Distribution Interpolation of the Radon Transforms for Shape Transformation of Gray-Scale Images and Volumes

Abstract

In this paper, we extend 1D distribution interpolation to 2D and 3D by using the Radon transform. Our algorithm is fundamentally different from previous shape transformation techniques, since it considers the objects to be interpolated as density distributions rather than level sets of density functions. First, we perform distribution interpolation on the precalculated Radon transforms of two different density functions, and then an intermediate density function is obtained by a consistent inverse Radon transform. This approach guarantees a smooth transition along all the directions the Radon transform is calculated for. Unlike the previous methods, our technique is able to interpolate between features that do not even overlap and it does not require a one dimension higher object representation. We will demonstrate that these advantageous properties can be well exploited for 3D modeling and metamorphosis.

Appendix

Theorem 1

Let us denote the Radon transforms of the 2D projections $p^0_\theta (t,z)$ and $p^1_\theta (t,z)$ of the density functions $f_0(x,y,z)$ and $f_1(x,y,z)$ by $p^0_{\theta , \varphi }(r)$ and $p^1_{\theta , \varphi }(r)$, respectively:

$$\begin{aligned} p^0_{\theta , \varphi }(r) = \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } p^0_\theta (t,z) \delta (t \cos (\varphi ) + z \sin (\varphi ) - r) dt dz, \end{aligned}$$

(11)

$$\begin{aligned} p^1_{\theta , \varphi }(r) = \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } p^1_\theta (t,z) \delta (t \cos (\varphi ) + z \sin (\varphi ) - r) dt dz. \end{aligned}$$

Additionally, let us introduce operator $D_w$, which generates an interpolated 1D projection $p_{\theta , \varphi }(r)$ from the 1D projections $p^0_{\theta , \varphi }(r)$ and $p^1_{\theta , \varphi }(r)$ using distribution interpolation:

$$\begin{aligned} p_{\theta , \varphi }(r) = D_w[p^0_{\theta , \varphi }(r), p^1_{\theta , \varphi }(r)]. \end{aligned}$$

(12)

The 2D inverse Radon transform $p_\theta (t,z)$ of $p_{\theta , \varphi }(r)$ yields consistent and valid projections of an intermediate density function f(x, y, z) in a sense that

$$\begin{aligned} \int _{-\infty }^{\infty } p_{\theta _0}(t,z) dt = \int _{-\infty }^{\infty } p_{\theta _1}(t,z) dt, \end{aligned}$$

(13)

for arbitrary pairs of $\theta _0$ and $\theta _1$ and for each value of z.

Proof

Note that $p_{\theta , \varphi }(r)$ is a consistent and valid 2D Radon transform of a projection $p_\theta (t,z)$ because

$$\begin{aligned} \int _{-\infty }^{\infty } p_{\theta , \varphi _0}(r) dr = \int _{-\infty }^{\infty } p_{\theta , \varphi _1}(r) dr = \end{aligned}$$

(14)

$$\begin{aligned} (1-w) \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } f_0(x,y,z) dx dy dz + w \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } f_1(x,y,z) dx dy dz, \end{aligned}$$

for arbitrary pairs of $\varphi _0$ and $\varphi _1$. Therefore, Eq. 13 can be expressed from $p_{\theta , \varphi }(r)$, where $\varphi = \pi / 2$ and $r = z$ correspond exactly to projections onto the z axis:

$$\begin{aligned} \int _{-\infty }^{\infty } p_{\theta _0}(t,z) dt = p_{\theta _0, {\pi \over 2}}(z) = D_w[p^0_{\theta _0, {\pi \over 2}}(z), p^1_{\theta _0, {\pi \over 2}}(z)], \end{aligned}$$

(15)

$$\begin{aligned} \int _{-\infty }^{\infty } p_{\theta _1}(t,z) dt = p_{\theta _1, {\pi \over 2}}(z) = D_w[p^0_{\theta _1, {\pi \over 2}}(z), p^1_{\theta _1, {\pi \over 2}}(z)]. \end{aligned}$$

Note that

$$\begin{aligned} p^0_{\theta _0, {\pi \over 2}}(z) = p^0_{\theta _1, {\pi \over 2}}(z) = \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } f_0(x,y,z) dx dy, \end{aligned}$$

(16)

and

$$\begin{aligned} p^1_{\theta _0, {\pi \over 2}}(z) = p^1_{\theta _1, {\pi \over 2}}(z) = \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } f_1(x,y,z) dx dy. \end{aligned}$$

(17)

Thus, $p_{\theta _0, {\pi \over 2}}(z)$ is equal to $p_{\theta _1, {\pi \over 2}}(z)$ because they are obtained by applying operator $D_w$ on the same 1D projections. Consequently, Eq. 13 is satisfied for arbitrary pairs of $\theta _0$ and $\theta _1$ and for each value of z.