SPHERICAL SHEARLET-BASED COMPRESSION AND RECONSTRUCTION METHOD FOR THREE-DIMENSIONAL SCALAR INFORMATION

Abstract

A spherical shearlet-based compression and reconstruction method for three-dimensional scalar information is disclosed. The method is used for processing data distributed in accordance to certain probability distribution in a three-dimensional space, and is especially suitable for processing random or deterministic scalar data having a spherical distribution feature under polar coordinates and being anisotropic on a sphere, including spatial data with physical significance and clinical observation data in biomedicine. In the present disclosure, on the basis of reasonably dividing the three-dimensional space into multiple concentric spherical layers, the three-dimensional data distribution is decomposed into multiple layers of spherical data. In each layer related spherical information is decomposed, and key information is extracted, compressed and stored by using the mathematical property of a spherical shearlet system, and the original three-dimensional data information can be reconstructed or approximately restored from extracted key data.

Description

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims priority to Chinese Patent Application No. 202310337079.6, filed on Mar. 31, 2023, the content of which is incorporated herein by reference in its entirety.

TECHNICAL FIELD

The present disclosure belongs to applied mathematics and computer graphics and particularly relates to a spherical shearlet-based compression and reconstruction method for three-dimensional scalar information.

BACKGROUND

Conventional methods such as Fourier analysis, spline analysis and wavelet analysis which have been utilized in data processing are mostly based on Cartesian coordinates. Recently, scientific researchers applied deep learning methods to the learning of a simply connected set represented in a two-dimensional polar coordinate and observed a significant improvement compared to a conventional learning method by adopting a Cartesian orthogonal coordinate system, which means that the polar coordinate representation has inherent advantages in analyzing some two-dimensional data. In the processing of three-dimensional data, some dataset features are concentrated within the neighborhood of a two-dimensional spherical surface, and their distribution has linear singularity with respect to the two-dimensional curved surface. If a representation system with a spherical anisotropic structure is adopted, combined with three-dimensional polar coordinates, key information from such data usually can be captured in a more accurate and efficient way and stored-a spherical shearlet representation is one representation with such characteristic. However, so far there is no technical method for three-dimensional information compression and reconstruction based on a spherical shearlet representation.

SUMMARY

Regarding the shortcomings of the existing technical methods, the present disclosure provides a spherical shearlet-based compression and reconstruction technical method for three-dimensional scalar information, which is used for decomposing, extracting, storing and reconstructing scalar data information meeting a certain distribution in a three-dimensional space, including data from a mass distribution with a spherical feature in the three-dimensional space or random data distributed according to certain probability distribution. In biomedicine, three-dimensional data of human heart, kidney and brain surface from ultrasound magnetic resonance and CT image are suitable for being processed by the spherical shearlet-based information compression and reconstruction method; and in nature, distribution data of earth surface ridges, submarine trenches and star surroundings may be analyzed by using this method as well.

A specific technical solution is as follows.

A spherical shearlet-based compression and reconstruction method for three-dimensional scalar information includes the following steps:

• • S1: decomposing a three-dimensional space or a set V in the three-dimensional space into nonintersecting concentric spherical layers, U_i∈IV_[ri,ri+1]=V, and partitioning, layer by layer, scalar data information X distributed in the three-dimensional space;
  • S2: setting a spherical layer selection mechanism F_S:X→F_SX={F_SX_i}_i∈I+ based on the type of the three-dimensional scalar data X, and extracting, layer by layer, spherical information to be processed by a discrete spherical shearlet system; and
  • S3: decomposing, extracting and storing the spherical data from each layer, by the discrete spherical shearlet system, to reconstruct three-dimensional data in the space. The discrete spherical shearlet system has the following expression:

$\begin{array}{cc}\left\{S_{j,k}:=S_{{\sigma }_{j,}{a}_k}^{\alpha }|{\sigma }_j\in G,a_k\in \left(0,\infty \right),0<\alpha <1\right\}& \left(1\right)\end{array}$

• •  where {a_k}_k≥1 represents a sampling of the positive real axis, a_k monotonically approaches zero; index α represents the degree of anisotropy, and the smaller a value of the index is, the higher the degree of anisotropy is; and G represents a finite or countable discrete subset of the orthogonal group SO(3), so that an integral, of a square-integrable function h on any sphere on the orthogonal group SO(3) has a discrete expression:

$\begin{array}{cc}{\langle h\rangle }_{\mathrm{SO}\left(3\right)}={\sum }_{{\sigma }_j\in G}h\left({\sigma }_j^{-1}{z}_0\right)w_j& \left(2\right)\end{array}$

• •  where z₀ represents a selected pole on the sphere, and w_j represents a weight depending on G. The discrete system may be obtained from a single or a finite number of generation functions S^α through spherical dilation transform D_a and spherical rotation on a discretized parameter set. P₁ is a projection onto the space spanned by spherical harmonic functions of degrees n=1, 2, . . . , 1, and S^α needs to satisfy, for example, the following restriction condition:

$\begin{array}{cc}\frac{1}{2l+1}{\int }_0^{\infty }{P_l{D}_a{S}^{\alpha }}_2^2{a}^{-2-\alpha }\mathrm{da}<\infty ,\forall l\ge 0& \left(3\right)\end{array}$

The discrete spherical shearlet system {S_j,k}_j,k has the ability of stable decomposition and reconstruction and has an adjustable anisotropic support, namely, in the case that S² is the two-dimensional sphere and R represents the real domain, inputted spherical information X_s:S²→R has the following reconstruction formula after certain normalization operation:

$\begin{array}{cc}{X}_s={\sum }_{l\le L}{P}_l{X}_s+{\sum }_{{\sigma }_j\in G}{\sum }_k{\sum }_{l>L}{X}_{s,j}{P}_l{S}_{j,k}\left({\sigma }_j^{-1}{z}_0\right)S_{j,k}& \left(4a\right)\end{array}$ $\begin{array}{cc}{X}_{s,j}=w_j{X}_s\left({\sigma }_l^{-1}{z}_0\right)& \left(4b\right)\end{array}$

where L represents a positive integer, X_s represents a distribution or a random variable that satisfies a square integrable condition. The spherical shearlet transform of X_s in discrete form has an expression:

$\begin{array}{cc}\mathrm{SH}\left(X_s;j,k\right):={\sum }_{l>L}{X}_{s,j}{P}_l{S}_{j,k}\left({\sigma }_j^{-1}{z}_0\right),& \left(5\right)\end{array}$

and its calculation concerns further discretization of P₁S_j,k, which can be accomplished through calculation prior to performing the spherical shearlet transform.

Further, in the step S1, a specific implementation step of partitioning, layer by layer, the set and the scalar data information is as follows: letting V=R³ be the entire three-dimensional real space or a bounded set including to-be-processed data. Prior information is unnecessary for the method of the present disclosure, but if prior information of data distribution of the entire space exists and how many blocks the data are partitioned is known, the entire space may be pre-partitioned into several parts by using an appropriate data classification method, and then each part is used as V to be processed, respectively. V is partitioned into concentric spherical layers V_[ri,ri+1], i∈I, that do not intersect with each other, according to the scale or size so of a local feature that needs to be extracted, so that r_i+1−r_i∝S₀, U_i∈I V_[ri,ri+1]=V, and at the same time, three-dimensional scalar data is partitioned correspondingly into {X_i}_i∈I, where X_i represents data of X in the concentric spherical layers V_[ri,ri+1]. The index set/represents a finite set or a countable set in the modeling sense and a finite set in the actual operation sense. Different probability measures u are selected to adapt to different data types, where the analyzed three-dimensional data distribution may be a continuous or discrete distribution. For example, a restriction of du onto the concentric spherical layers V_[ri,ri+1] may be defined to be dμ|v_[ri,ri+1]=X_idv.

Setting a threshold N_b<<μ(V) according to actual needs, directly abandoning data in the concentric spherical layers V_[ri,ri+1] and letting X_i=0, whenever a total data volume μ(V_[ri,ri+1])<N_b in V_[ri,ri+1]; or otherwise, further decomposing a data distribution in V_[ri,ri+1] into data distribution in each subdomain V_i,k, and V_[ri,ri+1]=U_kV_i,k. The selection of N_b shall ensure that the extraction of key information is not affected by abandoning data from a certain concentric spherical layer, while reducing the amount of calculation for analyzing data. The subdomains are located in a cone defined by the same radial sections with ∫X d{right arrow over (v)} as an apex if the subdomains V_i,k and V_i+1,k′ in adjacent layers V_[ri,ri+1] and V_[ri+1,ri+2] have a common area element.

Setting r_i,k,1=r_i,k,2=r_i,k,−1=r_i,k,−2=r_i on each subdomain V_i,k that satisfies μ (V_i,k)<δ_iμ (V_[ri,ri+1]), and δ_i<<1; otherwise, setting

$\begin{array}{cc}{c}_i^{\prime }={\mathrm{supX}}_j,d{\mu }^c{|}_{V_{i,k}}=c_i^{\prime }\mathrm{dv}-d\mu {|}_{V_{i,k}},& \left(6a\right)\end{array}$ $\begin{array}{cc}{\stackrel{\_}{r}}_{i,k}=\underset{r}{\mathrm{argmin}}{\int }_{V_{i,k}}{❘❘x❘-r❘}^{2}d{\mu }^c,& \left(6b\right)\end{array}$

and letting r_i,k,1=inf_r{r:μ({x∈V_i,k:r_i,k<|x|<r})>ε}, r_i,k,2=sup_r{r:μ({x∈V_i,k:r<|x|<r_i+1})>ε}, r_i,k,−1=sup_r{r:μ({x∈V_i,k:r<|x|<r_i,k})>ε}, and r_i,k,−2=inf_r{r:μ({x∈V_i,k:r_i<|x|<r})>ε}, where the value ε represents an appropriately chosen threshold. In particular, replacing the expression (6b) by

${\stackrel{\_}{r}}_{i,k}=\underset{r}{\mathrm{argmin}}{\sum }_{x\in V_{i,k}}{❘❘x❘-r❘}^2{X}_{i,k}\left(x\right)$

to reduce the amount of calculation when data in V_i,k include a small amount of discrete data points.

Partitioning the subdomain V_i,k into two along r=r_i,k to obtain two fine subdomains V′_i,k and V″_i,k in the case that a certain given positive constant c satisfies r_i,k,1−r_i,k,−1≥c·s₀, and letting r′_i,k,2=r_i,k,2, r′_i,k−2=r_i,k,1, r″_i,k,2=r_i,k,−1, and r′_i,k−2=r_i,k−2. While in other subdomains, letting r′_i,k,2=r_i,k,2, r′_i,k,−2=r″_i,k,2=r_i,k, and r″_i,k,−2=r_i,k,−2, or letting r′_i,k,2=r_i,k,2, r′_i,k,−2=r_i,k,−2, and r″_i,k,2=r″_i,k,−2=r_i. In the above subdivision, when μ(V′_i,k)<δ_iμ(V_[ri,ri+1]), data in V′_i,k may be abandoned directly while those in V″_i,k are retained. The steps are repeated to traverse all subdomains V_i,k, so that {V_[ri,ri+1]}_i∈1 is updated till r_i,k,1−r_i,k,−1 in each subdomain is a negligible quantity value and finally a group of subdomains {V_[ri,ri+1]}_i∈I+ is obtained. This process further subdivides a region where data distribution has a prominent geometric feature, for example, a region that contains unconnected parts is partitioned into two connected subdomains to be processed, respectively, which is an adaptive refinement.

After finite times of partitions, pairing and combining the subdomains of V_[ri,ri+1] together with their {r′_i,k,2, r′_i,k,−2}_V′i,k, {r″_i,k,2, r″_i,k,−2}_V″i,k . . . to obtain a finite set {(r′_i,2, r′_i,−2):r′_i,2|V_i,k=r′_i,k,2, r′_i,−2|v_i,k=r′_i,k,−2} in V_[ri,ri+1].

Further, the step S2 of setting an information selection mechanism F_S:X→F_SX={F_SX_i}_i∈I+ and decomposing data information in the three-dimensional space into multiple layers of spherical information F_SX for processing may, for example, adopt the following solution.

Defining 1_W as a characteristic function of a set W in the three-dimensional space, and considering a data distribution X=X_c=c_W·1_W, namely, X is a non-zero constant c_W on a locally connected space set W and almost everywhere zero on its complement W^c; and when each concentric spherical layer V_[ri,ri+1] of {V_[ri,ri+1] }_i∈I+ is fully refined, letting the to-be-processed data be:

$\begin{array}{cc}\{\begin{array}{c}{f}_i^{+}\left({\omega }_{i,k}\right)=\frac{r_{i,k,2}}{r_i}\\ f_i^{-}\left({\omega }_{i,k}\right)=\frac{r_{i,k,-2}}{r_i}\end{array}& \left(7\right)\end{array}$

where ω_i,k=(θ_i,k, φ_i,k) corresponds to directional coordinates of a subdomain V_i,k. The set W may correspond to, but not limited to, a three-dimensional cartoon figure with good local regularity.

Adopting the processing solution presented below, when data distributions X=Σ_m=1^m c_Wm·1_Wm or X=X_c+X_d include a non-negligible discrete component X_d=Σ_p∈Dd_pδ_p, where D represents a finite discrete subset in the three-dimensional space, d_p represents a positive integer, and δ_p represents the delta distribution at a point p. Defining Y to be a blockwise constant distribution that satisfies Y|v_i,k=∫y_i,kXdv, and letting {c_t, {V′_t}}_t≤tbe a set defined deductively according to c₁=minY|v_i,k together with a subdomain V′₁=argminY|v_i,k, c₂=min_vi,k≠v′1Y|_Vi,k−c₁ together with a subdomain V′₂=argmin_vi,k≠v′1Y|_vi,k, and so on; on each V_[ri,ri+1], defining distributions Y₁|v_i,k and Y_t|v_i,k deductively as:

$\begin{array}{cc}{Y}_1{|}_{V_{i,k}}=\{\begin{array}{cc}{c}_1& Y{|}_{V_{i,k}}\ne 0\\ 0& Y{|}_{V_{i,k}}=0\end{array}& \left(8a\right)\end{array}$ $⋮$ $\begin{array}{cc}{Y}_t{|}_{V_{i,k}}=\{\begin{array}{c}Y{|}_{V_{i,k}}-{\sum }_{s<t}{c}_s,Y{|}_{V_{i,k}}\ne 0\\ 0,Y{|}_{V_{i,k}}=0\mathrm{or}{V}_{i,k}\in U_{s<t}\left\{V_s^{\prime }\right\}\end{array}& \left(8b\right)\end{array}$

The to-be-processed data of a V_[ri,ri+1] layer satisfies the following expressions:

$\begin{array}{cc}\{\begin{array}{c}{f}_i^{+}={\sum }_{t\le \stackrel{\_}{t}}{\left(g_i^{+}\right)}_t\\ f_i^{-}={\sum }_{t\le \stackrel{\_}{t}}{\left(g_i^{-}\right)}_t\end{array}& \left(9\right)\end{array}$

Finally, step S3 is performed: denoting the spherical information F_SX part that includes the to-be-processed data depending on spherical coordinates in the expression (7) or the expression (9) as {xⁱ}_i∈I+={F_SX_i}_i∈I+, decomposing the spherical shearlet according to the expression (4a), and storing the coefficients {c_j,k^i,+}_j,k and {c_j,k^i,−}_j,k obtained from the spherical shearlet transform (5). The original three-dimensional data X is approximated by the following distribution:

$\begin{array}{cc}\tilde{X}=c_W·{\sum }_i\left(1_{U_i^{+}}-1_{U_i^{-}}\right)& \left(10\right)\end{array}$

where 1_Ui+ represents the characteristic function of a set U_i⁺={x=|x|ω_i,k∈R³:|x|≤{tilde over (x)}^i,+(ω_i,k)}, and 1_Ui−; represents the characteristic function of a set U_i⁻={x=|x|ω_i,k∈R³:|x|≤{tilde over (x)}^i,−(ω_i,k)}, and the following relations are satisfied:

$\begin{array}{cc}{\tilde{x}}^{i,+}={\tilde{x}}_L^{i,+}+{\tilde{x}}_H^{i,+}=r_i{\sum }_{l\le L}{P}_l{x}^{i,+}+r_i{\sum }_{j,k}{c}_{j,k}^{i,+}{S}_{j,k}& \left(11a\right)\end{array}$ $\begin{array}{cc}{\tilde{x}}^{i,-}={\tilde{x}}_L^{i,-}+{\tilde{x}}_H^{i,-}=r_i{\sum }_{l\le L}{P}_l{x}^{i,-}+r_i{\sum }_{j,k}{c}_{j,k}^{i,-}{S}_{j,k}& \left(11b\right)\end{array}$

so that on a finite (i, j, k) index set there are:

$\begin{array}{cc}{\sum }_i{x^{i,+}-x_L^{i,+}-{\sum }_{j,k}{c}_{j,k}^{i,+}{S}_{j,k}}_B\le {\epsilon }^{\prime }{\sum }_i{x^{i,+}}_B& \left(12a\right)\end{array}$ $\begin{array}{cc}{\sum }_i{x^{i,-}-x_L^{i,-}-{\sum }_{j,k}{c}_{j,k}^{i,-}{S}_{j,k}}_B\le {\epsilon }^{\prime }{\sum }_i{x^{i,-}}_B,& \left(12b\right)\end{array}$

where ε′<<1 is chosen according to required accuracy, ∥·∥_B represents a norm that may reflect data singular feature in space B adapted to the spherical shearlet system, and {tilde over (x)}_Hⁱ represents an approximation of a part of spherical data that is not of low degree. A corresponding shearlet coefficient has predictable sparsity in a case that F_SX has a linear singularity property on the sphere. The number of layers needing to be calculated is usually finite, so the calculation efficiency is determined by the efficiency of spherical shearlet representation.

Beneficial effects of the present disclosure are as follows:

In the present disclosure the three-dimensional data distribution is decomposed into multiple layers of spherical data through reasonably partitioning the three-dimensional space into a plurality of concentric spherical layers, and by using the spherical shearlet system the three-dimensional space scalar data in each layer are decomposed, compressed and reconstructed under a polar coordinate system. The spherical shearlet representation has foreseeable superiority over conventional methods in the aspect of processing scalar data with a spherical distribution feature in the three-dimensional space, especially those with a curvelinear singular distribution on the sphere.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic flowchart of the spherical shearlet-based compression and reconstruction method for three-dimensional scalar information provided by the present disclosure.

FIG. 2 is an illustration of data distribution in a concentric spherical layer.

FIG. 3 is a schematic diagram of a concentric spherical layer V_[ri,ri+1] with a cone contained inside.

DESCRIPTION OF EMBODIMENTS

The present disclosure is described below in detail according to main flows and functions of a method. Specific embodiments and accompanying conceptual drawings described here are merely for explaining the present disclosure instead of limiting the present disclosure, and optimization of a calculation quantity and the like in specific implementations is not discussed.

This method is mainly used for processing scalar data information in a three-dimensional space, including data from a mass distribution with a spherical feature in the three-dimensional space or random data distributed according to certain probability distribution. At the technical level, data whose three-dimensional space distribution coordinates can be inputted into a computer in matrix form and calculated, can be processed by this technical method, although the present disclosure is not described in a matrix language. In biomedicine, three-dimensional ultrasound, magnetic resonance and CT image data of structures such as heart, kidney and brain surface of people are suitable for being processed by the spherical shearlet-based information compression and reconstruction method; and in nature, distribution data of earth surface ridges, submarine trenches and star surroundings may be analyzed by using this method as well.

As shown in a schematic flowchart FIG. 1, a method of the present disclosure includes the following three steps:

• • S1: three-dimensional image data V is decomposed into a plurality of concentric spherical layers under polar coordinates, U_i∈IV_[ri,ri+1]=V, and acquired scalar data information X distributed in the three-dimensional space is partitioned layer by layer.
  • S2: a spherical layer selection mechanism F_S:X→F_SX={F_SX_i}_i∈I+ is set according to a type of the three-dimensional scalar data X, and spherical data information to be processed by a discrete spherical shearlet system is extracted layer by layer; and
  • S3: the spherical data from each layer are decomposed, extracted and stored by the discrete spherical shearlet system and then data in the three-dimensional space are reconstructed. The discrete spherical shearlet system has an expression:

$\begin{array}{cc}\left\{S_{j,k}:=S_{{\sigma }_{j,}{a}_k}^{\alpha }|{\sigma }_j\in G,a_k\in \left(0,\infty \right),0<\alpha <1\right\}& \left(1\right)\end{array}$

where {a_k}_k≥1 represents a sampling of the positive real axis, a_k monotonically approach zero, and δ′>0 exists, so that |a_k−a_k+1|<δ′; index α represents the degree of anisotropy, and the smaller a value of the index is, the higher the degree of anisotropy is; and G represents a finite or countable discrete subset of an orthogonal group SO(3), so that an integral of a square-integrable spherical function h on the orthogonal group has a discrete expression:

$\begin{array}{cc}{\langle h\rangle }_{\mathrm{SO}\left(3\right)}={\sum }_{{\sigma }_j\in G}h\left({\sigma }_j^{-1}{z}_0\right)w_j& \left(2\right)\end{array}$

where z₀ represents a selected pole on the sphere, and w; represents a weight depending on G. The discrete spherical shearlet system may be obtained from a single or a finite number of generation function S^α through spherical dilation transform D_a and spherical rotation on a discretized parameter set. P₁ is a projection onto the space spanned by spherical harmonic functions of degrees n=1, . . . , 1, and the generation functions S^α need to satisfy, for example, the following restriction condition:

$\begin{array}{cc}\frac{1}{2l+1}{\int }_0^{\infty }{P_l{D}_a{S}^{\alpha }}_2^2{a}^{-2-\alpha }\mathrm{da}<\infty ,\forall l\ge 0& \left(3\right)\end{array}$

The discrete spherical shearlet system {S_j,k}_j,k has the function of stably decomposing and reconstructing spherical information and has an adjustable anisotropic support, namely, in a case that S² is a two-dimensional sphere and Rsymbolizes real domain, inputted spherical information X_s:S²→R after normalization has a construction formula:

$\begin{array}{cc}{X}_s={\sum }_{l\le L}{P}_l{X}_s+{\sum }_{{\sigma }_j\in G}{\sum }_k{\sum }_{l>L}{X}_{s,j}{P}_l{S}_{j,k}\left({\sigma }_j^{-1}{z}_0\right)S_{j,k}& \left(4a\right)\end{array}$ $\begin{array}{cc}{X}_{s,j}=w_j{X}_s\left({\sigma }_j^{-1}{z}_0\right)& \left(4b\right)\end{array}$

where L represents a positive integer, X_s represents a distribution that is assumed to satisfy a square integrable condition ∫_s2|X_s|²dω<∞, and a spherical shearlet transform of X_s in discrete form has an expression:

$\begin{array}{cc}\mathrm{SH}\left(X_s;j,k\right)={\sum }_{l>L}{X}_{s,j}{P}_l{S}_{j,k}\left({\sigma }_j^{-1}{z}_0\right).& \left(5\right)\end{array}$

The specific calculation of the expression (5) concerns further discretization of P₁S_j,k, which may be obtained through calculation prior to performing the spherical shearlet transform.

In the step S1, a specific implementation step of partitioning the space set and the scalar information layer by layer is as follows:

V represents the entire three-dimensional real space R³ or a bounded set including to-be-analyzed data. Prior information is unnecessary for the method of the present disclosure, but if prior information of data distribution of the entire space exists and how many blocks the data are partitioned is known, the entire space may be pre-partitioned into several parts by using an appropriate data classification method, and then each part is used as V to be processed, respectively. V is partitioned into nonintersecting concentric spherical layers V_[ri,ri+1], where i∈I, according to the scale or size s₀ of a local feature that needs to be extracted, so that r_i+1−r_i∝S₀, U_i∈IV_[ri,ri+1]=V, and at the same time, three-dimensional data is partitioned correspondingly into {X_i}_i∈I, where X_i represents data of X contained in the concentric spherical layers V_[ri,ri+1]. The index set/is a finite set or a countable set in the model sense and is a finite set in the actual operation sense. If a spatial measure dv is given a certain coordinate to become a volume element d{right arrow over (v)} with a direction, a common center of the concentric spherical layers V_[ri,ri+1] can be chosen as ∫X d{right arrow over (v)}, and an origin of coordinates may be reset here for the convenience of subsequent calculation and operation. As analyzed data may form a continuous distribution or a discrete distribution, different probability measures u are selected according to different data types. For example, the restriction of dμ onto the concentric spherical layers V_[ri,ri+1] may be taken to be dμ|v_[ri,ri+1]=X_idv. FIG. 2 is an illustration of data distribution in a concentric spherical layer, in which the main part of data has a spherical distribution feature, and some non-negligible data set that is not necessarily connected to the main part, is distributed on the periphery.

A threshold N_b<<μ(V) is set according to actual needs, so that the concentric spherical layers V_[ri,ri+1] are abandoned directly by letting X_i=0 in the case that the total data volume μ(V_[ri,ri+1])<N_b in V_[ri,ri+1]. Otherwise, a data distribution in V_[ri,ri+1] may be further decomposed into data distribution in each subdomain V_i,k, with V_[ri,ri+1]=U_kV_i,k. Selection of N_b is supposed to ensure that the extraction of key information is not affected by abandoning data from a certain concentric layer, and meanwhile the amount of calculation for analyzing entire data is reduced. As shown in the schematic diagram of a concentric spherical layer V_[ri,ri+1] in FIG. 3, the cone part corresponds to a certain subdomain V_i,k in space division, and all cones share a common apex p₀. Subdomains V_i,k and V_i+1,k′ are located in a cone with p₀=∫X d{right arrow over (v)} as an apex defined by the same radial sections in the case that the subdomains V_i,k and V_i+1,k′ in the adjacent layers V_[ri,ri+1] and V_[ri+1,ri+2] have a common area element.

r_i,k,1=r_i,k,2=r_i,k,−1=r_i,k,−2=r_i is set on each subdomain V_i,k that satisfies μ(V_i,k)<δ_iμ(V_[ri,ri+1]), δ_i<<1; otherwise, setting

$\begin{array}{cc}{c_i^{\prime }={\mathrm{supX}}_i,d{\mu }^c{❘}_{V_{i,k}}=c_i^{\prime }\mathrm{dv}-d\mu ❘}_{V_{i,k}},& \left(6a\right)\end{array}$ $\begin{array}{cc}{\stackrel{\_}{r}}_{i,k}=\underset{r}{\mathrm{argmin}}{\int }_{V_{i,k}}{❘❘x❘-r❘}^{2}d{\mu }^c,& \left(6b\right)\end{array}$

and r_i,k,1=inf_r{r:μ({x∈V_i,k:r_i,k<|x|<r})>ε}, r_i,k,2=sup_r{r:μ({x∈V_i,k:r<|x|<r_i+1})>ε}, r_i,k,−1=sup_r{r:μ({x∈V_i,k:r<|x|<ri,k})>ε}, and r_i,k,−2=inf_r{r:μ({x∈V_i,k:r_i<|x|<r})>ε}, where ε represents an appropriately chosen threshold. In particular, the expression (6b) may be replaced by

${\stackrel{\_}{r}}_{i,k}=\underset{r}{\mathrm{argmin}}{\sum }_{x\in V_{i,k}}{❘❘x❘-r❘}^2{X}_{i,k}\left(x\right)$

locally to reduce the amount of calculation in the case that data in V_i,k comprise a small amount of discrete data points, where X_i,k(x)=0 if there exist no effective data at a point x.

If a certain given positive constant c has r_i,k,1−r_i,k,−1≥c·s₀, the subdomain V_i,k is partitioned into two along r=r_i,k to obtain two fine subdomains V′_i,k and V″_i,k, and r′_i,k,2=r_i,k,2, r′_i,k,−2=r_i,k,1, r″_i,k,2=r_i,k,−1, and r″_i,k,−2=r_i,k,−2. In other subdomains in V_[ri,ri+1]=r′_i,k,2=r_i,k,2, r′_i,k,−2=r″_i,k,2=r_i,k, and r″_i,k,−2=r_i,k,−2, or r′_i,k,2=r_i,k,2, r′_i,k,−2=r_i,k,−2, and r′_i,k,2=r″_i,k−2=r_i. In the above subdivision, in the case that μ(V′_i,k)<δ_iμ(V_[ri,ri+1]), V′_i,k may be abandoned directly and r′_i,k,2 and r′_i,k−2 may not exist, while data in V″_i,k are reserved, and r′_i,k,2=r′_i,k,−2=r_i is set. The steps are repeated to traverse all subdomains V_i,k, so {V_[ri,ri+1]}_i∈I is updated till r_i,k,1−r_i,k,−1 in each subdomain is a negligible quantity and finally a group of subdomains {V_[ri,ri+1]}_i∈i+ is obtained. This process further subdivides a region where data distribution has a prominent geometric feature. For example, a region that contains unconnected parts is partitioned into two connected subdomains to be processed, which is an adaptive refinement.

In the subsequent operation, the subdomains obtained from finite times of partitions of V_[ri,ri+1] finite number of times of and the corresponding {r′_i,k,2, r′_i,k,−2}_v′i,k and {r″_i,k,2, r″_i,k,−2}_v″i,k . . . are pair-wise combined, for example a pair of {r′_i,k,2, r′_i,k,−2}_v′i,k is selected from V_i,k to be pair-wise matched with {r′_i,k,2, r′_i,k,−2} vim in from V_j,m, leading to a finite number of combinations {(r′_i,2, r′_i,−2):r′_i,2|v_i,k=r′_i,k,2, r′_i,−2|v_i,k=r′_i,k,−2} in V_[ri,ri+1] are obtained, where the number of combinations does not exceed the largest number of partition times among V_i,k'S.

The step S2 of setting an information selection mechanism F_S:X→F_SX according to a type of data distribution and decomposing data information in the three-dimensional space into multiple layers of spherical information F_SX for processing may, for example, adopt the following solution:

• • 1_W is defined as the characteristic function of a set W in the three-dimensional space, a data distribution X=X_c=c_W·1_W is considered, namely, X is a non-zero constant c_W in a locally connected space set W and is 0 almost everywhere in its complement W^c. In the case that each concentric spherical layer V_[ri,ri+1] of {V_[ri,ri+1]}_i∈I+ is fully refined, let

$\begin{array}{cc}\{\begin{array}{c}{f}_i^{+}\left({\omega }_{i,k}\right)=\frac{r_{i,k,2}}{r_i}\\ f_i^{-}\left({\omega }_{i,k}\right)=\frac{r_{i,k,-2}}{r_i}\end{array}& \left(7\right)\end{array}$

be to-be-processed data, where ω_i,k=(θ_i,k, φ_i,k) corresponds to directional coordinates of a subdomain V_i,k, or briefly recorded as

$f_i^{+}=\frac{r_{i,2}}{r_i},f_i^{-}=\frac{r_{i,-2}}{r_i}.$

This is a particularly important type of data distribution. Just as the situation that planar shearlet system has been proved to be quite suitable for two-dimension cartoon-like images with certain regularity on a boundary, in the three-dimensional space, it can be assumed that W consists of a finite number of connected sets, and its boundary surface has good local regularity. For instance a three-dimensional cartoon figure, but is not limited to this type of data distribution in principle. In practice, the data to be analyzed usually exist in discrete forms, form a fitting or approximation of a three-dimensional figure with good regularity, while include some details and noise. For example, in biomedicine, when three-dimensional gray-scale image data of human body tissue is processed, usually much additional information is attached to a smooth organ image. A brain from inside to outside has apparent spherical distribution feature, especially its surface cortex, and lines and furrows of the cerebral surface exhibit anisotropic structures, which are suitable for being decomposed and reconstructed by the spherical shearlet. In addition to the biomedicine image data, it can be utilized when one performs physical simulation or models the Earth or distant stars. For example in a Saturn image captured by telescope, regular strips and rings surrounding the Saturn can be observed, which are suitable for being processed by the spherical shearlet system. As the magnification of the observing telescope increases, one gradually discovers asteroids or giant rocks distributed around celestial bodies-when only the distribution position of asteroids is interested, these scattered distributions can be treated as discrete data information.

When the noisy points are not a study object of interest, pre-processing denoising method including regularization models, neural networks and other means can be employed. However, some noisy points can be occasionally very interesting and important, and may exist in a discrete data form. Therefore, the case of including non-negligible discrete components X_d=Σ_p∈D d_pδ_p needs to be considered in the model sometimes, namely, X=X_c+X_d, where D is a bounded discrete subset in the three-dimensional space, d_p is a positive integer, and δ_p is the delta distribution at the point p. The spherical shearlet is capable of detecting discrete singular points, and as a decomposition and reconstruction process that concerns only summation and integral operations without a derivation operation, it is relatively noise-robust compared with some conventional methods.

When data including Σ_m=1^mc_Wm·1_Wm and the discrete components X_d are processed, the following solution may be adopted:

Y is defined to be a blockwise constant distribution that satisfies Y|V_i,k=∫_Vi,kXdv in the case that D is a finite set. A set of {c_t, {V′_t}}_t≤t is obtained deductively according to: c₁=minY|v_i,k together with a subdomain set {V′₁}=argminY|v_i,k, c₂=min_Vi,k≠V1, Y|_Vi,k−c₁ together with a subdomain set {V′₂}=argmin_Vi,k≠V1, Y|_Vi,k, and so on. Distributions Y₁|v_i,k and Y_t|v_i,k's are processed on each V_[ri,ri+1] in the same way as the data type X_c, to obtain a set of functions {((g_i⁺)_t, (g_i⁻)_t)}_t≤t, in which t=min{t:Y_t|v_i,k=0, ∀V_i,k ⊂V_[ri,ri+1]}−1, and distributions Y₁|v_i,k and Y_t|v_i,k are defined by the following expressions:

$\begin{array}{cc}{Y}_1{|}_{V_{i,k}}=\{\begin{array}{cc}{c}_1,& Y{|}_{V_{i,k}}\ne 0\\ 0,& Y{|}_{V_{i,k}}=0\end{array}& \left(8a\right)\end{array}$ $⋮$ $\begin{array}{cc}{Y}_t{|}_{V_{i,k}}=\{\begin{array}{cc}Y{|}_{V_{i,k}}-{\sum }_{s<t}{c}_s,& Y{|}_{V_{i,k}}\ne 0\\ 0,& Y{|}_{V_{i,k}}=0\mathrm{or}{V}_{i,k}\in U_{s<t}\left\{V_s^{\prime }\right\}\end{array}& \left(8b\right)\end{array}$

The to-be-processed data of a layer V_[ri,ri+1] have expressions:

$\begin{array}{cc}\{\begin{array}{c}{f}_i^{+}={\sum }_{t\le \stackrel{\_}{t}}{\left(g_i^{+}\right)}_t\\ f_i^{-}={\sum }_{t\le \stackrel{\_}{t}}{\left(g_i^{-}\right)}_t\end{array}& \left(9\right)\end{array}$

If the quantity of points in D is large and the distribution of X_d has a certain simple geometrical structure, an appropriate cost function may be set, and the discrete data are pre-processed to obtain a principal manifold or the main part of interest hidden under X_d, namely determine a dimensionality reduction mapping of X_d→X_c, to reduce corresponding data to the type of X_c, and to be processed by using the spherical shearlets.

A specific implementation of the step S3 is as follows:

The spherical information F_SX part that includes the to-be-processed data depending on the spherical coordinates in the expressions (7) or (9) is recorded as {xⁱ}_i∈I+={F_SX_i}_i∈I+, spherical shearlet is decomposed according to the expression (4), and the coefficients {c_j,k^i,+}_j,k and {c_j,k^i,−}_j,k obtained from the spherical shearlet transform (5) are stored. The original three-dimensional space data distribution X may be approximated by a following expression:

$\begin{array}{cc}\tilde{X}=c_W·{\sum }_i\left(1_{U_i^{+}}-1_{U_i^{-}}\right)& \left(10\right)\end{array}$

where 1_Ui+ represents a characteristic function of a set U_i⁺={x=|x|ω_i,k∈R³:|x|≤{tilde over (x)}^i,+(ω_i,k)}, and 1_Ui− represents a characteristic function of a set U_i⁻={x=|x|ω_i,k∈R³:|x|≤{tilde over (x)}^i,−(ω_i,k)}, and the following relations are satisfied:

$\begin{array}{cc}{\tilde{x}}^{i,+}={\tilde{x}}_L^{i,+}+{\tilde{x}}_H^{i,+}=r_i{\sum }_{l\le L}{P}_l{x}^{i,+}+r_l{\sum }_{j,k}{c}_{j,k}^{i,+}{S}_{j,k}& \left(11a\right)\end{array}$ $\begin{array}{cc}{\tilde{x}}^{i,-}={\tilde{x}}_L^{i,-}+{\tilde{x}}_H^{i,-}=r_i{\sum }_{l\le L}{P}_l{x}^{i,-}+r_l{\sum }_{j,k}{c}_{j,k}^{i,-}{S}_{j,k},& \left(11b\right)\end{array}$

so that on a finite (i, j, k) index set there are:

$\begin{array}{cc}{\sum }_l{x^{i,+}-x_L^{i,+}-{\sum }_{j,k}{c}_{j,k}^{i,+}{S}_{j,k}}_B\le {\epsilon }^{\prime }{\sum }_i{x^{i,+}}_B& \left(12a\right)\end{array}$ $\begin{array}{cc}{\sum }_l{x^{i,-}-x_L^{i,-}-{\sum }_{j,k}{c}_{j,k}^{i,-}{S}_{j,k}}_B\le {\epsilon }^{\prime }{\sum }_i{x^{i,-}}_B,& \left(12b\right)\end{array}$

where ε<<1 is chosen according to required accuracy, ∥·∥_B represents a norm that may reflect the data singular feature in space B adapted to the spherical shearlet system, and {tilde over (x)}_Hⁱ represents an approximation of the part of spherical data that is not of low degree. The corresponding shearlet coefficients have predictable sparsity in the case that F_SX has a curve linear singularity property on the sphere. The number of layers needing to be calculated is usually finite, so the overall calculation efficiency is determined by the efficiency of spherical shearlet representation.

Different from methods that are applied to data processing such as Fourier analysis, spline analysis and wavelet analysis under a conventional Cartesian coordinate system, the present disclosure provides a method for decomposing, compressing and reconstructing the three-dimensional space scalar data under the polar coordinate system by using the spherical shearlet system. Compared with conventional methods, the method based on spherical shearlet representation has predictable superiority in terms of processing the scalar data information with the spherical distribution feature in the three-dimensional space, especially with the linear singular distribution on the sphere.

Claims

1. A spherical shearlet-based compression and reconstruction method for three-dimensional scalar information, applied to decomposing, extracting, storing and reconstructing scalar data including three-dimensional geometric data having a spherical distribution feature and random data satisfying certain probability distributions, wherein the method comprises the following steps: step S1: decomposing a three-dimensional space or a set V into concentric spherical layers, Ui∈I V[ri,ri+1]=V, and partitioning, layer by layer, scalar data information X distributed in the three-dimensional space, wherein V represents an entire three-dimensional real space R3, or a bounded set comprising to-be-processed data; partitioning V into nonintersecting concentric spherical layers V[ri,ri+1], where i∈I, according to the scale or size s0 of a local feature that needs to be extracted, wherein ri+1−ri∝S0, Ui∈I V[ri,ri+1]=V, and partitioning three-dimensional scalar data correspondingly into {Xi}i∈I, where Xi represents data of X contained in the concentric spherical layers V[ri,ri+1], an index set/represents a finite set or a countable set in modeling sense and a finite set in actual operation sense; andselecting different probability measures u to adapt to different types of data, and letting the restriction of the probability measures u on the concentric spherical layers V[ri,ri+1] be dμ|v[ri,ri+1]=Xidv, wherein an analyzed three-dimensional data distribution is a continuous distribution or a discrete distribution;step S2: setting a spherical layer selection mechanism FS:X→FSX={FSXi}i∈I+ based on a type of the three-dimensional scalar data X, and extracting, layer by layer, spherical information to be processed by a discrete spherical shearlet system; anddefining 1W as a characteristic function of a set W in the three-dimensional space, and considering a data distribution X=Xc=cW·1W, namely X is a non-zero constant cW on a locally connected space set W and almost everywhere zero on its complement Wc, wherein when each concentric spherical layer V[ri,ri+1] of {V[ri,ri+1]}i∈I+ is fully refined, the to-be-processed data have expressions:

2. The spherical shearlet-based compression and reconstruction method for three-dimensional scalar information according to claim 1, wherein said partitioning, layer by layer, the set and the scalar data information in the step S1 comprises: setting a threshold Nb<<μ(V), directly abandoning data in the concentric spherical layer V[ri,ri+1] and letting Xi=0, whenever a total data volume μ(V[ri,ri+1])<Nb in V[ri,ri+1]; or otherwise, further decomposing a data distribution in V[ri,ri+1] into data distribution in each subdomain Vi,k, with V[ri,ri+1]=UkVi,k, wherein subdomains Vi,k and Vi+1,k′ are located in a cone defined by same radial sections with ∫X d{right arrow over (v)} as an apex when the subdomains Vi,k and Vi+1,k′ in adjacent layers V[ri,ri+1] and V[ri+1,ri+2] have a common area element;setting ri,k,1=ri,k,2=ri,k,−1=ri,k,−2=ri on each subdomain Vi,k that satisfies μ(Vi,k)<δiμ(V[ri,ri+1]), and δi<<1; otherwise, setting

3. The spherical shearlet-based compression and reconstruction method for three-dimensional scalar information according to claim 2, wherein one approach of the step S2 comprises: adopting a processing solution presented below, when data distributions X=Σm=1mcWm·1Wm or X=Xc+Xd comprises a non-negligible discrete component Xd=Σp∈Ddpδp, where D represents a finite discrete subset in the three-dimensional space, dp represents a positive integer, and δp represents the delta distribution at a point p:defining Y to be a blockwise constant distribution that satisfies Y|vi,x=∫Vi,kXdv, and letting {ct, {V′t}}t≤t, be a set defined deductively according to: c1=minY|Vi,k together with a subdomain set {V′1}=argminY|vi,k, c2=minVi,k≠V′1Y|Vi,k−c1 together with a subdomain set {V′2}=argminVi,k≠V′1Y|Vi,k, and so on; and on each V[ri,ri+1], defining distributions Y1|vi,k and Yt|vi,k deductively as

4. The spherical shearlet-based compression and reconstruction method for three-dimensional scalar information according to claim 3, wherein one approach of the step S3 comprises: denoting the spherical data information that comprises the to-be-processed data depending on spherical coordinates in the expression (1) or the expression (9) as {xi}i∈I+={FSXi}i∈I+, decomposing the spherical shearlet according to the expression (5a), and storing the coefficients {cj,ki,+}j,k and {cj,ki,−}j,k obtained from the spherical shearlet transform, wherein the original three-dimensional space data distribution X is approximated by: