![\"FMP\"](\"../data/FMP_Teaser_Cover.png\")
\n",
"![\"AudioLabs\"](\"../data/Logo_AudioLabs_Long.png\"){kind=logo}
\n",
"\n",
"\n",
"![\"C4\"](\"../data/C4_nav.png\")
\n",
"\n", "Following Section 4.4.1 of [Müller, FMP, Springer 2015], we introduce in this notebook a basic procedure for novelty detection. This approach was first introduced and applied to the audio domain by Foote.\n", "\n", "
![\"FMP_C4_F23a-b\"](\"../data/C4/FMP_C4_F23a-b.png\")
\n",
"\n",
"The two dark blocks on the main diagonal correspond to the regions of high similarity within the two sections. In contrast, the light regions outside these blocks express that there is a low cross-similarity between the sections. Thus, to find the boundary between the two sections one needs to identify the crux of the checkerboard. This can be done by correlating $\\mathbf{S}$ with a kernel that itself looks like a checkerboard. The simplest such kernel is the $(2\\times 2)$-unit kernel defined by\n",
"\n",
"\\begin{equation}\n",
" \\mathbf{K} = \\left[\\begin{array}{rr} -1 & 1\\\\ 1 & -1 \\end{array}\\right]\n",
" = \\left[\\begin{array}{rr} 0 & \\,\\,\\,\\,\\,1\\\\ 1 & 0 \\end{array}\\right] -\n",
" \\left[\\begin{array}{rr} 1 & \\,\\,\\,\\,\\,0\\\\ 0 & 1 \\end{array}\\right].\n",
"\\end{equation}\n",
"\n",
"This kernel can be written as the difference between a \"coherence\" and an \"anti-coherence\" kernel. The first kernel measures the self-similarity on either side of the center point and will be high when each of the two regions is homogeneous. The second kernel measures the cross-similarity between the two regions and will be high when there is little difference across the center point. The difference between the two values estimates the **novelty** of the feature sequence at the center point. The novelty is high when the two regions are self-similar but different from each other.\n",
"\n",
"In audio structure analysis, where one is typically interested in changes on a larger time scale, kernels of larger size are used. Adopting a centered view, where a physical time position is associated to the center of a window or kernel, we assume that the size of the kernel is odd given by $M=2L+1$ for some $L\\in\\mathbb{N}$. A **box-like checkerboard kernel** of size $M$ is an $(M\\times M)$ matrix $\\mathbf{K}_\\mathrm{Box}$, which is indexed by $[-L:L]\\times[-L:L]$. The matrix is defined by \n",
"\n",
"\\begin{equation}\n",
" \\mathbf{K}_\\mathrm{Box} = \\mathrm{sgn}(k)\\cdot \\mathrm{sgn}(\\ell),\n",
"\\end{equation}\n",
"\n",
"where $k,\\ell\\in[-L:L]$ and \"$\\mathrm{sgn}$'' is the sign function (being $-1$ for negative numbers, $0$ for zero, and $1$ for positive numbers). For example, in the case $L=2$, one obtains\n",
"\n",
"\\begin{equation}\n",
" \\mathbf{K}_\\mathrm{Box} = \\left[\\begin{array}{rrrrr} \n",
"\t\t\t\t\t\t-1 & -1 & \\,\\,\\,\\,\\,0 & 1 & 1 \\\\ \n",
"\t\t\t\t\t\t-1 & -1 & 0 &1 & 1 \\\\\n",
"\t\t\t\t\t\t 0 & 0 & 0 & 0 &0 \\\\\n",
"\t\t\t\t\t\t 1 & 1 & 0 & -1 & -1 \\\\\n",
"\t\t\t\t\t\t 1 & 1 & 0 & -1 & -1 \n",
" \\end{array}\\right]\n",
"\\end{equation}\n",
"\n",
"Note that the zero row and the zero column in the middle have been introduced more for theoretical reasons to ensure the symmetry of the kernel matrix. In the following code cell, we implement and visualize the box-like checkerboard kernel "
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"execution": {
"iopub.execute_input": "2024-02-15T08:53:51.260525Z",
"iopub.status.busy": "2024-02-15T08:53:51.260211Z",
"iopub.status.idle": "2024-02-15T08:53:51.418365Z",
"shell.execute_reply": "2024-02-15T08:53:51.417836Z"
}
},
"outputs": [
{
"data": {
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\n",
"text/plain": [
"![\"FMP_C4_F23c-d\"](\"../data/C4/FMP_C4_F23c-d.png\")
\n",
"\n",
"To compensate for the influence of the actual kernel size and of the tapering, one may normalize the kernel. This can be done by dividing the kernel by the sum over the absolute values of the kernel matrix:\n",
"\n",
"\\begin{equation}\n",
" \\mathbf{K}_\\mathrm{norm}(k,\\ell) = \\frac{\\mathbf{K}_\\mathrm{Gauss}(k,\\ell)}{\\sum_{k,\\ell\\in[-L:L]}|\\mathbf{K}_\\mathrm{Gauss}(k,\\ell)|}.\n",
"\\end{equation}\n",
"\n",
"The normalization becomes important when combining and fusing novelty information that is obtained from kernels of different size. In the following implementation, the taper parameter $\\varepsilon$ is specified in terms of the variance $\\sigma$ (normalized with respect to the kernel size $M=2L+1$)."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"execution": {
"iopub.execute_input": "2024-02-15T08:53:51.421120Z",
"iopub.status.busy": "2024-02-15T08:53:51.420922Z",
"iopub.status.idle": "2024-02-15T08:53:53.363078Z",
"shell.execute_reply": "2024-02-15T08:53:53.362496Z"
}
},
"outputs": [
{
"data": {
"image/png": 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nlJ7R3BIRf194H2YjaZxmNN06dTIbEz51Gq7rJT0oaWNNFvZ1krZJ2rbnyScLd8esG0qfOpWqgjDQvlLSs5Lek9OfRgONpHslbZ/mtQb4OPAK4ALgCeAj020jIjZExOqIWL3slJz6AWajbw7y0fSrGJwNbK5+fpGBKghX0LszcK2kczPXvwX4cm5nGp061aUSHCTpU8AXm+zLbD6Zg9vba4DXVt9vAv6LKeVWGKiCACCpXwXhO3XrS7oKeBT4ZW5nSt51WjHw49XA9lL7MhtFEYdqXw29qIoBkFsFoV+jadr1JZ1Ab8D54Gw6U/Ji8M2SLqCXVX0n8M6C+zIbMUFGGcHacistVUH4IL27yc9K060+vWIDTUT86axXSiW+yslOlIq8WrGivh3g4MH0MnWOOy65yOJdu+rbU8mzEsm3ckivTC7TNI9YKmkVpAPy5iJgL+ejlVPNMl8Ayc9ZbbmVlqogXAy8VdLN9PKvHZL0q4j4aN2BODm5WSsC+HXi1UiRKggR8ZqImIyISeAfgb9JDTLggcasJf0ZTd2rkVJVEI6KA/bMWtN4MJlRqSoIU5a5Kbc/HmjMWtE/dRoPHmjMWpF1MXje8EBj1oqs29vzhgcas1Z4RtOeY45pHkezbNnw+jOTBQvq23P6mQr+2Lu3vj2V4CvDJZekl0kdSiqOJqe4W+qtaNoO8NKX1re3E0fjazRmVpxnNGZW1CF8jcbM5kDjBydHhgcas1b4YrCZFefb22Y2JzyjMbOifHvbzIrzNZr2SLCwpks5EVM5kVdNLVpU355T9jBVMfOpp+rbh5BwdvWMKZVekHrLh1EBsuk2UsF4kK4ymfPRSsVpzs54zWicj8asFWXz0ZQstyLpfElfl/SwpIckJYdpDzRmrejfdap7NVKk3IqkhcBngXdFxHn0KiUkp2YeaMxaUzTD3hp6ZVKovl41zTKHy61ExH6gX26lbv3LgAcj4tvQS5AVEcnOeqAxa0XQiwyue/WqIAy81s1iB0XKrQDnACHpbknflPTenM5062Kw2djICtirrYLQUrmVhcDvAa8CngM2S7o/IjanVjKzOdf89nZL5VZ2AV+NiL3Vfu4ELqR3HWdGPnUya8VolluhVzHhfEnHVxeGf59eCd1a3ZrRpArIDUNdnE7uMqmgjFQ7pIND9u2rbx9CFfhVq9LLpN6K1J9rGDX/UtsYRtKqVGgUZNUFnKWiAXvrgdskXQc8BlwDvXIrwL9GxJURcUBSv9zKAmDjlHIrR6wfET+X9A/0BqkA7oyIL6U6062BxmxslM1HU7LcSkR8lt4t7mweaMxa40cQzKyo8XoEwQONWSv8UKWZzQkPNGZWlE+dzKw4nzqZWXHOGZxN0jXATcBvAxdFxLaBthuB6+gN238ZEXdnbLB54quUYQTspfqR089UlFoqIG8IAXuTk+llUm9Fqj0nEK5pUqphJK3KCcbL+ejk84xmNrYDfwh8cvCXVU6LtwHnAS8D7pV0Ts7j5GbjwddoskXEdwGkIx4CXQPcGhH7gB9I2kEv98XXm+zPbH4Zn/93S12jeTmwZeDnwTwXZuaSuC9Wl/MiIqZ7IhRmkeeiSuazDmDlGWdMt4jZPOUZzWF1OS9q1OW5mLr9DcAGgNUXXphKumM2T/Qz7I2HUvlo7gDeJmmRpDOBs4FvFNqX2Qgqnpy8UxoNNJKulrQL+F3gS5LuBqhyWtxGLyHOXcC7fcfJbFDZcitdo4junK1I2gP8cMqvJ4C9LXRnNkahj+B+DtvUfv5mRCzLWVHSXdX6dfZGxOVH27ku6dRAMx1J2+oSNHfBKPQR3M9hG5V+doFzBptZcR5ozKy4URhoNrTdgQyj0EdwP4dtVPrZus5fozGz0TcKMxozG3GdHGgkXSPpYUmHJK2e0najpB2SvifpTW31cSpJN0n6saRvVa8r02vNHUmXV+/ZDkk3tN2fmUjaKemh6j3cll6jPEkbJe2WtH3gd6dIukfS96uvL2mzj13XyYGGF9JP3Df4yynpJy4H/kVSItPInLolIi6oXkfUymlL9R59DLgCOBe4tnovu+p11XvYlVvHn6b3eRt0A7A5Is6mVw62s4N3F3RyoImI70bE96ZpOpx+IiJ+APTTT1i9i4AdEfFoROwHbqX3XlqGiLgPeHLKr9cAm6rvNwFXzWWfRk0nB5oaLwd+NPBz19JPXC/pwWqq3aWpdNfft0EBfEXS/dWT/V21PCKeAKi+ntpyfzqttZzBpdNPlFDXZ+DjwIeq/nwI+AjwjrnqW0Kr79ssvToiHpd0KnCPpEeqGYWNsNYGmtLpJ0rI7bOkTwFfLNyd2Wj1fZuNqjY0EbFb0u30Tvu6OND8VNKKiHhC0gpgd9sd6rJRO3XqbPqJ6sPWdzW9C9pdsRU4W9KZko6ld0H9jpb7dARJJ0g6qf89cBndeh8H3QGsrb5fC8w0Czc6Wm5F0tXAPwPL6KWf+FZEvCkiHpbUTz9xgG6ln7hZ0gX0Tkl2Au9stTcDIuKApOuBu4EFwMYqlUfXLAdur3JQLwQ+FxF3tdslkPR54LXARJUW5QPAeuA2SdcBjwHXtNfD7nNksJkVN2qnTmY2gjzQmFlxHmjMrDgPNGZWnAcaMyvOA42ZFeeBxsyK80BjZsX9P8HmhwZk2Z/kAAAAAElFTkSuQmCC\n",
"text/plain": [
"![\"FMP_C4_F24_color\"](\"../data/C4/FMP_C4_F24_color.png\")
\n",
"\n",
"The following code cell provides an implementation for computing a novelty function. "
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"execution": {
"iopub.execute_input": "2024-02-15T08:53:53.365853Z",
"iopub.status.busy": "2024-02-15T08:53:53.365646Z",
"iopub.status.idle": "2024-02-15T08:53:53.468549Z",
"shell.execute_reply": "2024-02-15T08:53:53.468019Z"
}
},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"![\"C0\"](\"../data/C0_nav.png\") | \n",
" ![\"C1\"](\"../data/C1_nav.png\") | \n",
" ![\"C2\"](\"../data/C2_nav.png\") | \n",
" ![\"C3\"](\"../data/C3_nav.png\") | \n",
" | \n",
" ![\"C5\"](\"../data/C5_nav.png\") | \n",
" ![\"C6\"](\"../data/C6_nav.png\") | \n",
" ![\"C7\"](\"../data/C7_nav.png\") | \n",
" ![\"C8\"](\"../data/C8_nav.png\") | \n",
"