Spherical Tensor Algebra: A Toolkit for 3D Image Processing

Abstract

With the advent of novel 3D image acquisition techniques, their efficient and reliable analysis becomes more and more important. In particular in 3D, the amount of data is enormous and requires for an automated processing. The tasks are manifold, starting from simple image enhancement, image reconstruction, image description and object/feature detection to high-level contextual feature extraction. One important property that most of these tasks have in common is their covariance to rotations. Spherical Tensor Algebra (STA) offers a general framework to fulfill these demands. STA transfers theories from mathematical physics and harmonic analysis into the domain of image analysis and pattern recognition. The main objects of interest are orientation fields. The interpretations of the fields are manifold. Depending on the application, they can represent local image descriptors, features, orientation scores or filter responses. STA deals with the processing of such fields in the domain of the irreducible representations of the rotation group. Two operations are fundamental: the extraction/projection of the features by convolution-like procedures and the nonlinear covariant combination by spherical products. In this paper, we propose an open-source toolbox that implements, in addition to fundamental STA operators, advanced functions for feature detection and image enhancement and makes them accessible to the 3D image processing community. The core features are implemented in C (CPU and GPU) with APIs in C++ and MATLAB. As examples, we show applications for medical and biological images.

Appendices

Appendix: Cartesian Tensors

Definition 6

(Epsilon Tensor) The epsilon tensor is defined by

$$\begin{aligned} \epsilon _{ijk}:={\left\{ \begin{array}{ll} 0, \text { if any two labels are identical}\\ 1, \text { if i, j, k is an even permutation of 1,2,3}\\ -1, \text { if i, j, k is an odd permutation of 1,2,3}\\ \end{array}\right. } \end{aligned}$$

Spherical two Cartesian Transformation: Let $\mathbf {T}^2$ be a Cartesian tensor of order 2. Then

$$\begin{aligned}&\mathbf {b}^0= {\frac{-\left( t^{(2)}_{00}+t^{(2)}_{11}+t^{(2)}_{22} \right) }{\sqrt{3}}}, \mathbf {b}^1= { \begin{pmatrix} \frac{1}{2} \left( t^{(2)}_{20}-t^{(2)}_{02}+{\mathbf {i}}\left( t^{(2)}_{12}-t^{(2)}_{21}\right) \right) \\ \frac{{\mathbf {i}}}{\sqrt{2}}\left( t^{(2)}_{01}-t^{(2)}_{10}\right) \\ \frac{1}{2} \left( t^{(2)}_{20}-t^{(2)}_{02}-{\mathbf {i}}\left( t^{(2)}_{12}-t^{(2)}_{21}\right) \right) \end{pmatrix}} \\&~\mathbf {b}^2={\begin{pmatrix} \frac{1}{2} \left( t^{(2)}_{00}-t^{(2)}_{11}+{\mathbf {i}}\left( t^{(2)}_{01}+t^{(2)}_{10}\right) \right) \\ \frac{1}{2} \left( (t^{(2)}_{02}+t^{(2)}_{20})+{\mathbf {i}}\left( t^{(2)}_{12}+t^{(2)}_{21}\right) \right) \\ \frac{-1}{\sqrt{6}}\left( t^{(2)}_{00}+t^{(2)}_{11}-2t^{(2)}_{22} \right) \\ \frac{1}{2} \left( -(t^{(2)}_{02}+t^{(2)}_{20})+{\mathbf {i}}\left( t^{(2)}_{12}+t^{(2)}_{21}\right) \right) \\ \frac{1}{2} \left( t^{(2)}_{00}-t^{(2)}_{11}-{\mathbf {i}}\left( t^{(2)}_{01}+t^{(2)}_{10}\right) \right) \end{pmatrix}} ~. \end{aligned}$$

are the irreducible representations of $\mathbf {T}^2$ according to (10). The inverse, see (11), is

$$\begin{aligned}&\mathbf {T}^2_\text {tr}= {-\frac{ b^0_0}{\sqrt{3}} \mathbf {I}_{3 \times 3}} ,\nonumber \\&\mathbf {T}^2_\text {anti}= { \begin{pmatrix} 0 &{} - {\mathbf {i}}\frac{b^1_0}{\sqrt{2}} &{} -\frac{1}{2}\left( b^1_{-1}+b^1_{1}\right) \\ {\mathbf {i}}\frac{b^1_0}{\sqrt{2}} &{} 0 &{} -{\mathbf {i}}\frac{1}{2}\left( b^1_{-1}-b^1_{1}\right) \\ \frac{1}{2}\left( b^1_{-1}+b^1_{1}\right) &{} {\mathbf {i}}\frac{1}{2}\left( b^1_{-1}-b^1_{1}\right) &{} 0 \end{pmatrix}, } \nonumber \\&\mathbf {T}^2_\text {sym}= { \begin{pmatrix} \frac{1}{6}\left( 3 b^2_{-2} - \sqrt{6} b^2_{0}+ 3 b^2_{2}\right) &{}\quad \frac{1}{2} {\mathbf {i}}\left( b^2_{-2} - b^2_{2}\right) &{}\quad \frac{1}{2} \left( b^2_{-1} - b^2_{1}\right) \\ \frac{1}{2} {\mathbf {i}}\left( b^2_{-2} - b^2_{2}\right) &{}\quad \frac{1}{6}(-3 b^2_{-2} - \sqrt{6} b^2_{0}- 3 b^2_{2}) &{} \quad \frac{1}{2} {\mathbf {i}}(b^2_{-1} + b^2_{1}) \\ \frac{1}{2} \left( b^2_{-1} - b^2_{1}\right) &{} \quad \frac{1}{2} {\mathbf {i}}\left( b^2_{-1} + b^2_{1}\right) &{}\quad \sqrt{\frac{2}{3}}b^2_0 \\ \end{pmatrix}}. \end{aligned}$$

(58)

1.1 Irreducible Components

In this paragraph, we give a brief sketch about the composition of Cartesian tensors with a full DOF in terms of their irreducible counterparts. For a proof and further details, we refer to section 4.4 in [15].

The order of all irreducible components of an order $j$ Cartesian is $\le j$. We start with the most simple tensor with directional information, a spherical tensor of order one. The idea is to, based on order one tensors, recursively construct all possible spherical tensors up to order $j$. The number and orders of the set of tensors are identical to the number and orders of the irreducible components of an order $j$ Cartesian tensor.

In Sect. 3.1, we have seen that there are, for an order one tensor, three possible operations in each step: decreasing the order (inner product-like operation), keeping the order (cross product-like) and increasing the order (an outer product). With these three operations, we set up Algorithm 1. Table 1 shows the results for tensors up to order 5.

Table 1 Relation between Cartesian and spherical (irreducible) tensors

Spherical Harmonics

We always use Racah-normalized spherical harmonics such that $\mathbf {Y}^\ell (\mathbf {r})^\top \mathbf {Y}^\ell (\mathbf {r}) = 1$, or $\mathbf {Y}^\ell (\mathbf {r})^\top \mathbf {Y}^\ell (\mathbf {r}') = P_\ell (\cos (\mathbf {r},\mathbf {r}'))$, where the $P_\ell$ are the Legendre polynomials:

$$\begin{aligned} P_\ell (t) = \frac{1}{2^\ell \ell !} \partial _t^\ell (t^2-1)^\ell . \end{aligned}$$

(59)

In terms of the associated Legendre polynomials, the components $Y^\ell _m$ of the spherical harmonics are written as

$$\begin{aligned} Y^\ell _m (\phi ,\theta ) = \sqrt{\frac{(l-m)!}{(l+m)!}} P_\ell ^m(\cos (\theta )) e^{{\mathbf {i}}m\phi }. \end{aligned}$$

(60)

The Racah-normalized spherical harmonics are orthogonal with respect to

$$\begin{aligned} \int _{S_2} \overline{Y^{\ell _1}_{m_1}} (\mathbf {n}) {Y^{\ell _2}_{m_2}}(\mathbf {n}) d \mathbf {n} = \frac{4 \pi }{2 j+1}\delta _{j_1,j_2} \delta _{m_1,m_2}. \end{aligned}$$

(61)

They can be turned into the orthonormal spherical harmonics via $\sqrt{\frac{2 j+1}{4 \pi }} Y^{\ell }_{m}$.

Mostly, we write $\mathbf {r} \in S^2$ instead of $(\phi ,\theta )$. The Racah-normalized solid harmonics⁴ can be written as

$$\begin{aligned}&R^\ell _m(\mathbf {r}) =\sqrt{(\ell +m)!(\ell -m)!}\nonumber \\&\quad \times \sum _{i,j,k} \frac{\delta _{i+j+k,\ell }\delta _{i-j,m}}{i!j!k! 2^i 2^j} (x-{\mathbf {i}}y)^j (-x-{\mathbf {i}}y)^i z^k, \end{aligned}$$

(62)

where $\mathbf {r} = (x,y,z)$. They are related to spherical harmonics⁵ by $R^\ell _m(\mathbf {r}) /r^\ell = Y^\ell _m(\mathbf {r})$.

The spherical harmonics rotate according to

$$\begin{aligned} \mathbf {Y}^j(\mathbf {U}(g) \mathbf {n}) = \mathbf {\mathbf {D}}({g})^j \mathbf {Y}^j( \mathbf {n}). \end{aligned}$$

(63)

Coupling two spherical harmonics with each other gives another spherical harmonic of desired order:

$$\begin{aligned}&\mathbf {Y}^{j_1}(\mathbf {n})\circ _{\!j}\mathbf {Y}^{j_2}(\mathbf {n}) = \langle j 0 | j_1 0, j_2 0\rangle \mathbf {Y}^{j}(\mathbf {n}) ~. \end{aligned}$$

(64)

$$\begin{aligned}&\langle J 0 | \ell _1 0,\ell _2 0 \rangle Y^J_M \nonumber \\&\quad = \sum _{m_1,m_2} \langle J M | \ell _1 m_1,\ell _2 m_2 \rangle Y^{\ell _1}_{m_1} Y^{\ell _2}_{m_2} \end{aligned}$$

(65)

$$\begin{aligned}&\langle \ell _2 0 | \ell _1 0,J 0 \rangle Y^J_M \nonumber \\&\quad = \sum _{m_1,m_2} \langle \ell _2 m_2 | \ell _1 m_1, J M \rangle \overline{Y^{\ell _1}_{m_1}} Y^{\ell _2}_{m_2} \end{aligned}$$

(66)

$$\begin{aligned}&Y^{\ell _1}_{m_1} Y^{\ell _2}_{m_2} \nonumber \\&\quad = \sum _{J,M} \langle J M | \ell _1 m_1,\ell _2 m_2 \rangle \langle J 0 | \ell _1 0,\ell _2 0 \rangle Y^{J}_{M} \end{aligned}$$

(67)

$$\begin{aligned}&\overline{Y^{\ell _1}_{m_1}} Y^{\ell _2}_{m_2} \nonumber \\&\quad = \sum _{J,M} \frac{2J+1}{2\ell _1+1} \langle \ell _1 m_1 | \ell _2 m_2,J M \rangle \langle \ell _1 0 | \ell _2 0,J 0 \rangle Y^{J}_{M} \end{aligned}$$

(68)

Spherical Expansion of the Dirac Delta Function Let $\mathbf {n},\mathbf {n}'\in S_2$ and $\delta ^2_{\mathbf {n}}:S_2\rightarrow {\mathbb {R}}$ the delta function on the 2-sphere, whereas $\delta _{\mathbf {n}}^2(\mathbf {n}')=\delta (\theta -\theta ') \delta (\phi -\phi ')$ and $\int _{S_2} \delta ^2_{\mathbf {n}}(\mathbf {n}') d \mathbf {n}'=1$. According to [7], page 792,

$$\begin{aligned} \delta ^2_{\mathbf {n}}(\mathbf {n}'):=\sum _{j=0}^\infty \textstyle {\frac{(2j+1)}{4\pi }} {({\mathbf {Y}^j}(\mathbf {n}'))}^T {\mathbf {Y}^j}(\mathbf {n}). \end{aligned}$$

(69)

The Plane Wave The plane wave expansion (see e.g [88], p. 136) in terms of spherical harmonics:

$$\begin{aligned} e^{{\mathbf {i}}{\mathbf {k}}^T \mathbf {r}}=\sum _{j=0}^\infty ({\mathbf {i}})^j(2j+1) J_{j}(k r){\mathbf {Y}^j}(\mathbf {r})^T \mathbf {Y}^j(\mathbf {k}). \end{aligned}$$

(70)

Clebsch–Gordan Coeffcients

The Clebsch–Gordan coefficients written in terms binomial coefficients⁶ are defined by

(71)

see also [1].

The Clebsch–Gordan coefficients of SO(3) fulfill several orthogonality relations:

$$\begin{aligned}&\sum _{j,m} \langle j m | j_1 m_1, j_2 m_2 \rangle \langle j m | j_1 m'_1, j_2 m'_2 \rangle \nonumber \\&\quad =\delta _{m_1,m_1'} \delta _{m_2,m_2'} \end{aligned}$$

(72)

$$\begin{aligned}&\sum _{j,m} \frac{2j+1}{2j_1+1} \langle j_1 m_1 | j m, j_2 m_2 \rangle \langle j_1 m_1' | j m, j_2 m_2' \rangle \nonumber \\&\quad = \delta _{m_1,m_1'} \delta _{m_2,m_2'} \end{aligned}$$

(73)

$$\begin{aligned}&\sum _{{m = m_1 + m_2} } \langle j m | j_1 m_1, j_2 m_2 \rangle \langle j' m' | j_1 m_1, j_2 m_2 \rangle \nonumber \\&\quad = \delta _{j,j'} \delta _{m,m'} \end{aligned}$$

(74)

$$\begin{aligned}&\sum _{m_1,m } \langle j m | j_1 m_1, j_2 m_2 \rangle \langle j m | j_1 m_1, j'_2 m'_2 \rangle \nonumber \\&\quad = \frac{2j+1}{2j'_2+1} \delta _{j_2,j'_2} \delta _{m_2,m_2'} \end{aligned}$$

(75)

For particular combinations, there are simple, explicit formulas:

$$\begin{aligned}&\langle \ell m | (\ell -\lambda )(m-\mu ), \lambda \mu \rangle =\nonumber \\&\quad \left( \begin{array}{c} \ell +m \\ \lambda + \mu \end{array} \right) ^{1/2} \left( \begin{array}{c} \ell -m \\ \lambda - \mu \end{array}\right) ^{1/2} \left( \begin{array}{c} 2\ell \\ 2\lambda \end{array}\right) ^{-1/2} \end{aligned}$$

(76)

$$\begin{aligned}&\langle \ell m | (\ell +\lambda )(m-\mu ), \lambda \mu \rangle =\nonumber \\&\quad \times (-1)^{\lambda +\mu }\left( \begin{array}{c} \ell +\lambda -m + \mu \\ \lambda + \mu \end{array} \right) ^{1/2} \nonumber \\&\quad \times \left( \begin{array}{c} \ell +\lambda +m-\mu \\ \lambda - \mu \end{array}\right) ^{1/2} \left( \begin{array}{c} 2\ell + 2\lambda + 1 \\ 2\lambda \end{array}\right) ^{-1/2} \end{aligned}$$

(77)

There are several symmetry relations

$$\begin{aligned}&\langle j m | j_1 m_1, j_2 m_2 \rangle = \langle j_1 m_1, j_2 m_2| j m \rangle \end{aligned}$$

(78)

$$\begin{aligned}&\langle j m | j_1 m_1, j_2 m_2 \rangle = (-1)^{j+j_1+j_2} \langle j m | j_2 m_2, j_1 m_1 \rangle \end{aligned}$$

(79)

$$\begin{aligned}&\langle j m | j_1 m_1, j_2 m_2 \rangle \nonumber \\&\quad =(-1)^{j+j_1+j_2} \langle j (-m) | j_1 (-m_1), j_2 (-m_2) \rangle \end{aligned}$$

(80)

$$\begin{aligned}&\langle j m | j_1 m_1, j_2 m_2 \rangle \nonumber \\&\quad =\sqrt{\frac{2j+1}{2j_2+1} }(-1)^{j_1+m_1} \langle j_2 m_2 | j m, j_1 (-m_1) \rangle , \end{aligned}$$

(81)

and associativity relations:

$$\begin{aligned}&\langle J,M|j_1+j_2,m_1+m_2,j_3,m_3\rangle \nonumber \\&\quad \times \langle j_1+j_2,m_1+m_2|j_1,m_1,j_2,m_2\rangle \nonumber \\&=\langle J,M|j_1+j_3,m_1+m_3,j_2,m_2\rangle \nonumber \\&\quad \quad \times \langle j_1+j_3,m_1+m_3|j_1,m_1,j_3,m_3 \rangle \end{aligned}$$

(82)

where $J=j_1+j_2+j_3$ and $M=m_1+m_2+m_3$. For $j_3 > j_1+j_2$, there exist further associativities, namely

$$\begin{aligned}&\langle j_3-j_1-j_2,m_1+m_2+m_3| \nonumber \\&\quad \quad J-j_1,m_1+m_3,j_2,m_2\rangle \nonumber \\&\quad \times \langle j_3-j_1,m_1+m_3|j_1,m_1,J,m_3\rangle \nonumber \\&=\langle j_3-j_1-j_2,m_1+m_2+m_3| \nonumber \\&\quad \quad J-j_2,m_2+m_3,j_1,m_1\rangle \nonumber \\&\quad \quad \times \langle j_3-j_2,m_2+m_3|j_2,m_2,J,m_3\rangle \end{aligned}$$

(83)

Wigner 6j-Symbols

Definition 7

(Wigner 6j-Symbols) The Wigner 6j-Symbols are defined by

$$\begin{aligned}&\left\{ \begin{matrix} j_1 &{} j_2 &{} j_4 \\ J &{} j_3 &{} j_5 \end{matrix}\right\} =\frac{(-1)^{j_1+j_2+j_3+J}}{\sqrt{(2j_4+1)(2j_5+1)}} \nonumber \\&\quad \times \sum _{\begin{array}{c} {m_1},{m_2},{m_3}\\ {m_4},{m_5} \end{array}} \langle j_1 m_1, j_2 m_2 | j_4 m_4 \rangle \langle j_3 m_3, j_4 m_4 | J M \rangle \nonumber \\&\quad \times \langle j_1 m_1, j_3 m_3 | j_5 m_5 \rangle \langle j_2 m_2, j_5 m_5 | J M \rangle ~. \end{aligned}$$

(84)

(see, e.g., [68], page 1, eq. (D.2))

The permutation of any pairs

$$\begin{aligned} \begin{matrix} j_1 \\ j_2 \end{matrix} \left\lbrace \begin{matrix} ~ \\ ~ \end{matrix} \rightleftarrows \begin{matrix} ~ \\ ~ \end{matrix}\right\rbrace \begin{matrix} j_3 \\ j_4 \end{matrix} \text {or} \begin{matrix} j_1 \\ j_2 \end{matrix} \left\lbrace \begin{matrix} ~ \\ ~ \end{matrix} \rightleftarrows \begin{matrix} ~ \\ ~ \end{matrix}\right\rbrace \begin{matrix} j_2 \\ j_1 \end{matrix} \end{aligned}$$

(85)

leaves the value of the 6j symbol unaltered (see [68]). Similar to the Clebsch–Gordan coefficients, there exist simple, explicit expressions for some special cases:⁷

$$\begin{aligned}&\left\{ \begin{matrix} j_1 &{} j_2 &{} (j_1+j_2) \\ (j_1+j_2+j_5) &{} j_5 &{} (j_1+j_5) \end{matrix}\right\} \nonumber \\&\quad =\frac{(-1)^{2j_1+2j_2+2j_5}}{\sqrt{2(j_1+j_2)+1}\sqrt{2(j_1+j_5)+1}} ~, \end{aligned}$$

(86)

$$\begin{aligned}&\left\{ \begin{matrix} j_1 &{} j_2 &{} j_3 \\ 0 &{} j_5 &{} j_6 \end{matrix}\right\} =\frac{(-1)^{j_1+j_2+j_5}\delta _{j_2,j_6}\delta _{j_3,j_5}}{\sqrt{2j_2+1}\sqrt{j_3+1}} \end{aligned}$$

(87)

and

$$\begin{aligned}&\left\{ \begin{matrix} j_1 &{} j_2 &{} (j_1+j_2) \\ (j_3-j_2-j_1) &{} j_3 &{} (j_3-j_1) \end{matrix}\right\} \nonumber \\&\quad =\frac{1}{\sqrt{2(j_1+j_2)+1}\sqrt{2(j_3-j_1)+1}} ; \end{aligned}$$

(88)

a conclusion from Theorem  1.

Wigner-D Matrix

The irreducible representation of SO(3) is called Wigner-D matrices [115, 116]. We denote them by the matrix $\mathbf {D}^j(g)\in {\mathbb {C}}^{(2j+1)\times (2j+1)}$, where $j\in {\mathbb {N}}_0$, with $j=\{0,\ldots ,\infty \}$. The $j$th order representation works on a ${\mathbb {C}}^{2j+1}$-dimensional vector space. We denote the components of $\mathbf {D}^j(g)$ by $D^j_{mn}(g)$. In Euler angles in ZYZ convention, we have

$$\begin{aligned} D^j_{mn}(\gamma ,\beta ,\alpha ) = e^{-{\mathbf {i}}m\gamma } d^j_{mn}(\beta ) e^{{\mathbf {i}}n \alpha }, \end{aligned}$$

(89)

where $d^j_{mn}(\beta )$ is the ’small’ Wigner-d matrix, which is real-valued and explicitly written as

$$\begin{aligned}&d^j_{mn}(\beta ) = [(j+m)!(j-m)!(j+n)!(j-n)!]^{1/2}\nonumber \\&\quad \sum _s \frac{(-1)^{m-n+s}}{(j+n-s)!s!(m-n+s)!(j-m-s)!} \nonumber \\&\quad \quad \times \left( \cos \frac{\beta }{2}\right) ^{2j+n-m-2s}\left( \sin \frac{\beta }{2}\right) ^{m-n+2s}. \end{aligned}$$

(90)

The representations of different orders are connected via the Clebsch–Gordan coefficients by:

$$\begin{aligned}&D^{j_1}_{m_1 n_1}D^{j_2}_{m_2 n_2} \nonumber \\&\quad = \sum _{l,m,n} D^{j}_{m n} \langle jm | j_1 m_1, j_2 m_2 \rangle \langle jn | j_1 n_1, j_2 n_2 \rangle ; \end{aligned}$$

(91)

see equation 2.3.2 in [15].

Another important equality is

$$\begin{aligned}&\int _{SO(3)} \overline{D^{j_1}_{m_1 n_1}} (g) \overline{D^{j_2}_{m_2 n_2}} (g) D^{j_3}_{m_3 n_3}(g) dg\nonumber \\&\quad = \frac{8 \pi ^2}{2 j_3 + 1} \langle j_3 m_3 | j_2 m_2, j_1 m_1 \rangle \langle j_3 n_3 | j_2 n_2 ,j_1 n_1 \rangle \end{aligned}$$

(92)

Relation to Spherical Harmonics The Wigner-D matrices build an orthogonal basis for functions in SO(3). Let $f(\theta ,\phi ,\psi ):SO(3)\rightarrow {\mathbb {C}}$ be a function with $f(\theta ,\phi ,\psi )=f(\theta ,\phi )$. Let $A^j_{m,n}\in {\mathbb {C}}$ be the expansion coefficients of f in terms of the Wigner-D matrices. Then the expansion

$$\begin{aligned} f(\varphi ,\theta ,\psi )&=\sum _{j=0}^\infty \sum _{m,n=-j}^j\overline{A^j_{m,n}} D^j_{mn}(\varphi ,\theta ,\psi )\nonumber \\&=c(j) \sum _{j=0}^\infty \sum _{m=-j}^j\overline{a}^j_{m} Y^j_{m}(\varphi ,\theta ) \end{aligned}$$

(93)

is the spherical harmonic expansion (up to a constant $c(j)\,{\in }\,{\mathbb {R}}$).

Real and Imaginary Tensor Fields

For all spherical tensors, $\mathbf {v}^j\in {\mathbb {C}}^{2j+1}$ exits a conjugated counterpart $\left( \mathbf {v}^j\right) ^\ddagger \in {\mathbb {C}}^{2j+1}$, with $\left( v^j_m\right) ^\ddagger :=(-1)^m \overline{v^j_{-m}}$. The tensor conjugation induces a unique decomposition of the spherical tensor space ${\mathbb {C}}^{2j+1}$ into two vector spaces $V_j,{{\mathbf {i}}V}_j\subset {\mathbb {C}}^{2j+1}$. Let $\mathbf {v}^j\in {\mathbb {C}}^{2j+1}$, then

$$\begin{aligned} \mathbf {v}^j=\underset{\in V_j}{\underbrace{{\frac{(\mathbf {v}^j+\left( \mathbf {v}^j\right) ^\ddagger )}{2}}}}\oplus \underset{\in {{\mathbf {i}}V}_j}{\underbrace{{\frac{(\mathbf {v}^j-\left( \mathbf {v}^j\right) ^\ddagger )}{2}}}}. \end{aligned}$$

(94)

Despite the fact that these vector spaces are complex valued, we treat them as real-valued vector spaces, because they are closed under weighted superposition for the real numbers; i.e., if $\mathbf {v}^j\in V_j$, then $\forall \alpha \in {\mathbb {R}}: \alpha \mathbf {v}^j\in V_j$. Same for ${{\mathbf {i}}V}_j$. With this assumption, the spherical tensor space ${\mathbb {C}}^{2j+1}$ is a direct sum of these two subspaces, that is ${\mathbb {C}}^{2j+1}=V_j\oplus {{\mathbf {i}}V}_j$. For the sake of consistency to standard complex numbers, we call the vector space $V_j\subset {\mathbb {C}}^{2j+1}$ the real spherical tensor space and ${{\mathbf {i}}V}_j\subset {\mathbb {C}}^{2j+1}$ the imaginary spherical tensor space, i.e., we can always represent an $\mathbf {v}^j\in {{\mathbf {i}}V}_j$ in terms of an ${\mathbf {i}}\mathbf {w}^j$, where $\mathbf {w}^j\in V_j$ (and vice versa).

Corollary 2

( Symmetry) Let $\mathbf {v}^j\in V_j$ and $\mathbf {w}^j\in {{\mathbf {i}}V}_j$. The tensors $\mathbf {v}^j$ and $\mathbf {w}^j$ have the following symmetries

$$\begin{aligned} v^j_m=(-1)^m \overline{v^j_{-m}}~ {and}&~{(real\, \,tensor\, space)} \end{aligned}$$

(95)

$$\begin{aligned} w^j_m=(-1)^{m+1} \overline{w^j_{-m}}~.&~{(imaginary\, tensor\, space)} \end{aligned}$$

(96)

This is a direct conclusion from the tensor conjugation property.

Fig. 26

(Coupling Three Spherical Tensors) The products defined in each third of this circle are spanning the tensor space of the products of the remaining two thirds. Which means they are mutually linear dependent according to Theorem 1

Tensor Triple Products

Theorem 1

(Coupling Three Spherical Tensors) We have the following identity when coupling three spherical tensors $\mathbf {u}^{j_1}\in {\mathbb {C}}^{2j_1+1}$, $\mathbf {v}^{j_2}\in {\mathbb {C}}^{2j_2+1}$ and $\mathbf {w}^{j_3}\in {\mathbb {C}}^{2j_3+1}$ to form a tensor of rank J based on an intermediate rank ${ |j_1-j_2 |}\le L_{12}\le j_1+j_2$ :

$$\begin{aligned}&\displaystyle ((\mathbf {u}^{j_1} \circ _{L_{12}} \mathbf {v}^{j_2}) \circ _{J} \mathbf {w}^{j_3})\nonumber \\&\quad =\sum _{L_{23}} (\mathbf {u}^{j_1} \circ _{J} (\mathbf {v}^{j_2} \circ _{L_{23}} \mathbf {w}^{j_3})) \sqrt{(2L_{12}+1)(2L_{23}+1)} \nonumber \\&\quad \times (-1)^{j_1+j_2+j_3+J} \left\{ \begin{matrix} j_2 &{} j_1 &{} L_{12} \\ J &{} j_3 &{} L_{23} \end{matrix}\right\} . \end{aligned}$$

(97)

With $\left\{ \begin{matrix} j_1 &{} j_2 &{} j_4 \\ J &{} j_3 &{} j_5 \end{matrix}\right\} \in {\mathbb {R}}$, we denote the Wigner 6j-symbol (see Sect. 1), which are the weighting factors playing a role when coupling three spherical tensors. With Theorem 1, we can identify the symmetries that exist when coupling three spherical tensors. By exchanging the coupling order of the three tensors, we see that each of the following sets of tensors, $\{(\mathbf {u}^{j_1} \circ _{J} (\mathbf {v}^{j_2} \circ _{L_{23}} \mathbf {w}^{j_3}))\}_{\forall L_{23}}$, $\{(\mathbf {w}^{j_3} \circ _{J} (\mathbf {u}^{j_1} \circ _{L_{12}} \mathbf {v}^{j_2}))\}_{\forall L_{12}}$ and $\{(\mathbf {v}^{j_2} \circ _{J} (\mathbf {u}^{j_1} \circ _{L_{13}} \mathbf {w}^{j_3}))\}_{\forall L_{13}}$, can be formed via linear combination of tensors of only one of the remaining sets. That is, they are mutually linearly dependent. This fact is illustrated in Fig. 26. That is, regarding the computation of linearly independent features it is sufficient (and essential) to compute only one set of features out of those three linearly dependent sets. We use this property for computing an linearly independent set of bi-spectrum features in our applications.

Proof

In the following, we derive Eq. (97). According to [117], page 17, Eq. (90), there exists the recoupling rule

$$\begin{aligned}&\displaystyle \sum _{M_{12}}\langle j_1 m_1, j_2 m_2 | L_{12} M_{12} \rangle \langle L_{12} M_{12}, j_3 m_3 | J M \rangle \nonumber \\&\quad = \sum _{L_{23},M_{23}} \sqrt{(2L_{12}+1)(2L_{23}+1)}W(j_1 j_2 J j_3,L_{12}L_{23})\nonumber \\&\quad \times \langle j_1 m_1, L_{23} M_{23} | J M \rangle \langle j_2 m_2, j_3 m_3 | L_{23} M_{23} \rangle , \end{aligned}$$

(98)

where $W(j_1 j_2 J j_3,L_{12}L_{23})\in {\mathbb {R}}$ is a Racah W-coefficient [78]. Moreover, the following relation to the Wigner 6j-symbols is known (see, e.g., [105], p. 17, Eqs. (93) and (94))

(99)

By just writing out the tensor product of three spherical tensors, and by substituting Eqs. (99) into (98), we can derive the equation in Theorem 1, namely

$$\begin{aligned}&\left[ ((\mathbf {u}^{j_1} \circ _{L_{12}} \mathbf {v}^{j_2})\circ _{J} \mathbf {w}^{j_3})\right] _{M}= \sum _{M=M_{12}+m_3} u^{j_1}_{m_1}v^{j_2}_{m_2}w^{j_3}_{m_3} \nonumber \\&\quad \times \langle L_{12} M_{12}, j_3 m_3 | J M \rangle \langle j_1 m_1, j_2 m_2 | L_{12} M_{12} \rangle \nonumber \\&=\sum _{m_3}\sum _{M_{12}}\langle L_{12} M_{12}, j_3 m_3 | J M \rangle \nonumber \\&\quad \times \langle j_1 m_1, j_2 m_2 | L_{12} M_{12} \rangle u^{j_1}_{m_1}v^{j_2}_{m_2}w^{j_3}_{m_3}\nonumber \\&=\sum _{m_3,L_{23},M_{23},L_{23}} \sqrt{(2L_{12}+1)(2L_{23}+1)} (-1)^{j_1+j_2+j_3+J}\nonumber \\&\quad \times \left\{ \begin{matrix} j_2 &{} j_1 &{} L_{12} \\ J &{} j_3 &{} L_{23} \end{matrix}\right\} \langle j_1 m_1, L_{23} M_{23} | J M \rangle \nonumber \\&\quad \times \langle j_2 m_2, j_3 m_3 | L_{23} M_{23} \rangle u^{j_1}_{m_1}v^{j_2}_{m_2}w^{j_3}_{m_3}\nonumber \\&= \sum _{{L_{23},m_3,M_{23}}} \langle j_1 m_1, L_{23} M_{23} | J M \rangle \nonumber \\&\quad \times \langle j_2 m_2, j_3 m_3 | L_{23} M_{23} \rangle u^{j_1}_{m_1}v^{j_2}_{m_2}w^{j_3}_{m_3} \nonumber \\&\quad \times \sqrt{(2L_{12}+1)(2L_{23}+1)} \nonumber \\&\quad (-1)^{j_1+j_2+j_3+J}\left\{ \begin{matrix} j_2 &{} j_1 &{} L_{12} \\ J &{} j_3 &{} L_{23} \end{matrix}\right\} \nonumber \\&=\sum _{L_{23}} [(\mathbf {u}^{j_1} \circ _{J} (\mathbf {v}^{j_2} \circ _{L_{23}} \mathbf {w}^{j_3}))]_M \nonumber \\&\quad \times \sqrt{(2L_{12}+1)(2L_{23}+1)} \nonumber \\&\quad (-1)^{j_1+j_2+j_3+J}\left\{ \begin{matrix} j_2 &{} j_1 &{} L_{12} \\ J &{} j_3 &{} L_{23} \end{matrix}\right\} ~. \end{aligned}$$

(100)

$\square$

Proving corollary 1

We show all three equalities using the recoupling rule Eq. (97). In the first scenario, we have

and in the second scenario

Similarly, the third case can be shown:

$\square$