Discrete curve symmetry detection
First Claim
1. A computer accessible memory medium that stores program instructions for detecting symmetries of discrete curves, wherein the program instructions are executable by a processor to perform:
- applying a first mapping operator to a first discrete curve, thereby generating a first mapped discrete curve, wherein the mapping operator operates to amplify features of the first discrete curve;
computing a correlation of the first mapped discrete curve with each of a plurality of rotationally shifted versions of a second mapped discrete curve, thereby generating a corresponding plurality of correlation values, wherein each correlation value indicates a respective degree of correlation between the first mapped discrete curve and a respective rotationally shifted version of the second mapped discrete curve;
determining a minimum period of the correlation of the first and second discrete curves based on the plurality of correlation values;
determining a symmetry group of the first and second discrete curves based on the minimum period; and
outputting the determined symmetry group.
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Abstract
System and method for detecting symmetries of discrete curves. A mapping operator is applied to a first discrete curve to amplify its features, generating a first mapped discrete curve. A correlation of the first mapped discrete curve with each of a plurality of rotationally shifted versions of a second mapped discrete curve is computed, generating a corresponding plurality of correlation values. A minimum period of the two curves is determined based on the correlation values, and, based on the minimum period, a symmetry group (SG) of the two curves is determined and output. If the two curves are the same curve, the SG is the rotational SG of the discrete curve. If the second curve is a reflection of the first, the SG is the mutual reflection SG of the first. If the first and second curves are different curves, the SG is the mutual SG of the two curves.
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Citations
58 Claims
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1. A computer accessible memory medium that stores program instructions for detecting symmetries of discrete curves, wherein the program instructions are executable by a processor to perform:
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applying a first mapping operator to a first discrete curve, thereby generating a first mapped discrete curve, wherein the mapping operator operates to amplify features of the first discrete curve; computing a correlation of the first mapped discrete curve with each of a plurality of rotationally shifted versions of a second mapped discrete curve, thereby generating a corresponding plurality of correlation values, wherein each correlation value indicates a respective degree of correlation between the first mapped discrete curve and a respective rotationally shifted version of the second mapped discrete curve; determining a minimum period of the correlation of the first and second discrete curves based on the plurality of correlation values; determining a symmetry group of the first and second discrete curves based on the minimum period; and outputting the determined symmetry group. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 56, 57, 58)
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42. A method for detecting symmetries of discrete curves, the method comprising:
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applying a first mapping operator to a first discrete curve, thereby generating a first mapped discrete curve, wherein the mapping operator operates to amplify features of the first discrete curve; computing a correlation of the first mapped discrete curve with each of a plurality of rotationally shifted versions of a second mapped discrete curve, thereby generating a corresponding plurality of correlation values, wherein each correlation value indicates a respective degree of correlation between the first mapped discrete curve and a respective rotationally shifted version of the second mapped discrete curve; determining a minimum period of the correlation of the first and second discrete curves based on the plurality of correlation values; determining a symmetry group of the first and second discrete curves based on the minimum period; and outputting the determined symmetry group. - View Dependent Claims (43, 44, 45, 46, 47, 48, 49, 50, 51, 52)
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53. A method for detecting reflection symmetries of discrete curves, the method comprising:
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applying a mapping operator to a discrete curve, thereby generating a mapped discrete curve, wherein the mapping operator operates to amplify features of the discrete curve; computing a correlation of the mapped discrete curve with each of a plurality of rotationally shifted versions of a reflected image of the mapped discrete curve, thereby generating a corresponding plurality of correlation values, wherein each correlation value indicates a respective degree of correlation between the mapped discrete curve and a respective rotationally shifted version of the reflected image of the mapped discrete curve; determining a minimum period of the correlation based on the plurality of correlation values; determining a reflection symmetry group of the discrete curve based on the minimum period; and outputting the determined reflection symmetry group.
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54. A computer accessible memory medium that stores program instructions for detecting rotational symmetries of discrete curves, wherein the program instructions are executable by a processor to perform:
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applying a mapping operator to a discrete curve, thereby generating a mapped discrete curve, wherein the mapping operator operates to amplify features of the discrete curve; computing a correlation of the mapped discrete curve with each of a plurality of rotationally shifted versions of the mapped discrete curve, thereby generating a corresponding plurality of correlation values, wherein each correlation value indicates a respective degree of correlation between the mapped discrete curve and a respective rotationally shifted version of the mapped discrete curve; determining a minimum period of the correlation of the discrete curve based on the plurality of correlation values; and determining a symmetry group of the discrete curve based on the minimum period; and outputting the determined rotational symmetry group.
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55. A computer accessible memory medium that stores program instructions for detecting reflection symmetries of a discrete curve, wherein the program instructions are executable by a processor to perform:
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applying a mapping operator to a discrete curve, thereby generating a mapped discrete curve, wherein the mapping operator operates to amplify features of the discrete curve; computing a correlation of the mapped discrete curve with each of a plurality of rotationally shifted versions of a reflected image of the mapped discrete curve, thereby generating a corresponding plurality of correlation values, wherein each correlation value indicates a respective degree of correlation between the mapped discrete curve and a respective rotationally shifted version of the reflected image of the mapped discrete curve; determining a minimum period of the correlation of the discrete curve based on the plurality of correlation values; and determining a reflection symmetry group of the discrete curve based on the minimum period; and outputting the determined reflection symmetry group.
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Specification