AIMS CDT - Signal Processing Michaelmas Term 2020 Xiaowen Dong - PowerPoint PPT Presentation

AIMS CDT - Signal Processing Michaelmas Term 2020 Xiaowen Dong Department of Engineering Science

Representation of Signals 2 /70

What is a representation of a signal? • Sum of delta functions in time or space (sampling domain) - good for display or playback - not good for analysis (e.g., denoising, compression) t T 3 /70

What is a representation of a signal? • In a time-series setting, useful representation could be past samples • More generally, it involves transformation of the signal into a new domain - where useful characteristics are revealed example: Fourier coe ffi cients reveal rate of change of the signal - Source: http://www.physik.uni-kl.de • Usefulness of the representation depends on the analysis goal which may vary but all share the core desire for simpli fi cation - 4 /70

Example: Denoising 10 Original Signal Denoised Signal 5 0 -5 -10 0 200 400 600 800 1000 1200 goal: separate signal and noise 5 /70

Example: Compression original JPEG 2000 (comp. ratio: 10) JPEG 2000 (comp. ratio: 100) goal: compress fi le size without sacri fi cing (too much) quality 6 /70

Example: Recognition samples true “causes” PCA ICA sparse coding KSVD [Tosic11] goal: capture true “causes” of signals given noisy input 7 /70

<latexit sha1_base64="9T3rpMd4lDavbSX3a5i0kGx3PY=">ACRXicbZDNS8MwGMbT+TXn19Sjl+AQvDgaEfQiDL14kgnuA9Yy0jRdw9KmJKkwSv3jvHj35n/gxYMiXjXdnCbL4Q8/J73JW8eL+FMadt+tUpLyura+X1ysbm1vZOdXevrUQqCW0RwYXselhRzmLa0kxz2k0kxZHacbXhd+54FKxUR8r0cJdSM8iFnACNYG9auO4wnuq1FkrsxphiyHl7A3A5OQ9e38cYEhw4gvtFq0stsTlOduv1qz6/a4KJAU1ED02r2qy+OL0ga0VgTjpXqITvRboalZoTvOKkiaYDPGA9oyMcUSVm41TyOGRIT4MhDQn1nBM/05kOFLFkqYzwjpU814B/N6qQ4u3IzFSapTCYPBSmHWsAiUugzSYnmIyMwkczsCkmIJSbaBF8xIaD5Ly+K9mkd2XV0d1ZrXE3jKIMDcAiOAQLnoAFuQBO0AFP4A18gE/r2Xq3vqzvSWvJms7sg5myfn4BNsm1DQ=</latexit> <latexit sha1_base64="9T3rpMd4lDavbSX3a5i0kGx3PY=">ACRXicbZDNS8MwGMbT+TXn19Sjl+AQvDgaEfQiDL14kgnuA9Yy0jRdw9KmJKkwSv3jvHj35n/gxYMiXjXdnCbL4Q8/J73JW8eL+FMadt+tUpLyura+X1ysbm1vZOdXevrUQqCW0RwYXselhRzmLa0kxz2k0kxZHacbXhd+54FKxUR8r0cJdSM8iFnACNYG9auO4wnuq1FkrsxphiyHl7A3A5OQ9e38cYEhw4gvtFq0stsTlOduv1qz6/a4KJAU1ED02r2qy+OL0ga0VgTjpXqITvRboalZoTvOKkiaYDPGA9oyMcUSVm41TyOGRIT4MhDQn1nBM/05kOFLFkqYzwjpU814B/N6qQ4u3IzFSapTCYPBSmHWsAiUugzSYnmIyMwkczsCkmIJSbaBF8xIaD5Ly+K9mkd2XV0d1ZrXE3jKIMDcAiOAQLnoAFuQBO0AFP4A18gE/r2Xq3vqzvSWvJms7sg5myfn4BNsm1DQ=</latexit> <latexit sha1_base64="9T3rpMd4lDavbSX3a5i0kGx3PY=">ACRXicbZDNS8MwGMbT+TXn19Sjl+AQvDgaEfQiDL14kgnuA9Yy0jRdw9KmJKkwSv3jvHj35n/gxYMiXjXdnCbL4Q8/J73JW8eL+FMadt+tUpLyura+X1ysbm1vZOdXevrUQqCW0RwYXselhRzmLa0kxz2k0kxZHacbXhd+54FKxUR8r0cJdSM8iFnACNYG9auO4wnuq1FkrsxphiyHl7A3A5OQ9e38cYEhw4gvtFq0stsTlOduv1qz6/a4KJAU1ED02r2qy+OL0ga0VgTjpXqITvRboalZoTvOKkiaYDPGA9oyMcUSVm41TyOGRIT4MhDQn1nBM/05kOFLFkqYzwjpU814B/N6qQ4u3IzFSapTCYPBSmHWsAiUugzSYnmIyMwkczsCkmIJSbaBF8xIaD5Ly+K9mkd2XV0d1ZrXE3jKIMDcAiOAQLnoAFuQBO0AFP4A18gE/r2Xq3vqzvSWvJms7sg5myfn4BNsm1DQ=</latexit> <latexit sha1_base64="9T3rpMd4lDavbSX3a5i0kGx3PY=">ACRXicbZDNS8MwGMbT+TXn19Sjl+AQvDgaEfQiDL14kgnuA9Yy0jRdw9KmJKkwSv3jvHj35n/gxYMiXjXdnCbL4Q8/J73JW8eL+FMadt+tUpLyura+X1ysbm1vZOdXevrUQqCW0RwYXselhRzmLa0kxz2k0kxZHacbXhd+54FKxUR8r0cJdSM8iFnACNYG9auO4wnuq1FkrsxphiyHl7A3A5OQ9e38cYEhw4gvtFq0stsTlOduv1qz6/a4KJAU1ED02r2qy+OL0ga0VgTjpXqITvRboalZoTvOKkiaYDPGA9oyMcUSVm41TyOGRIT4MhDQn1nBM/05kOFLFkqYzwjpU814B/N6qQ4u3IzFSapTCYPBSmHWsAiUugzSYnmIyMwkczsCkmIJSbaBF8xIaD5Ly+K9mkd2XV0d1ZrXE3jKIMDcAiOAQLnoAFuQBO0AFP4A18gE/r2Xq3vqzvSWvJms7sg5myfn4BNsm1DQ=</latexit> <latexit sha1_base64="+A6VEtos3HL/1r2wKYFonhC20Mo=">AB8XicbVDLSsNAFL2pr1pfVZduBovgqiQi6LoxmUF+8A2lMl0g6dTMLMjVBC/8KNC0Xc+jfu/BsnbRbaemDgcM69zLknSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlslrHuBtRwKRvoUDJu4nmNAok7wST29zvPHFtRKwecJpwP6IjJULBKFrpsR9RHAdhxmaDas2tu3OQVeIVpAYFmoPqV38YszTiCpmkxvQ8N0E/oxoFk3xW6aeGJ5RN6Ij3LFU04sbP5oln5MwqQxLG2j6FZK7+3shoZMw0CuxkntAse7n4n9dLMbz2M6GSFLli4/CVBKMSX4+GQrNGcqpJZRpYbMSNqaMrQlVWwJ3vLJq6R9Ufcund/WvcFHWU4QRO4Rw8uIG3ETWsBAwTO8wptjnBfn3flYjJacYucY/sD5/AHdUJEH</latexit> <latexit sha1_base64="+A6VEtos3HL/1r2wKYFonhC20Mo=">AB8XicbVDLSsNAFL2pr1pfVZduBovgqiQi6LoxmUF+8A2lMl0g6dTMLMjVBC/8KNC0Xc+jfu/BsnbRbaemDgcM69zLknSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlslrHuBtRwKRvoUDJu4nmNAok7wST29zvPHFtRKwecJpwP6IjJULBKFrpsR9RHAdhxmaDas2tu3OQVeIVpAYFmoPqV38YszTiCpmkxvQ8N0E/oxoFk3xW6aeGJ5RN6Ij3LFU04sbP5oln5MwqQxLG2j6FZK7+3shoZMw0CuxkntAse7n4n9dLMbz2M6GSFLli4/CVBKMSX4+GQrNGcqpJZRpYbMSNqaMrQlVWwJ3vLJq6R9Ufcund/WvcFHWU4QRO4Rw8uIG3ETWsBAwTO8wptjnBfn3flYjJacYucY/sD5/AHdUJEH</latexit> <latexit sha1_base64="+A6VEtos3HL/1r2wKYFonhC20Mo=">AB8XicbVDLSsNAFL2pr1pfVZduBovgqiQi6LoxmUF+8A2lMl0g6dTMLMjVBC/8KNC0Xc+jfu/BsnbRbaemDgcM69zLknSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlslrHuBtRwKRvoUDJu4nmNAok7wST29zvPHFtRKwecJpwP6IjJULBKFrpsR9RHAdhxmaDas2tu3OQVeIVpAYFmoPqV38YszTiCpmkxvQ8N0E/oxoFk3xW6aeGJ5RN6Ij3LFU04sbP5oln5MwqQxLG2j6FZK7+3shoZMw0CuxkntAse7n4n9dLMbz2M6GSFLli4/CVBKMSX4+GQrNGcqpJZRpYbMSNqaMrQlVWwJ3vLJq6R9Ufcund/WvcFHWU4QRO4Rw8uIG3ETWsBAwTO8wptjnBfn3flYjJacYucY/sD5/AHdUJEH</latexit> <latexit sha1_base64="+A6VEtos3HL/1r2wKYFonhC20Mo=">AB8XicbVDLSsNAFL2pr1pfVZduBovgqiQi6LoxmUF+8A2lMl0g6dTMLMjVBC/8KNC0Xc+jfu/BsnbRbaemDgcM69zLknSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlslrHuBtRwKRvoUDJu4nmNAok7wST29zvPHFtRKwecJpwP6IjJULBKFrpsR9RHAdhxmaDas2tu3OQVeIVpAYFmoPqV38YszTiCpmkxvQ8N0E/oxoFk3xW6aeGJ5RN6Ij3LFU04sbP5oln5MwqQxLG2j6FZK7+3shoZMw0CuxkntAse7n4n9dLMbz2M6GSFLli4/CVBKMSX4+GQrNGcqpJZRpYbMSNqaMrQlVWwJ3vLJq6R9Ufcund/WvcFHWU4QRO4Rw8uIG3ETWsBAwTO8wptjnBfn3flYjJacYucY/sD5/AHdUJEH</latexit> <latexit sha1_base64="P/78WoKPqTPMAmrnJ7K8sbyWLsQ=">AB8XicbVBNS8NAFHypX7V+VT16WSyCp5KIoMeiF48VbCu2oWy2m3bpZhN2X8QS+i+8eFDEq/Gm/GTZuDtg4sDPvsfMmSKQw6LrfTmldW19o7xZ2dre2d2r7h+0TZxqxlslrG+D6jhUijeQoGS3yea0yiQvBOMr3O/8i1EbG6w0nC/YgOlQgFo2ilh15EcRSE2dO0X625dXcGsky8gtSgQLNf/eoNYpZGXCGT1Jiu5yboZ1SjYJPK73U8ISyMR3yrqWKRtz42SzxlJxYZUDCWNunkMzU3xsZjYyZRIGdzBOaRS8X/O6KYaXfiZUkiJXbP5RmEqCMcnPJwOhOUM5sYQyLWxWwkZU4a2pIotwVs8eZm0z+qeW/duz2uNq6KOMhzBMZyCBxfQgBtoQgsYKHiGV3hzjPivDsf89GSU+wcwh84nz/9OZEc</latexit> <latexit sha1_base64="P/78WoKPqTPMAmrnJ7K8sbyWLsQ=">AB8XicbVBNS8NAFHypX7V+VT16WSyCp5KIoMeiF48VbCu2oWy2m3bpZhN2X8QS+i+8eFDEq/Gm/GTZuDtg4sDPvsfMmSKQw6LrfTmldW19o7xZ2dre2d2r7h+0TZxqxlslrG+D6jhUijeQoGS3yea0yiQvBOMr3O/8i1EbG6w0nC/YgOlQgFo2ilh15EcRSE2dO0X625dXcGsky8gtSgQLNf/eoNYpZGXCGT1Jiu5yboZ1SjYJPK73U8ISyMR3yrqWKRtz42SzxlJxYZUDCWNunkMzU3xsZjYyZRIGdzBOaRS8X/O6KYaXfiZUkiJXbP5RmEqCMcnPJwOhOUM5sYQyLWxWwkZU4a2pIotwVs8eZm0z+qeW/duz2uNq6KOMhzBMZyCBxfQgBtoQgsYKHiGV3hzjPivDsf89GSU+wcwh84nz/9OZEc</latexit> <latexit sha1_base64="P/78WoKPqTPMAmrnJ7K8sbyWLsQ=">AB8XicbVBNS8NAFHypX7V+VT16WSyCp5KIoMeiF48VbCu2oWy2m3bpZhN2X8QS+i+8eFDEq/Gm/GTZuDtg4sDPvsfMmSKQw6LrfTmldW19o7xZ2dre2d2r7h+0TZxqxlslrG+D6jhUijeQoGS3yea0yiQvBOMr3O/8i1EbG6w0nC/YgOlQgFo2ilh15EcRSE2dO0X625dXcGsky8gtSgQLNf/eoNYpZGXCGT1Jiu5yboZ1SjYJPK73U8ISyMR3yrqWKRtz42SzxlJxYZUDCWNunkMzU3xsZjYyZRIGdzBOaRS8X/O6KYaXfiZUkiJXbP5RmEqCMcnPJwOhOUM5sYQyLWxWwkZU4a2pIotwVs8eZm0z+qeW/duz2uNq6KOMhzBMZyCBxfQgBtoQgsYKHiGV3hzjPivDsf89GSU+wcwh84nz/9OZEc</latexit> <latexit sha1_base64="P/78WoKPqTPMAmrnJ7K8sbyWLsQ=">AB8XicbVBNS8NAFHypX7V+VT16WSyCp5KIoMeiF48VbCu2oWy2m3bpZhN2X8QS+i+8eFDEq/Gm/GTZuDtg4sDPvsfMmSKQw6LrfTmldW19o7xZ2dre2d2r7h+0TZxqxlslrG+D6jhUijeQoGS3yea0yiQvBOMr3O/8i1EbG6w0nC/YgOlQgFo2ilh15EcRSE2dO0X625dXcGsky8gtSgQLNf/eoNYpZGXCGT1Jiu5yboZ1SjYJPK73U8ISyMR3yrqWKRtz42SzxlJxYZUDCWNunkMzU3xsZjYyZRIGdzBOaRS8X/O6KYaXfiZUkiJXbP5RmEqCMcnPJwOhOUM5sYQyLWxWwkZU4a2pIotwVs8eZm0z+qeW/duz2uNq6KOMhzBMZyCBxfQgBtoQgsYKHiGV3hzjPivDsf89GSU+wcwh84nz/9OZEc</latexit> Signal representation via dictionaries • Orthogonal dictionaries signal dictionary coe ffi cients = X Φ = [ φ 0 φ 1 · · · φ N − 1 ] x c “atoms” 8 /70