# Interior reconstruction

In image reconstruction, interior reconstruction, also known as limited field of view (LFV) reconstruction, is a technique to correct truncation artifacts caused by limiting image data to a small field of view. The reconstruction focuses on an area called the region of interest (ROI). Interior reconstruction can be applied to either dental or cardiac CT images, but the concept is not limited to CT. Interior reconstruction is applied using one of various methods.

## Methods

The purpose of each method is to solve for vector $x$ in the following problem:

$\begin{bmatrix} f \\ g \end{bmatrix}= \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}.$
Region of interest (ROI) of an image showing an object

Let $X$ be the region of interest (ROI) and $Y$ be the region outside of $X$. Assume $A$, $B$, $C$, $D$ are known matrices; $x$ and $y$ are unknown vectors of the original image, while $f$ and $g$ are vector measurements of the responses, $f$ being known and $g$ unknown. Further, $x$ is inside of the region $X$, ($x \in X$) and $y$, in the region $Y$, ($y \in Y$), is outside of region $X$. Also, $f$ is inside of a region in the measurement corresponding to $X$. This region is denoted as $F$, ($f \in F$), while $g$ is outside of the region $F$. This region corresponds to $Y$, and is denoted as $G$, ($g \in G$).

For CT image reconstruction purposes, $C = 0$.

In order to simplify the concept of the interior reconstruction, the matrices $A$, $B$, $C$, $D$ are applied to image reconstruction instead of using complicated operators.

The first interior reconstruction method under consideration below is the extrapolation method. That is a local tomography method which eliminates the truncation artifacts but introduces another kind of artifact, i.e. "bowl effect". Then we examine an improvement called the adaptive extrapolation method. The iterative extrapolation method below also improves the reconstruction results. In some special cases, the exact reconstruction can be found for the interior reconstruction. The local inverse method explained below modifies the local tomography method and perhaps can improve the reconstruction result of the local tomography. Further, the iterative reconstruction method can be applied to interior reconstruction. Among the above methods, extrapolation is often applied.

### Extrapolation method

1) Projections of Sheep-Logan phantoms. 2) Trucated projections(Zero extrapolation). 3) Constant extraplation.4) expontional extrapolation. 5). Qudratical extrapolation. 6 Mixed extrapolation of 4) and 5).
region of Interest of the image
$\begin{bmatrix} f \\ g \end{bmatrix}= \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$

Assume $A$, $B$, $C$, $D$ are known matrices; $x$ and $y$ are unknown vectors; $f$ is a known vector; $g$ is an unknown vector. We desire to know the vector $x$. Here it is assumed that $x$ and $y$ are the original image, while $f$ and $g$ are measurements of the responses. Vector $x$ is at the inside of the region $X$, ($x \in X$) where $X$ is also called the Region of interest (ROI). Vector $y$ is at the outside of the region $X$. The outside region is referred as $Y$, ($y \in Y$) and $f$ is at the inside of a region in the measurement corresponding to $X$. This region is denoted as $F$, ($f \in F$). Also, the region of vector $g$, which is at the outside of the region $F$, corresponds to $Y$ and is denoted as $G$, ($g \in G$). In CT image reconstruction, it has

$C = 0$

The question is: What is the solution of this problem in the region of $x$?

In order to simplify the concept of the interior reconstruction, the matrices $A$, $B$, $C$, $D$ are applied to image reconstruction instead of using a complicated operator.

The response in the outside region can be a guess $G$; for example, assume it is $g_{ex}$

$\begin{bmatrix} x_0 \\ y_0 \end{bmatrix}= \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} \begin{bmatrix} f \\ g_{ex} \end{bmatrix}$
(a) Shepp-Logan head phantom. (b) The crop of the phantom. (c) The reconstruction without extrapolation. (d) The reconstruction with constant extrapolation. (e) The reconstruction with quadratic extrapolation. (f) the reconstruction with mixed extrapolation; here L = 256, m = 1, α = 0.73. (g) the reconstruction with mixed extrapolation; here L = 256, m = 2, α = 0.5. (h) the crop of the reconstruction with FBP method using non-truncated projections, which was utilized for comparison.

In the above formula a simple solution of $x$ is obtained; it is written as $x_0$. This is called the extrapolation method. The result is dependent on how good the guess function or extrapolation function $g_{ex}$ is. A frequent choice is

$g_{ex}|_{\partial G}=f |_{\partial F}$

at the boundary of the two regions. The examples of the extrapolation method can be seen in the cited references [1] [2] [3] .[4] The extrapolation method is often combined with a priori knowledge. [5] [6]

(There is also a fast extrapolation method to reduce calculation time, shown below.)

Assume a rough solution, $x_0$ and $y_0$, is obtained from the extrapolation method described above. The response in the outside region $g_1$ can be calculated as follows:

$g_1 = C x_0+D y_0$

The reconstructed image can be calculated as following,

$\begin{bmatrix} x_1 \\ y_1 \end{bmatrix}= \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} \begin{bmatrix} f \\ g_1+g_{1ex} \end{bmatrix}$

It is assumed that

$f |_{\partial F}=(g_1+g_{1ex})|_{\partial G}$

at the boundary of the interior region. Here $x_1$ is a solution of this problem. This method is referred to as the adaptive extrapolation method. $g_{1ex}$ is the adaptive extrapolation function. The adaptive extrapolation method can be seen in the cited references [7] [8] [9] [10] .[5]

### Iterative extrapolation method

It is assumed that a rough solution, $x_0$ and $y_0$, is obtained from the extrapolation method described above.

$\begin{bmatrix} f_1 \\ g_1 \end{bmatrix}= \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} 0 \\ y_0 \end{bmatrix}$

or

$f_1=B y_0$

The reconstruction can be obtained as following,

$\begin{bmatrix} x_1 \\ y_1 \end{bmatrix}= \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} \begin{bmatrix} f - f_1 \\ g_{ex} \end{bmatrix}$

Here $g_{1ex}$ is an extrapolation function. It is assumed that

$(f-f_1)|_{\partial F}=g_{1ex}|_{\partial G}$

$x_1$ is one solution of this problem. The iterative extrapolation method can be seen in the cited reference.[11]

### Local tomography

Local tomography is also referred as lambda tomography. The filter used in local tomography is very short.[12][13]

### Local inverse method

The local inverse method extends the concept of local tomography. The response in the outside region can be calculated as follows:

$f = A x + B y$

consider the generalized inverse $B^+$ satisfying

$B B^+ B = B$

Define

$Q=[I-BB^+]$

so that

$QB = 0$

Hence,

$Q f = Q A x$

The above equation can be solved as

$x_1 = A^+ Q^+ Q f$

Considering

$QQ = Q$
$QQQ = Q$

$Q$ is the generalized inverse of $Q$, i.e.

$Q^+ = Q$

The solution can be simplified as

$x_1 = A^+ Q f$

The matrix $A^+Q = A^+ [I-BB^+]$ is referred to as the local inverse of a matrix ${\begin{bmatrix} A & B \\ C & D \\ \end{bmatrix}}$ corresponding to $A$. This method is referred to as the local inverse method.[11]

### Iterative reconstruction method

Here a goal function is defined and this method Iteratively achieves the goal. If the goal function can be some kind of normal, this is known as a minimal norm method. We have

$\min( R\|x\| + S\|y\|+T\|g\|)$

subject to

$\begin{bmatrix} x \\ y \end{bmatrix}= \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} \begin{bmatrix} f \\ g \end{bmatrix}$

and $f$ is known.

where $R$, $S$ and $T$ are weighting constants of the minimization and $\|\cdot\|$ is some kind of norm. The often used norms are $L_0$, $L_1$, $L_2$, $L_{+\infty}$ or total variation (TV) norm, or some combination of the above norms. An example of this method is also referred to as the projection onto convex sets (POCS) method. [14] [15]

### Analytical solution

In special situations, the interior reconstruction can be obtained as an analytical solution. The solution of $x$ is exact in such cases.

### Fast extrapolation

The extrapolated data often convolutes to a kernel function. After data is extrapolated, the size of the data is increased N times, where N = 2 ~ 3. If this data needs to be convoluted to a known kernel function, the numerical calculations will increase log(NN times even with the fast Fourier transform (FFT). There exists an algorithm; it analytically calculates the contribution from the part of the extrapolated data. The calculation time can be omitted compared to the original convolution calculation. Hence with this algorithm the calculation of a convolution using the extrapolated data is not increased noticeably. This is referred to as the fast extrapolation.[19]

## Comparison of the above interior reconstruction methods

• The extrapolation method is suitable to the situation where
$|x| > |y|$ and $|X| > |Y|$
i.e. a small truncation artifacts situation.
• The adaptive extrapolation method is suitable to the situation where
$|x| \sim |y|$ and $|X| \sim |Y|$
i.e. a normal truncation artifacts situation. This method also offers a rough solution for the exterior region.
• The iterative extrapolation method is suitable to the situation where
$|x| \sim |y|$ and $|X| \sim |Y|$
i.e. a normal truncation artifacts situation. This method gets better interior reconstruction compared to the adaptive reconstruction, but at the price of totally missing the result in the exterior region.
• The local tomography is suitable to the situation where
$|x| \ll |y|$ and $|X| \ll |Y|$
i.e. a largest truncation artifacts situation. There are no truncation artifacts for this method. However there is a fixed error in the reconstruction. The error is not dependent on the value of $|y|$.
• The local inverse method is the same as local tomography. It is suitable to the situation where
$|x| \ll |y|$ and $|X| \ll |Y|$
i.e. a largest truncation artifacts situation. There are no truncation artifacts for this method. However there is a fixed error in the reconstruction. The error is not dependent on the value of $|y|$. The error is perhaps smaller than with local tomography.
• The iterative reconstruction method achieves a good result, but at the price of huge calculations.
• The analytic method achieves the exact result, but it is only functional for some special situations.
• The fast extrapolation method can get the same results as the other extrapolation methods. It can be applied to the above interior reconstruction methods to reduce the calculation.

## Notes

1. ^ M.M. Seger, Rampﬁlter implementation on truncated projection data. Application to 3D linear tomography for logs,Proceedings SSAB02, Symposium on Image Analysis, Lund, Sweden, 7–8 March 2002. Editor Astrom.
2. ^ F .Rashid-Farrokhi, K.J.R. Liu, C.A. Berenstein and D.Walnut,Wavelet-based Multiresolution Local Tomography, IEEE Transactions on Image Processing 6 (1997), 1412–1430.
3. ^ M. Nilsson, Local Tomography at a Glance, Licentiate Theses in Mathematical Sciences 2003:3 ISSN 1404-028X, ISBN 91-628-5741-X, LUTFMA-2007-2003. Printed in Sweden by KFS AB Lund, 2003.
4. ^ P.S. Cho, A.D. Rudd and R.H. Johnson, Cone-beam CT from width truncated projections, Computerized Medical Imaging and Graphics 20(1) (1996), 49–57, 49–57.
5. ^ a b J. Hsieh, E. Chao, J. Thibault, B. Grekowicz, A. Horst, S. McOlash and T.J. Myers, A novel reconstruction algorithm to extend the CT scan fieldofview, Medical Phys 31 (2004), 2385–2391.
6. ^ K.J. Ruchala, G.H. Olivera, J.M. Kapatoes, P.J. Reckwerdt and T.R. Mack, Methods for improving limited fieldofview radiotherapy reconstructions using imperfect a priori images, Med Phys 29 (2002), 2590–2605.
7. ^ M. Nassi,W.R. Brody, B.P.Medoff and A.Macovski, Iterative reconstruction reprojection: an algorithm for limited data cardiac computed tomography, IEEE trans Biomed Engineering 295 (1982), 333–340.
8. ^ J.H. Kim, K.Y. KWAK, S.B. Park and Z.H. Cho, Projection space iteration reconstruction reprojection, IEEE transaction on Medical Imaging 4 (1983), 139–143
9. ^ P.S.Cho, A.D. Rudd and R.H. Johnson, Conebeam CT from width truncated projections Computerized,Medical Imaging and Graphics 20 (1996), 49–57.
10. ^ B. Ohnesorge, T. Flohr, K. Schwarz, J.P. Heiken and K.T. Bae, 2000 Efficient correction for CT image artifacts caused by objects extending outside the scan field of view, Med Phys 27, 39–46.
11. ^ a b Shuangren Zhao, Kang Yang, Dazong Jiang, Xintie Yang, Interior reconstruction using local inverse, J Xray Sci Technol. 2011; 19(1): 69-90
12. ^ A. Faridani, E.L. Ritman and K.T. Smith, Local tomography, SIAM J APPL MATH 52 (1992), 459–484.
13. ^ A. Katsevich, 1999 Cone beam local tomography, SIAM J APPL MATH 59, 2224–2246.
14. ^ Ye. Yangbo, Yu. 1 Hengyong 2 and GeWang, Exact Interior Reconstruction from Truncated Limited Angle Projection Data, International Journal of Biomedical Imaging (2008), 1–6.
15. ^ L. Zeng, B. Liu, L. Liu and C. Xiang, A new iterative reconstruction algorithm for 2D exterior fanbeam CT, Journal of XRay Science and Technology 18 (2010), 267–277.
16. ^ Y. Zou and X. Pan, 2004, Exact image reconstruction on PIlines from minimum data in helical conebeam CT, Physics in Medicine and Biology 49(6), 941–959.
17. ^ M. Defrise, F. Noo, R. Clackdoyle and H. Kudo, Truncated Hilbert transform and image reconstruction from limited tomographic data. IOPscience.iop.org, 2006
18. ^ F. Noo, R. Clackdoyle and J.D. Pack, A twostep Hilbert transform method for 2D image reconstruction, Phys Med Biol 49 (2004), 3903–3923.
19. ^ S Zhao, K Yang, X Yang, Reconstruction from truncated projections using mixed extrapolations of exponential and quadratic functions, Journal of X-ray Science and Technology, 2011, 19(2) pp 155–72