms and essentially over samples the wavelet frame system starting from level J — 1 and downwards. In this paper, we focus on the quasi-affine system because it has been shown to work better in image restoration [12]. We set J = 0 and consider only l < 0.

The construction of Ф is usually based on a multiresolution analysis (MRA) [12] that is generated by some reflnable function ф with refinement mask a_{0}2 '_{2}(Z) satisfying

The key is to find the masks a_{r}2 '_{2}(Z) that gives

The finite sequences, ai,..., a_{R} are called wavelet frame masks, or the high pass filters of the system. The refinement mask a0 is also known as the low pass filter. The two equations above can be combined by defining ф_{0} := ф.

The unitary extension principle (UEP) [11] provides a general theory for constructing MRA-based tight wavelet frames. As long as fa_{i}a_{R}} are finitely supported and their Fourier series satisfy

for all v 2 f0, ж} and ? 2 [—ж, ж], the quasi-affine systemX(Ф) forms a tight frame in L2.R).

Consider the centered B-splines of order p, i.e.,

with j = 0 when p is even and j = i when p is odd. The corresponding refinement mask is given as

and the p wavelet masks as

It is straightforward to show that the UEP conditions (5) are satisfied. Wavelet frame masks for p = 1,2,4 are shown in Table 1. It is worth noting that these masks correspond to differential operators of various orders. For example, for piecewise

Table 1 Wavelet frame masks

Piecewise constant, p = 1

Piecewise linear, p = 2

Piecewise cubic, p = 4

a_{0} = ±[1, 1]

a_{0} = j[l.2, 1]

aо = ?[1.4, 6, 4,1]

a = j[l.-l]

ал = Jl-1.2,-1]

«1 = ?[1.-4. 6, -4,1]

7

o'

7

и

§

a_{2} = |[—1,2,0,-2, 1]

a_{3} = ^[1.0,-2, 0,1]

o_{4} = |[—1, —2,0,2,1]

linear B-spline, the masks a and a_{2} correspond to the first order and second order difference operators respectively up to a scaling factor.

When a tight wavelet frame is used, the given data is considered to be sampled as a local average u[k] = (f, ф(- — k)}. Noting that [12]

where the dilated sequence is defined as

the decomposition and reconstruction down to level — L [12], i.e.,
where

can be realized with convolution using the masks. We denote by W the L-level framelet decomposition, i.e.,

with B_{L} : = {(1,1), (1,2),..., (1,R), (2,1),..., (L,R)} U {(L, 0)}, where the level l and band r framelet decomposition is given by

If we use W> to denote the framelet reconstruction, we have W>W = I, i.e., u = W>Wu.

Given a 1-dimensional framelet system for Ь_{2}(Ш), the d-dimensional tight wavelet frame system for L_{2}(R^{d}) can be easily constructed by using tensor products of 1-dimensional framelets [12].