Compressive sensing with local geometric features
Methods and apparatuses for compressive sensing that enable efficient recovery of features in an input signal based on acquiring a few measurements corresponding to the input signal. One method of compressive sensing includes folding an image to generate first and second folds, and recovering a feature of the image based on the first and second folds without reconstructing the image. One example of a compressive sensing apparatus includes a lens, a focal plane array coupled to the lens and configured to generate first and second folds based on the image, and a decoder configured to receive the first and second folds and to recover a feature of the image without reconstructing the image. The feature may be a local geometric feature or a corner. Compressive sensing methods and apparatuses for determining translation and rotation between two images are also disclosed.
1. A method of compressive sensing, comprising:
compressing a first image into a first sketch and a second sketch at least in part by partitioning the first image into a first plurality of non-overlapping subsections and linearly combining pixel values from each subsection of the first plurality of non-overlapping subsections, the first sketch and the second sketch each being undersampled representations of and smaller in size than the first image;
compressing a second image into a third sketch and a fourth sketch at least in part by partitioning the second image into a plurality of non-overlapping subsections and linearly combining pixel values from each subsection of the second plurality of non-overlapping subsections, the third sketch and the fourth sketch each being undersampled representations of and smaller in size than the second image; and
determining a translation between the first image and the second image, using a processor, based on the first sketch, the second sketch, the third sketch and the fourth sketch, without reconstructing each of the first image and the second image.
2. The method of claim 1 , wherein each of the first sketch and the third sketch has a first dimension and each of the second sketch and the fourth sketch has a second dimension, the second dimension being different from the first dimension.
3. The method of claim 1 , wherein determining the translation includes:
calculating a first phase correlation function based on the first sketch and the third sketch;
calculating a second phase correlation function based on the second sketch and the fourth sketch;
determining a first peak based on the first phase correlation function;
determining a second peak based on the second phase correlation function; and
determining the translation based on the first peak and the second peak.
4. The method of claim 3 , wherein compressing each of the first image and the second image includes encoding the first sketch and the second sketch based on a Chinese Remainder Theorem code and wherein determining the translation based on the first peak and the second peak includes decoding the first peak and the second peak based on the Chinese Remainder Theorem code.
5. A method of compressive sensing, comprising:
compressing a first image into a first sketch at least in part by partitioning the first image into a first plurality of non-overlapping subsections and linearly combining pixel values from each subsection of the first plurality of non-overlapping subsections, the first sketch being an undersampled representation of and smaller in size than the first image;
compressing a second image into a second sketch at least in part by partitioning the second image into a second plurality of non-overlapping subsections and linearly combining pixel values from each subsection of the second plurality of non-overlapping subsections, the second sketch being an undersampled representation of and smaller in size than the second image; and
determining a rotation between the first image and the second image, using a processor, based on the first sketch and the second sketch, without reconstructing each of the first image and the second image.
6. The method of claim 5 , wherein determining the rotation includes:
calculating a first distribution of gradient angles based on the first sketch;
calculating a second distribution of gradient angles based on the second sketch; and
correlating the first distribution and the second distribution.
7. An apparatus for compressive sensing, comprising:
an encoder configured to receive a first image and a second image, to compress the first image into a first sketch and a second sketch at least in part by partitioning the first image into a first plurality of non-overlapping subsections and linearly combining pixel values from each subsection of the first plurality of non-overlapping subsections, and to compress the second image into a third sketch and a fourth sketch at least in part by partitioning the second image into a second plurality of non-overlapping subsections and linearly combining pixel values from each subsection of the second plurality of non-overlapping subsections, the first sketch and the second sketch each being undersampled representations of and smaller in size than the first image, and the third sketch and the fourth sketch each being undersampled representations of and smaller in size than the second image; and
a processor configured to determine a translation between the first image and the second image, based on the first sketch, the second sketch, the third sketch and the fourth sketch, without reconstructing each of the first image and the second image.
8. The apparatus of claim 7 , wherein the encoder includes a focal plane array.
9. The apparatus of claim 7 , wherein each of the first sketch and the third sketch has a first dimension and each of the second sketch and the fourth sketch has a second dimension, the second dimension being different from the first dimension.
10. The apparatus of claim 7 , wherein the processor is further configured to:
calculate a first phase correlation function based on the first sketch and the third sketch;
calculate a second phase correlation function based on the second sketch and the fourth sketch;
determine a first peak based on the first phase correlation function;
determine a second peak based on the second phase correlation function; and
determine the translation based on the first peak and the second peak.
11. An apparatus for compressive sensing, comprising:
an encoder configured to receive a first image and a second image, to compress the first image into a first sketch at least in part by partitioning the first image into a first plurality of non-overlapping subsections and linearly combining pixel values from each subsection of the first plurality of non-overlapping subsections, and to compress the second image into a second sketch at least in part by partitioning the second image into a second plurality of non-overlapping subsections and linearly combining pixel values from each subsection of the second plurality of non-overlapping subsections, the first sketch being an undersampled representation of and smaller in size than the first image and the second sketch being an undersampled representation of and smaller in size than the second image; and
a processor configured to determine a rotation between the first image and the second image, based on the first sketch and the second sketch, without reconstructing each of the first image and the second image.
12. The apparatus of claim 11 , wherein the encoder includes a focal plane array.
13. The apparatus of claim 11 , wherein the processor is further configured to:
calculate a first distribution of gradient angles based on the first fold;
calculate a second distribution of gradient angles based on the second fold; and
correlate the first distribution and the second distribution to determine a rotation between the first image and the second image.
14. The method of claim 1 , wherein linearly combining the pixel values from each subsection of the first plurality of non-overlapping subsections includes adding the pixel values from each subsection.
15. The method of claim 1 , wherein partitioning the first image into the first plurality of non-overlapping subsections and linearly combining the pixel values includes multiplying the image by a sketching matrix, the sketching matrix defining an undersampling pattern.
16. The method of claim 1 , wherein compressing the first image into a first sketch and a second sketch includes folding the first image into a first fold and a second fold.
17. The method of claim 16 , wherein compressing the second image into a third sketch and a fourth sketch includes folding the second image into a third fold and a fourth fold.