Obscuring Surface Anatomy in Volumetric Imaging Data

Abstract

The identifying or sensitive anatomical features in MR and CT images used in research raise patient privacy concerns when such data are shared. In order to protect human subject privacy, we developed a method of anatomical surface modification and investigated the effects of such modification on image statistics and common neuroimaging processing tools. Common approaches to obscuring facial features typically remove large portions of the voxels. The approach described here focuses on blurring the anatomical surface instead, to avoid impinging on areas of interest and hard edges that can confuse processing tools. The algorithm proceeds by extracting a thin boundary layer containing surface anatomy from a region of interest. This layer is then “stretched” and “flattened” to fit into a thin “box” volume. After smoothing along a plane roughly parallel to anatomy surface, this volume is transformed back onto the boundary layer of the original data. The above method, named normalized anterior filtering, was coded in MATLAB and applied on a number of high resolution MR and CT scans. To test its effect on automated tools, we compared the output of selected common skull stripping and MR gain field correction methods used on unmodified and obscured data. With this paper, we hope to improve the understanding of the effect of surface deformation approaches on the quality of de-identified data and to provide a useful de-identification tool for MR and CT acquisitions.

Appendix

Generating the “internal” and “external” Surfaces

Given the original volume V with dimensions {m, n, k}, we can represent an anatomical surface S as a height field F ⁰(${\cal x}$,y)over plane XY (Fig. 1). Using ray casting with discrete step w along X and Y axes, values of F are obtained in points

$$ F_{{i,j}}^0 = {F^0}\left( {iw,jw} \right),\;i = 0, \ldots, \left[ {{{m} \left/ {w} \right.}} \right] - 1,\;j = 0, \ldots, [{{n} \left/ {w} \right.}] - 1. $$

(3)

Denoting for $s_{{i,j}}^0$ a vertex with coordinates$\left( {iw,jw,F_{{i,j}}^0} \right)$ belonging to an anatomical surface, a “natural” triangulation T of S ⁰ composed of triangles $T_{{ij}}^1 = \left( {{s_{{i,j}}},\;{s_{{i,j + 1}}},\;{s_{{i + 1,j}}}} \right)$ and $T_{{ij}}^2 = \left( {{s_{{i,j + 1}}},\;{s_{{i + 1,j + 1}}},\;{s_{{i + 1,j}}}} \right)$ can be selected (Fig. 10). Using T, it is now possible to describe the thin layer L that encloses anatomical surface and has roughly constant thickness. For that purpose, we construct triangulation meshes T ^t and T ^b of “external” S ^t (“above” S) and “deep” S ^b (“below” S) surfaces. Noting that $\sum\limits_{{k = 1}}^4 {{{( - 1)}^k}n_{{i,j}}^k = 0}$, an averaged unit normal ${\bar{n}_{{i,j}}}$ at this vertex is computed as

Fig. 10

Triangulation S of the anatomical surface S ⁰

$$ {\bar{n}_{{i,j}}} = {{{\left( {n_{{i,j}}^1 + n_{{i,j}}^3} \right)}} \left/ {{\left\| {n_{{i,j}}^1 + n_{{i,j}}^3} \right\|}} \right.}, $$

(4)

where $n_{{i,j}}^k,\;k = 1, \ldots, 4$ are outer unit normals of four triangular faces Θ¹,…,Θ⁴ of T that have a common vertex (Fig. 11).

Fig. 11

The average normal in vertex S _i,j (left) and construction of upper triangulation Θ _t (right)

If we travel a fixed distance h along ${\overline n_{{i,j}}}$ in “outer” (“upward”) direction, we will arrive at a vertex of the “external” triangulation T ^t sharing the XY parameterization with T, and similarly for “internal” surface triangulation T ^b:

$$ s_{{i,j}}^t = {s_{{i,j}}} + v_{{i,j}}^{{edge}},\quad s_{{i,j}}^b = {s_{{i,j}}} - v_{{i,j}}^{{edge}}, $$

(5)

where

$$ v_{{i,j}}^{{edge}} = \tfrac{1}{4}h{\bar{n}_{{i,j}}}\sum\limits_{{k = 1}}^4 {\frac{1}{{({{\bar{n}}_{{i,j}}},n_{{i,j}}^k)}}} . $$

(6)

Thus, T ^t and T ^b are fully determined by (9–10). Since these surfaces have triangular faces that are nearly parallel to the corresponding faces of S, the thickness of L is maintained about 2 h.

Boundary Volume Projection

The purpose of this derivation is to describe projection of L onto “thin” rectangular volume with dimensions m×n×h (Fig. 3b). Consider the quasi-block Ω _i,j formed by two space quadrilateral faces of S ^t and S ^b(Fig. 12). Each Ω _i,j can be consistently partitioned into six tetrahedra, as illustrated on Fig. 3e and f. Since cuboid box is an instance of octahedral element, establishing one-to-one correspondence between quasi-block Ω _i,j and a proper block is equivalent to establishing correspondence between each pair of matching tetrahedra.

Fig. 12

Quasi-block partition element Ω _i,j of the anterior layer L

Combining all transformed boxes together constitutes the new “flattened” rectangular volume (Fig. 3). Thus, one-to-one correspondence between L and is established via piecewise linear transform.

Consider the box volume with dimensions m×n×h (Fig. 3b). Consider the partition of into proper blocks , where indices i and j have the same meaning as in Figs. 10, 11 and 12. Our aim is to establish a transformation f between these partitions such that:

(7)

Since each Ω _i,j has eight vertices, it is possible to partition it into six tetrahedra $T_{{i,j}}^1,\, \ldots, \,T_{{i,j}}^6$ (As illustrated on Fig. 3e), and because there is a one to one correspondence between Ω _i,j and , a matching tetrahedral partition can be also selected for (Fig. 3f) and matching affine transformation be computed. Since Ω _i,j provides a partition of L, combining individual affine transformations$f_{{i,j}}^k$ for each pair of $\Omega_{{i,j}}^b$ and Ω _i,j (Fig. 3c and d) will define a continuous non-degenerate piecewise linear transformation f between L and :

(8)

Denoting the 4×4 affine transformation matrix from $T_{{i,j}}^k$ to for $A_{{i,j}}^k$,  $f_{{i,j}}^k$ for an arbitrary point$\ P\left( {\matrix{ {{x_p}} &{{y_p}} &{{z_p}} & 1 \\ }<!end array> } \right)$ in Ω _i,j can be expressed as

$$ f_{{i,j}}^k(P) = A_{{i,j}}^k\left( {\matrix{ {{x_p}} &{{y_p}} &{{z_p}} & 1 \\ }<!end array> } \right),\quad P \in T_{{i,j}}^k $$

(9)

Coefficients of $A_{{i,j}}^k$ are calculated from the system of 12 linear equations with 12 unknowns matching the vertices of T _k and $T_k^b$. Once this transform is calculated, intensities in target tetrahedron $T_k^b$ are assigned by discrete 3D scanline filling (Kaufman and Shimony 1987) of its bounding box, and applying ${\left( {A_{{i,j}}^k} \right)^{{ - 1}}}$ to each voxel that is determined to be inside of $T_k^b$ to find the matching voxel in T _k . Thus, for transformation from L to , the following discrete stepping procedure is applied:

• a. For each Ω _i,j , do steps b–d:

• b. For each k, form a system of 12 equations expressing the affine transform for each member tetrahedron of Ω _i,j ;

• c. Determine an affine transformation matrix $A_{{i,j}}^k$ and its inverse ;

• d. For each voxel in , determine the matching tetrahedron (k can be one of 1,…,6) and find the corresponding voxel from $T_{{i,j}}^k$ using , and set its intensity to the intensity of its prototype.

The transform back from to L is performed by scanning Ω _i,j instead of , and following steps a-c, switching symbols with hat with symbols without hat.