Increasing accuracy of discrete curve transform estimates for curve matching in four or more dimensions
View Patent ↗System and method for estimating a rotational shift between a first discrete curve and a second discrete curve, where the second discrete curve is a rotationally shifted version of the first discrete curve. First and second discrete curves are received. A rotational shift between the first discrete curve and the second discrete curve is estimated based on the first discrete curve and the second discrete curve. A cumulative rotational shift is updated based on the estimated rotational shift. A rotationally shifted version of the second discrete curve is generated based on the cumulative rotational shift. The estimating, updating, and generating are performed in an iterative manner using the respective rotationally shifted discrete curve for each iteration until a stopping condition occurs, thereby determining a final estimate of the rotational shift between the first discrete curve and the second discrete curve. The final estimate may be used to perform curve matching.
1. A computer-accessible memory medium that stores program instructions for estimating a rotational shift between a first discrete curve and a second discrete curve, wherein the program instructions are executable to perform:
receiving a first discrete curve and a second discrete curve, wherein the second discrete curve is a rotationally shifted version of the first discrete curve, wherein the first discrete curve and the second discrete curve each comprises a respective curve in at least four dimensions;
estimating a rotational shift between the first discrete curve and the second discrete curve based on the first discrete curve and the second discrete curve, wherein said estimating the rotational shift between the first discrete curve and the second discrete curve based on the first discrete curve and the second discrete curve comprises calculating:
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wherein y and z refer respectively to points in the first discrete curve and the second discrete curve, indexed by n and d, wherein n denotes a particular point of N points, and d denotes dimension up to a maximum of D;
updating a cumulative rotational shift based on the estimated rotational shift;
generating a rotationally shifted version of the second discrete curve based on the cumulative rotational shift;
performing said estimating, said updating, and said generating in an iterative manner using the respective rotationally shifted discrete curve for each iteration until a stopping condition occurs, thereby determining a final estimate of the rotational shift between the first discrete curve and the second discrete curve; and
storing the final estimate of the rotational shift, wherein the final estimate of the rotational shift is useable to perform discrete curve matching.
2. The memory medium of claim 1 , wherein each point of the first and second discrete curves comprises generalized coordinates.
3. The memory medium of claim 1 , wherein the first discrete curve and the second discrete curve each comprises a respective spatial curve.
4. The memory medium of claim 1 , wherein each point of the first and second curves comprises:
at least three 3D spatial coordinates; and
a respective one or more additional coordinates, each based on respective additional data associated with the point, wherein the one or more additional coordinates are interpreted as additional spatial coordinates for the point.
5. The memory medium of claim 1 , wherein the first discrete curve and the second discrete curve each comprises a respective sequence of points, wherein each point comprises:
a respective two or more spatial coordinates; and
a respective two or more additional coordinates, each based on respective additional data associated with the point, wherein the one or more additional coordinates are interpreted as additional spatial coordinates for the point.
6. The memory medium of claim 5 , wherein the respective two or more additional coordinates comprises image data, comprising one or more of:
a gray-scale value, wherein the gray-scale value is interpreted as another spatial coordinate; or
one or more color values, wherein the one or more color values are interpreted as other spatial coordinates.
7. The memory medium of claim 1 , wherein the stopping condition occurs when a magnitude of the estimated rotational shift is less than a specified threshold value.
8. The memory medium of claim 1 , wherein the stopping condition occurs when a specified number of iterations have elapsed.
9. The memory medium of claim 1 , wherein the first discrete curve and the second discrete curve each comprise a respective sequence of points in a complex space.
10. The memory medium of claim 1 , wherein the first discrete curve and the second discrete curve comprise substantially smooth discrete curves.
11. The memory medium of claim 1 , wherein the first discrete curve and the second discrete curve each comprises a respective closed curve.
12. The memory medium of claim 1 ,
wherein the first discrete curve and the second discrete curve each comprises the same number of points, wherein for each curve, each point corresponds to a value of an index; and
wherein the value of the estimated rotational shift is normalized with respect to the index.
13. The memory medium of claim 12 ,
wherein an estimated a rotational shift value of 1 corresponds to a cyclic shift of one point in the second discrete curve.
14. The memory medium of claim 1 , wherein the second discrete curve is rotationally shifted from the first discrete curve by less than a rotational distance between successive points in the second discrete curve.
15. The memory medium of claim 1 , wherein the second discrete curve is rotationally shifted from the first discrete curve by less than or equal to 5 degrees.
16. The memory medium of claim 1 , wherein the second discrete curve is rotationally shifted from the first discrete curve by less than or equal to 1 degree.
17. The memory medium of claim 1 , wherein said rotational shifting the second discrete curve based on the estimated rotational shift to generate a respective rotationally shifted discrete curve further comprises:
interpolating the second discrete curve.
18. The memory medium of claim 1 , wherein the program instructions are further executable to perform:
outputting the final estimate of the rotational shift between the first discrete curve and the second discrete curve.
19. The memory medium of claim 1 , wherein the program instructions are further executable to perform:
performing curve matching on the first discrete curve and the second discrete curve using the final estimate of the rotational shift, thereby generating results indicating a degree of match between the first discrete curve and the second discrete curve.
20. A method for estimating a rotational shift between a first discrete curve and a second discrete curve, the method comprising:
receiving a first discrete curve and a second discrete curve, wherein the second discrete curve is a rotationally shifted version of the first discrete curve;
estimating a rotational shift between the first discrete curve and the second discrete curve based on the first discrete curve and the second discrete curve, wherein said estimating the rotational shift between the first discrete curve and the second discrete curve based on the first discrete curve and the second discrete curve comprises calculating:
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=
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n
=
1
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d
=
1
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wherein y and z refer respectively to points in the first discrete curve and the second discrete curve, indexed by n and d, wherein n denotes a particular point of N points, and d denotes dimension up to a maximum of D;
updating a cumulative rotational shift based on the estimated rotational shift;
generating a rotationally shifted version of the second discrete curve based on the cumulative rotational shift; and
performing said estimating, said updating, and said generating in an iterative manner using the respective rotationally shifted discrete curve for each iteration until a stopping condition occurs, thereby determining a final estimate of the rotational shift between the first discrete curve and the second discrete curve; and
storing the final estimate of the rotational shift, wherein the final estimate of the rotational shift is useable to perform discrete curve matching.
21. A system for estimating a rotational shift between a first discrete curve and a second discrete curve, the system comprising:
a processor;
a memory medium coupled to the processor;
an input coupled to the processor and the memory medium; and
an output coupled to the processor and the memory medium;
wherein the input is operable to receive a first discrete curve and a second discrete curve, wherein the second discrete curve is a rotationally shifted version of the first discrete curve, and wherein the first discrete curve and the second discrete curve each comprises a respective sequence of points in at least three dimensions;
wherein the memory medium stores program instructions which are executable by the processor to:
estimate a rotational shift between the first discrete curve and the second discrete curve based on the first discrete curve and the second discrete curve, wherein said estimating the rotational shift between the first discrete curve and the second discrete curve based on the first discrete curve and the second discrete curve comprises calculating:
α
=
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n
=
1
N
∑
d
=
1
D
(
z
nd
-
y
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)
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n
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wherein y and z refer respectively to points in the first discrete curve and the second discrete curve, indexed by n and d, wherein n denotes a particular point of N points, and d denotes dimension up to a maximum of D;
update a cumulative rotational shift based on the estimated rotational shift;
generate a rotationally shifted version of the second discrete curve based on the cumulative rotational shift; and
perform said estimating, said updating, and said generating in an iterative manner using the respective rotationally shifted discrete curve for each iteration until a stopping condition occurs, thereby determining a final estimate of the rotational shift between the first discrete curve and the second discrete curve;
wherein the output is operable to output the final estimate of the rotational shift between the first discrete curve and the second discrete curve, and wherein the final estimate of the rotational shift is useable to perform discrete curve matching.
22. A system for estimating a rotational shift between a first discrete curve and a second discrete curve, the system comprising:
means for receiving a first discrete curve and a second discrete curve, wherein the second discrete curve is a rotationally shifted version of the first discrete curve, wherein the first discrete curve and the second discrete curve each comprises a respective curve in at least three dimensions;
means for estimating a rotational shift between the first discrete curve and the second discrete curve based on the first discrete curve and the second discrete curve, wherein said estimating the rotational shift between the first discrete curve and the second discrete curve based on the first discrete curve and the second discrete curve comprises calculating:
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=
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n
=
1
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d
=
1
D
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wherein y and z refer respectively to points in the first discrete curve and the second discrete curve, indexed by n and d, wherein n denotes a particular point of N points, and d denotes dimension up to a maximum of D;
means for updating a cumulative rotational shift based on the estimated rotational shift;
means for generating a rotationally shifted version of the second discrete curve based on the cumulative rotational shift;
means for performing said estimating, said updating, and said generating in an iterative manner using the respective rotationally shifted discrete curve for each iteration until a stopping condition occurs, thereby determining a final estimate of the rotational shift between the first discrete curve and the second discrete curve; and
means for storing the final estimate of the rotational shift, wherein the final estimate of the rotational shift is useable to perform discrete curve matching.
23. A programmable hardware element configured for estimating a rotational shift between a first discrete curve and a second discrete curve, wherein the programmable hardware element is configured to perform:
receiving a first discrete curve and a second discrete curve, wherein the second discrete curve is a rotationally shifted version of the first discrete curve, wherein the first discrete curve and the second discrete curve each comprises a respective curve in at least three dimensions;
estimating a rotational shift between the first discrete curve and the second discrete curve based on the first discrete curve and the second discrete curve, wherein said estimating the rotational shift between the first discrete curve and the second discrete curve based on the first discrete curve and the second discrete curve comprises calculating:
α
=
∑
n
=
1
N
∑
d
=
1
D
(
z
nd
-
y
nd
)
(
y
(
n
+
1
)
d
-
y
nd
)
∑
n
=
1
N
∑
d
=
1
D
(
y
(
n
+
1
)
d
-
y
nd
)
2
,
wherein y and z refer respectively to points in the first discrete curve and the second discrete curve, indexed by n and d, wherein n denotes a particular point of N points, and d denotes dimension up to a maximum of D;
updating a cumulative rotational shift based on the estimated rotational shift;
generating a rotationally shifted version of the second discrete curve based on the cumulative rotational shift;
performing said estimating, said updating, and said generating in an iterative manner using the respective rotationally shifted discrete curve for each iteration until a stopping condition occurs, thereby determining a final estimate of the rotational shift between the first discrete curve and the second discrete curve; and
storing the final estimate of the rotational shift, wherein the final estimate of the rotational shift is useable to perform discrete curve matching.