1. An imaging set-up for reverse projection to obtain quantitative X-ray images from a sample and to quantitatively extract both absorption and phase information from the sample, the imaging set-up comprising:
an X-ray source generating an X-ray beam;
gratings including a beam splitter grating and an analyzer grating having their respective lines parallel to each other, said beam splitter grating being a phase grating and said analyzer grating is a line absorption grating with high X-ray absorption;
a mechanism for placing the sample to be investigated either between said X-ray source and the said beam splitter grating or between said beam splitter grating and said analyzer grating;
a position-sensitive detector with spatially modulated detection sensitivity having a number of individual pixels;
means for recording images of said position-sensitive detector, a series of M images is collected by continuously or stepwise rotating from zero (0) to pi (\u03c0) or 2pi (2\u03c0) either the sample or said gratings and said X-ray source relative to the sample, wherein each image taken at an angle 0\u2266\u03c6\u2266\u03c0 contains a corresponding reverse projection image taken at an angle \u03c0\u2266\u03c6+\u03c0\u22662\u03c0, yielding in total a number of M2 pairs of specular images;
means for calculating pixel-wise an absorption image M and an refraction angle \u03b8r image out of the pairs of specular images without a need for phase stepping according to:
M
\u2061
(
x
r
,
\u03d5
,
z
)
=
\u222b
–
\u221e
\u221e
\u2062
\u03bc
\u2061
(
x
,
y
,
z
)
\u2062
\u2146
y
r
=
ln
(
2
\u2062
S
\u2061
(
x
g
D
)
\u2062
I
0
I
\u2061
(
x
r
,
\u03d5
,
z
)
+
I
\u2061
(
–
x
r
,
\u03d5
+
\u03c0
,
z
)
)
\u03b8
r
\u2061
(
x
r
,
\u03d5
,
z
)
=
–
\u222b
–
\u221e
\u221e
\u2062
\u2202
\u03b4
\u2061
(
x
,
y
,
z
)
\u2202
x
r
\u2062
\u2146
y
r
=
1
C
\u2062
I
\u2061
(
x
r
,
\u03d5
,
z
)
–
I
\u2061
(
–
x
r
,
\u03d5
+
\u03c0
,
z
)
I
\u2061
(
x
r
,
\u03d5
,
z
)
+
I
\u2061
(
–
x
r
,
\u03d5
+
\u03c0
,
z
)
where:
(x, y, z) are first spatial coordinates associated with the sample;
(xr, yr, z) are second spatial coordinates associated to the X-ray beam, the first and second coordinates being linked by a rotation matrix:
(
x
y
)
=
(
cos
\u2062
\u2062
\u03d5
–
sin
\u2062
\u2062
\u03d5
sin
\u2062
\u2062
\u03d5
cos
\u2062
\u2062
\u03d5
)
\u2062
(
x
r
y
r
)
,
where \u03c6 is a rotation angle between xr axis and x axis around a z axis;
Io is an incident X-ray intensity;
I(xr ,\u03c6,z) is intensity recorded at said position-sensitive detector for a beam decided by xr, z and the rotation angle \u03c6;
xg denotes a relative displacement between said phase grating and said analyzer grating along a direction perpendicular to both an incoming beam and a line of said gratings;
D is a distance between said phase grating and said analyzer grating;
S
\u2061
(
x
g
D
)
\u2003is a shifting curve;
C is a constant; and
M(xr,\u03c6,z) and \u03b8r(xr,\u03c6,z) are inline definitions representing an absorption signal and a refraction angle, respectively, for a given coordinate xr, z, and the rotation angle \u03c6.
2. The imaging setup according to claim 1, wherein the intensity I recorded by said position-sensitive detector is expressed as:
I
=
I
0
\xb7
exp
\u2061
–
\u222b
–
\u221e
\u221e
\u2062
\u03bc
\u2061
(
x
,
y
,
z
)
\u2062
\u2146
y
r
\xb7
S
\u2061
(
x
g
D
+
\u03b8
r
)
where \u03bc(x, y, z) is a linear absorption coefficient at the spatial coordinate (x, y, z), xg denotes the relative displacement between said phase grating and said analyzer grating along the direction perpendicular to both the incoming beam and the line of said gratings, \u03b8r is the refraction angle, D is the distance between said phase grating and said analyzer grating,
S
\u2061
(
x
g
D
)
is the shifting curve.
3. The imaging set-up according to claim 1, wherein said analyzer grating has a one-dimensional grating structure with high X-ray absorption contrast and is placed in front of said position sensitive detector with its lines parallel to those of said beam splitter grating.
4. The imaging set-up according to claim 1, wherein said analyzer grating has a one-dimensional grating structure with high X-ray absorption contrast, its period is a same as that of a self image of said beam splitter grating, is placed in front of said position sensitive detector with its lines parallel to those of said beam splitter grating.
5. The imaging set-up according to claim 1, wherein a distance between said beam splitter grating and said analyzer grating is chosen to be an odd fractional Talbot distance, given by equation
D
n
,
sph
=
L
\xb7
D
n
L
–
D
n
=
L
\xb7
n
\xb7
p
1
2
2
\u2062
\u03b7
2
\u2062
\u03bb
L
–
n
\xb7
p
1
2
2
\u2062
\u03b7
2
\u2062
\u03bb
,
where n=1, 3, 5 . . . , and
\u03b7
=
{
1
if
\u2062
\u2062
the
\u2062
\u2062
phase
\u2062
\u2062
shift
\u2062
\u2062
of
\u2062
\u2062
G
1
\u2062
\u2062
is
\u2062
\u2062
(
2
\u2062
l
–
1
)
\u2062
\u03c0
2
,
p
2
=
L
+
D
n
,
sph
L
\u2062
p
1
2
if
\u2062
\u2062
the
\u2062
\u2062
phase
\u2062
\u2062
shift
\u2062
\u2062
of
\u2062
\u2062
G
1
\u2062
\u2062
is
\u2062
\u2062
(
2
\u2062
l
–
1
)
\u2062
\u03c0
,
p
2
=
L
+
D
n
,
sph
L
\u2062
p
1
2
,
where l=1, 2, 3 . . . Dn is the odd fractional Talbot distance when a parallel X-ray beam is used, while Dn,sph is that when a fan or cone X-ray beam is used, L is a distance between said x-ray source and said phase grating, and p1 and p2 represent a period of said beam splitter grating and said analyzer grating, respectively.
6. The imaging set-up according to claim 1, wherein said beam splitter grating is a line phase grating with low X-ray absorption, but with considerable X-ray phase shift \u03a6, the X-ray phase shift defined as either
\u03a6
\u2208
(
(
2
\u2062
l
–
1
)
\u2062
\u03c0
2
–
arc
\u2062
\u2062
sin
\u2062
\u2062
0.8
,
(
2
\u2062
l
–
1
)
\u2062
\u03c0
2
+
arc
\u2062
\u2062
sin
\u2062
\u2062
0.8
)
or
\u03a6\u03b5((2l\u22121)\u03c0\u2212arcsin 0.8, (2l\u22121)\u03c0+arcsin 0.8), where l=1, 2, 3 . . . .
7. The imaging set-up according to claim 1, wherein said beam splitter grating is a line phase grating with low X-ray absorption and made from a material selected from the group consisting of silicon and a polymer.
8. The imaging set-up according to claim 1, wherein said analyzer grating is either placed front of said position sensitive detector or with its one-dimensional grating structure integrated into said position sensitive detector, a pixel of said position sensitive detector is from 2 to 10 times a size of a period of said analyzer grating, half lines with sensor in a pixel are sensitive to X-ray and half lines without sensor let X-ray go through.
9. The imaging set-up according to claim 1, further comprising a collimator disposed between said X-ray source and said beam splitter grating, said collimator limiting a spatial extent of illuminating X-rays to a fan beam, a line-array detector is used, and said mechanism rotates stepwise or continuously the sample relative to a rest of the apparatus, a rotational axis being perpendicular to an opening angle of a fan, and at a same time allows to translate either stepwise or continuously the sample relative to the rest of the apparatus along a direction parallel to a rotational axis.
10. A method for reverse projection to obtain quantitative X-ray images from a sample and to quantitatively extract both absorption and phase information from the sample, which comprises the steps of:
providing an X-ray source;
providing gratings including a beam splitter grating and an analyzer grating having their respective lines parallel to each other, wherein the beam splitter grating is a line grating selected from the group consisting of an absorption grating with high X-ray absorption and a phase grating with low X-ray absorption, and the analyzer grating is a line absorption grating with high X-ray absorption;
providing a position-sensitive detector with spatially modulated detection sensitivity having a number of individual pixels;
positioning at least one of the gratings relative to a probe in a direction xg being substantially perpendicular to both an incoming beam and an orientation of the lines of grating to make an imaging set-up on a center of a linear region of a shifting curve
S
\u2061
(
x
g
D
)
;
placing the sample to be investigated either between the X-ray source and the beam splitter grating or between the beam splitter grating and the analyzer grating, applying shots of the X-ray source to the sample and recording the images of the position-sensitive detector;
recording the images of the position-sensitive detector, wherein a series of M images is collected by continuously or stepwise rotating from zero (0)to pi (\u03c0) or 2pi (2\u03c0) either the sample or the gratings and the X-ray source relative to the sample, wherein each image taken at an angle 0\u2266\u03a6\u2266\u03c0 contains a corresponding reverse projection image taken at an angle \u03c0\u2266\u03a6+\u03c0\u22662\u03c0, yielding in total a number of M2 pairs of specular images; and
means for calculating pixel-wise an absorption image M and an refraction angle \u03b8r image out of the pairs of specular images without a need for phase stepping according to:
M
\u2061
(
x
r
,
\u03d5
,
z
)
=
\u222b
–
\u221e
\u221e
\u2062
\u03bc
\u2061
(
x
,
y
,
z
)
\u2062
\u2146
y
r
=
ln
\u2061
(
2
\u2062
S
\u2061
(
x
g
D
)
\u2062
I
0
I
\u2061
(
x
r
,
\u03d5
,
z
)
+
I
\u2061
(
–
x
r
,
\u03d5
+
\u03c0
,
z
)
)
\u03b8
r
\u2061
(
x
r
,
\u03d5
,
z
)
=
–
\u222b
–
\u221e
\u221e
\u2062
\u2202
\u03b4
\u2061
(
x
,
y
,
z
)
\u2202
x
r
\u2062
\u2146
y
r
=
1
C
\u2062
I
\u2061
(
x
r
,
\u03d5
,
z
)
–
I
\u2061
(
–
x
r
,
\u03d5
+
\u03c0
,
z
)
I
\u2061
(
x
r
,
\u03d5
,
z
)
+
I
\u2061
(
–
x
r
,
\u03d5
+
\u03c0
,
z
)
where:
(x, y, z) are first spatial coordinates associated with the sample;
(xr, yr, z) are second spatial coordinates associated to the X-ray beam, the first and second coordinates being linked by a rotation matrix:
(
x
y
)
=
(
cos
\u2062
\u2062
\u03d5
–
sin
\u2062
\u2062
\u03d5
sin
\u2062
\u2062
\u03d5
cos
\u2062
\u2062
\u03d5
)
\u2062
(
x
r
y
r
)
,
where \u03c6 is a rotation angle between xr axis and x axis around a z axis;
Io is an incident X-ray intensity;
I(xr, \u03c6,z) is intensity recorded at said position-sensitive detector for a beam decided by xr,z and the rotation angle \u03c6;
D is a distance between said phase grating and said analyzer grating;
C is a constant; and
M(xr, \u03c6,z) and \u03b8r(xr, \u03c6,z) are inline definitions representing an absorption signal and a refraction angle, respectively, for a given coordinate xr, z, and the rotation angle \u03c6.
11. The method according to claim 10, wherein if the beam splitter grating is a line phase grating with low X-ray absorption, a thickness of a grating line will be with considerable X-ray phase shift \u03a6, the X-ray phase shift being either
\u03a6
\u2208
(
(
2
\u2062
l
–
1
)
\u2062
\u03c0
2
–
arc
\u2062
\u2062
sin
\u2062
\u2062
0.8
,
(
2
\u2062
l
–
1
)
\u2062
\u03c0
2
+
arc
\u2062
\u2062
sin
\u2062
\u2062
0.8
)
or
\u03a6\u03b5((2l\u22121)\u03c0\u2212arcsin 0.8, (2l\u22121)\u03c0+arcsin 0.8), where l=1, 2, 3 . . . .
12. The method according to claim 10, wherein if the beam splitter grating is a line phase grating with low X-ray absorption, it will be made from a material selected from the group consisting of silicon and polymer.
13. The method according to claim 10, wherein the analyzer grating has a one-dimensional grating structure with high X-ray absorption contrast, its period is a same as that of the image of the beam splitter grating, and is placed in front of the position-sensitive detector with its lines parallel to those of the phase grating, the one-dimensional grating structure serving as an anti-scatter grid, or an anti-scatter grid is used as a modulation mask.
14. The method according to claim 10, wherein a distance between the beam splitter grating and the analyzer grating is chosen to be an odd fractional Talbot distance, given by equation
D
n
,
sph
=
L
\xb7
D
n
L
–
D
n
=
L
\xb7
n
\xb7
p
1
2
2
\u2062
\u03b7
2
\u2062
\u03bb
L
–
n
\xb7
p
1
2
2
\u2062
\u03b7
2
\u2062
\u03bb
,
where n=1, 3, 5 . . . , and
\u03b7
=
{
1
if
\u2062
\u2062
the
\u2062
\u2062
phase
\u2062
\u2062
shift
\u2062
\u2062
of
\u2062
\u2062
G
1
\u2062
\u2062
is
\u2062
\u2062
(
2
\u2062
l
–
1
)
\u2062
\u03c0
2
,
p
2
=
L
+
D
n
,
sph
L
\u2062
p
1
2
if
\u2062
\u2062
the
\u2062
\u2062
phase
\u2062
\u2062
shift
\u2062
\u2062
of
\u2062
\u2062
G
1
\u2062
\u2062
is
\u2062
\u2062
(
2
\u2062
l
–
1
)
\u2062
\u03c0
,
p
2
=
L
+
D
n
,
sph
L
\u2062
p
1
2
,
where l=1, 2, 3 . . . .
Dn is the odd fractional Talbot distance when a parallel X-ray beam is used, while Dn,sph is that when a fan or cone X-ray beam is used, L is a distance between the X-ray source and the phase grating, and p1 and p2 represent a period of the beam splitter grating and the analyzer grating, respectively.
15. The method according to claim 10, which further comprises disposing a collimator between the X-ray source and the beam splitter grating for limiting a spatial extent of illuminating X-rays to a fan beam, a line-array detector is used, and a mechanism is comprised that allows to rotate either stepwise or continuously the sample relative to the rest of the apparatus, the rotational axis being perpendicular to an opening angle of the fan, and at a same time allows to translate either stepwise or continuously the sample relative to the rest of the apparatus along a direction parallel to a rotational axis.
16. The method according to claim 10, which further comprises disposing a collimator between the x-ray source and the beam splitter grating for limiting a spatial extent of illuminating X-rays to a cone beam, a pixel-array detector is used, and a mechanism is comprised that allows to rotate the sample relative to the rest of the apparatus, perpendicular to an opening angle of the fan.
17. The method according to claim 10, which further comprises disposing the analyzer grating either in front of the position-sensitive detector or with its one-dimensional grating structure integrated into the position-sensitive detector, the pixel of the detector is from 2 to 10 times the size of the period of the grating, half lines with sensor in a pixel are sensitive to X-ray and half lines without sensor let X-ray go through.
The claims below are in addition to those above.
All refrences to claim(s) which appear below refer to the numbering after this setence.
1. A method for generating a result for a query of a document of elements using pre-computed step queries and pre-computed step query results stored in a database, the method comprising:
receiving the query, wherein the query comprises a path of elements in the document of elements;
reducing the query into a plurality of step queries, wherein a step query comprises a relationship between a plurality of elements determined from a part of the path of elements;
for each of the plurality of step queries, retrieving a pre-computed step query result for a step query in the plurality of step queries by querying the database using the step query, wherein the step query corresponds to a pre-computed step query for the pre-computed step query result; and
generating the result for the query using the step query results.
2. The method of claim 1, wherein generating the result comprises taking the intersection of the step query results.
3. The method of claim 1, wherein the result of the query comprises a location in the document of elements that includes the path of elements for the query.
4. The method of claim 1, wherein the result of the query comprises the path of elements for the query.
5. The method of claim 1, further comprising optimizing the query, wherein optimizing the query comprises generating sequences from the path of elements that interpolate the path.
6. The method of claim 1, wherein the plurality of step queries comprise at least one of a one-step query, two-step query, three-step query, and four-step query.
7. The method of claim 1, wherein reducing the query into the plurality of step queries comprises reducing the query into at least one two-step query.
8. The method of claim 1, wherein reducing the query into the plurality of step queries comprises reducing the query into at least one three-step query.
9. The method of claim 1, further comprising
computing a hash key for queries in the pre-computed step queries and plurality of step queries; and
storing the hash keys for the pre-computed step queries and the corresponding pre-computed step query results in the database.
10. The method of claim 9, wherein retrieving the pre-computed step query result comprises using the stored hash keys for the step queries to retrieve the pre-computed step query results corresponding to the hash keys.
11. The method of claim 9, wherein the step query results comprise a ID for one or more elements in the document of elements.
12. The method of claim 9, further comprising post-processing the intersection of the step query results to generate the result for the query.
13. The method of claim 12, wherein post-processing the result comprises matching each step query in the step query results to the query.
14. The method of claim 9, wherein the relationship between the plurality of elements comprises a parentchild relationship.
15. The method of claim 9, wherein the document of elements comprise an XML document.
16. The method of claim 9, wherein elements in the document of elements comprise at least one of element, word, attribute, and string elements.
17. A method for creating a database of step queries and step query results for a document of elements, the method comprising:
determining relationships between a plurality of elements from the document of elements;
generating step queries from the relationships;
generating step query results for the step queries, wherein a step query result for a step query corresponds to one or more elements in the document of elements for the step query; and
storing the step queries and corresponding step query results in the database, wherein the stored step query results are usable to generate a result for a main query, wherein the main query can be reduced to a plurality of step queries that correspond to the stored step queries.
18. The method of claim 17, further comprising generating an index for the step queries, the index pointing to the corresponding step query results for each step query.
19. The method of claim 17, wherein the step query results comprise a ID for one or more elements in the document of elements.
20. The method of claim 17, wherein the plurality of step queries and corresponding step query results are stored in a PostingList.