Interior reconstruction

In iterative reconstruction in digital imaging, 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 known as the region of interest (ROI). Although interior reconstruction can be applied to dental or cardiac CT images, the concept is not limited to CT. It is applied with one of several methods. In iterative reconstruction in digital imaging, 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 known as the region of interest (ROI). Although interior reconstruction can be applied to dental or cardiac CT images, the concept is not limited to CT. It is applied with one of several methods. The purpose of each method is to solve for vector x {displaystyle x} in the following problem: Let X {displaystyle X} be the region of interest (ROI) and Y {displaystyle Y} be the region outside of X {displaystyle X} .Assume A {displaystyle A} , B {displaystyle B} , C {displaystyle C} , D {displaystyle D} are known matrices; x {displaystyle x} and y {displaystyle y} are unknown vectors of the original image, while f {displaystyle f} and g {displaystyle g} are vector measurements of the responses ( f {displaystyle f} is known and g {displaystyle g} is unknown). x {displaystyle x} is inside region X {displaystyle X} , ( x ∈ X {displaystyle xin X} ) and y {displaystyle y} , in the region Y {displaystyle Y} , ( y ∈ Y {displaystyle yin Y} ), is outside region X {displaystyle X} . f {displaystyle f} is inside a region in the measurement corresponding to X {displaystyle X} . This region is denoted as F {displaystyle F} , ( f ∈ F {displaystyle fin F} ), while g {displaystyle g} is outside of the region F {displaystyle F} . It corresponds to Y {displaystyle Y} and is denoted as G {displaystyle G} , ( g ∈ G {displaystyle gin G} ). For CT image-reconstruction purposes, C = 0 {displaystyle C=0} . To simplify the concept of interior reconstruction, the matrices A {displaystyle A} , B {displaystyle B} , C {displaystyle C} , D {displaystyle D} are applied to image reconstruction instead of complex operators. The first interior-reconstruction method listed below is extrapolation. It is a local tomography method which eliminates truncation artifacts but introduces another type of artifact: a bowl effect. An improvement is known as the adaptive extrapolation method, although the iterative extrapolation method below also improves reconstruction results. In some cases, the exact reconstruction can be found for the interior reconstruction. The local inverse method below modifies the local tomography method, and may improve the reconstruction result of the local tomography; the iterative reconstruction method can be applied to interior reconstruction. Among the above methods, extrapolation is often applied. A {displaystyle A} , B {displaystyle B} , C {displaystyle C} , D {displaystyle D} are known matrices; x {displaystyle x} and y {displaystyle y} are unknown vectors; f {displaystyle f} is a known vector, and g {displaystyle g} is an unknown vector. We need to know the vector x {displaystyle x} . x {displaystyle x} and y {displaystyle y} are the original image, while f {displaystyle f} and g {displaystyle g} are measurements of responses. Vector x {displaystyle x} is inside the region of interest X {displaystyle X} , ( x ∈ X {displaystyle xin X} ). Vector y {displaystyle y} is outside the region X {displaystyle X} . The outside region is called Y {displaystyle Y} , ( y ∈ Y {displaystyle yin Y} ) and f {displaystyle f} is inside a region in the measurement corresponding to X {displaystyle X} . This region is denoted F {displaystyle F} , ( f ∈ F {displaystyle fin F} ). The region of vector g {displaystyle g} (outside the region F {displaystyle F} ) also corresponds to Y {displaystyle Y} and is denoted as G {displaystyle G} , ( g ∈ G {displaystyle gin G} ).In CT image reconstruction, it has To simplify the concept of interior reconstruction, the matrices A {displaystyle A} , B {displaystyle B} , C {displaystyle C} , D {displaystyle D} are applied to image reconstruction instead of a complex operator. The response in the outside region can be a guess G {displaystyle G} ; for example, assume it is g e x {displaystyle g_{ex}}