added find_dark2 (improved find_dark method)
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@ -276,8 +276,8 @@
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"def estimate_dark_lens_fluo(I, A, P, R, f_1, F_1):\n",
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" lenses = compute_lenses(A, P, f_1)\n",
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" fluos = estimate_fluos(f_1, F_1, R)\n",
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" lens = lenses[0]\n",
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" fluo = fluos[:,0]\n",
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" lens = np.mean(lenses, axis=0)\n",
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" fluo = np.mean(fluos, axis=1)\n",
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" dark = I - lens.reshape(1, len(lens)) * fluo.reshape(len(fluo), 1)\n",
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" return dark, lens, fluo\n",
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"\n",
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@ -326,8 +326,9 @@
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" dark_matrix, lens, fluo = estimate_dark_lens_fluo(I, A, P, R, f_1, F_1)\n",
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" error = np.var(dark_matrix)\n",
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" return error\n",
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" \n",
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" minimize_result = optimize.minimize(fun=function_to_minimize, x0=(-0.7, 0.5), method = 'Nelder-Mead')\n",
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" start_f_1 = -0.7\n",
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" start_F_1 = 0.5\n",
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" minimize_result = optimize.minimize(fun=function_to_minimize, x0=(start_f_1, start_F_1), method = 'Nelder-Mead')\n",
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" if not minimize_result.success:\n",
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" print(type(minimize_result))\n",
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" print(minimize_result)\n",
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@ -370,7 +371,111 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Even if the values of $f_1$ and $F_1$ are not correct, the exact value of for the dark $D$ has been found, which proves that this method is able to estimate the dark. We can also observe that $F$ and $L$ signals are recovered up to a scale factor."
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"Even if the values of $f_1$ and $F_1$ are not correct, the exact value of for the dark $D$ has been found, which proves that this method is able to estimate the dark. We can also observe that $F$ and $L$ signals are recovered up to a scale factor:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"lens_ratio = estimated_lens / lens\n",
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"fluo_ratio = estimated_fluo / fluo\n",
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"print('lens_ratio:',lens_ratio)\n",
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"print('fluo_ratio:',fluo_ratio)\n",
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"print('lens_ratio*fluo_ratio:', np.mean(lens_ratio)*np.mean(fluo_ratio))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## adding the constraint `F_1=fluo[0]`\n",
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"\n",
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"When we think about it, it makes sense $F$ and $L$ signals are recovered up to a scale factor only; indeed scaling $L$ up a factor $r$ has the same effect on $I$ as scaling $F$ up a factor $r$. In other words, the couple `(lens, fluo)` produce the same image as the couple `(estimated_lens, estimated_fluo)`. As a result, the sclaing factor $r$ is arbitrary. So, we need to reduce the degree of freedom. As a convention we then decide to fix the value of $F_1=1$. The corresponding code becomes:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"def find_dark2(I, F_1=1.0):\n",
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" (num_times, num_sites) = I.shape\n",
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" A = I[1:num_times] - I[0:num_times-1]\n",
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" R = A[1:num_times-1] / A[0:num_times-2]\n",
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" P = np.multiply.accumulate(R)\n",
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" \n",
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" def function_to_minimize(x):\n",
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" f_1 = x[0]\n",
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" dark_matrix, lens, fluo = estimate_dark_lens_fluo(I, A, P, R, f_1, F_1)\n",
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" error = np.var(dark_matrix)\n",
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" return error\n",
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" start_f_1 = -0.7\n",
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" minimize_result = optimize.minimize(fun=function_to_minimize, x0=(start_f_1), method = 'Nelder-Mead')\n",
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" if not minimize_result.success:\n",
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" print(type(minimize_result))\n",
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" print(minimize_result)\n",
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" raise Exception(minimize_result.message)\n",
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" f_1 = minimize_result.x[0]\n",
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" print('estimated value of f_1 : %f' % f_1)\n",
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" dark, lens, fluo = estimate_dark_lens_fluo(I, A, P, R, f_1, F_1)\n",
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" return dark, lens, fluo\n",
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"\n",
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"print('exact value of f_1 : %f' % (fluo[1]-fluo[0]))\n",
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"\n",
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"estimated_dark, estimated_lens, estimated_fluo = find_dark2(i, fluo[0])\n",
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"print('estimated dark value : ', estimated_dark)\n",
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"print('exact lens : ', lens)\n",
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"print('estimated lens : ', estimated_lens)\n",
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"pyplot.plot(lens)\n",
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"pyplot.plot(estimated_lens)\n",
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"pyplot.xlabel('$s$ (pixel position)')\n",
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"pyplot.ylabel('$L(s)$')"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"print('exact fluo :', fluo)\n",
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"print('estimated fluo', estimated_fluo)\n",
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"pyplot.plot(fluo)\n",
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"pyplot.plot(estimated_fluo)\n",
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"pyplot.xlabel('$t$ (time)')\n",
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"pyplot.ylabel('$F(s)$')\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Now that `F_1` is fixed to the exact value `fluo[0]`, both $L$ and $F$ estimations are the same as the ones that were used to construct $I$.\n",
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"\n",
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"Let's see how `find_dark2` is tolerant to noise:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"estimated_dark, estimated_lens, estimated_fluo = find_dark2(i + np.random.rand(i.shape[0], i.shape[1]) * 0.1, F_1=fluo[0])\n",
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"print('estimated dark:', np.mean(estimated_dark))\n",
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"print('exact dark:', real_dark)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## conclusion\n",
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"`find_dark2` function is able to estimate $D$, $F$ and $L$ from $I$, which is the goal of this study."
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]
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},
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{
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