Shearlets, like their curvelet predecessors, can capture variation over low dimensional sets more effectively than wavelets, which can capture pointwise variation succinctly, but require more coefficients to encode variation along a curve, surface, etc. In contrast to curvelets, shearlets have a group structure generated by parabolic dilation, translation, and shear, and are akin to wavelets in this sense. Earlier work of the author showed that an affine shear tight frame can be regarded as a subsampled system of an affine wavelet tight frame that has an MRA structure.
One writes \(f_{U;k}=|\det{U}|^{1/2} f(Ux-k)\) for \(k, x\in\mathbb{R}^d\) and \(U\) a \(d\times d\) matrix. Specific matrices appearing in wavelet and shearlet systems include \(A_\lambda=\text{diag}(\lambda^2,\lambda I_{d-1})\); \(M_\lambda=\lambda^2 I_d\); \(E_n\), a block matrix whose \(n\times n\) upper-left block has \((n-k)\)-th column equal to the \(k\)th column of \(I_n\), whose lower right \((d-n)\times (d-n)\) block is \(I_{d-n}\), and whose remaining blocks are zero; \(D_\lambda=\text{diag}(1, \lambda I_{d-1})\); and \(S^{\vec\tau}\), the \(d\times d\) matrix whose first row is \((1,\vec\tau)\) and whose other rows are the corresponding rows of \(I_d\) where \(\vec\tau=(\tau_2,\dots, \tau_d)\in \mathbb{R}^{d-1}\).
A (quasi-stationary) \(d\)-dimensional affine wavelet system consists of all shifts by \(k\in\mathbb{Z}^d\) of the generators \[ \text{AS}(\varphi, \{\Psi_j\}_{J}^\infty )= \{\varphi_{M_\lambda^J}\, \} \cup\{\psi_{S^{-\vec\ell}A^{j}_\lambda E_n}^{j,\vec{\ell}}: n=1,\dots, d,|\vec\ell|\leq \vec{s}_j\}_{j=J}^\infty \] with particular choices \(\vec{s}_j\in \mathbb{N}_0^{d-1}\) and \(\lambda>0\).
A (quasi-stationary) \(d\)-dimensional affine wavelet system consists of all shifts by \(k\in\mathbb{Z}^d\) of the generators \[ \text{WS}(\varphi^{J},{\Psi}_j\}_{j=J}^\infty =\{\varphi_{M_{\lambda}^J},\, \cup \psi_{M_\lambda^J E_{n}}^{j,\vec{\ell}}: n=1,\dots, d,|\vec\ell|\leq \vec{s}_j\}_{j=J}^\infty \]
It is shown in Theorem 2.1 that the affine shearlet system \(\text{AS}(\varphi, \{\Psi_j\}_{j=J}^\infty\) forms a tight frame if certain Fourier domain identities apply to the generators. In Sect. 2.2 systems of generators satisfying the desired identities are constructed starting from a regular partition of unity in frequency. Theorem 2.3 shows that \(\text{AS}(\varphi, \{\Psi_j\}_{j=J}^\infty)\) generates an affine shearlet tight frame if and only if \(\text{WS}(\varphi^{J},\Psi_j\}_{j=J}^\infty)\) generates an affine wavelet tight frame, starting from some base scale \(J_0\), provided frequency supports of different generators are disjoint. The relation between wavelet and shearlet generators is \(\psi^{j,\vec\ell}=\lambda^{-(d-1)j}\psi^{j,\vec\ell}(S^{-\vec\ell}D_\lambda^{-j}\cdot)\).
Section 3 focuses on developing the frame criteria in discrete coordinates starting from a fixed sample scale. The decomposition and reconstruction steps for the digital affine shearlet stransform are laid out in Algorithms 1 and 2 and redundancy rates are estimated essentially by \(d 2^d\), but are shown to be lower in practice. Main computations are done in the discrete Fourier domain. Performance in denoising or inpainting applications is reported in terms of PNSR, defined as \(\text{PNSR}(u,\tilde u)= 10\log_{10} \Bigl(\frac{255^2}{\text{MSE}(u,\tilde u)}\Bigr)\) where \(u\) is a denoised or inpainted approximate reconstruction. In Sect. 4, the author’s implementation (DAS) is compared with the dual-tree complex wavelet transform (DT-\(\mathbb{C}\)WT), standard shearlet transform (DNST), curvelets (FDCT) and another inpainting system (TP-\(\mathbb{C}\)TF\(_6\)), and shows consistent results with these systems at different noise levels with moderate improvement in most cases studied.