From 035e0079db3080859260205e3fbbb1f29ccdce7f Mon Sep 17 00:00:00 2001 From: Fadi Nammour Date: Tue, 22 Sep 2020 17:18:27 +0200 Subject: [PATCH 01/12] first commit Added paper summary in the README.md Added a weak lensing scheme in keynote --- .DS_Store | Bin 0 -> 6148 bytes README.md | 66 +++++++++++++++++++++++++++++++++++++++- weak_lensing_scheme.key | Bin 0 -> 475436 bytes 3 files changed, 65 insertions(+), 1 deletion(-) create mode 100644 .DS_Store create mode 100755 weak_lensing_scheme.key diff --git a/.DS_Store b/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..f6ba9ce1f262b351b75a1cba727c18f99545d192 GIT binary patch literal 6148 zcmeHKJx>EM41I=56tQ$cFlIyym6%yVxf?101N{Moeh3g(I#dX;-QVF?!gCy|9^Ax$ z5V9rb`4Zbno^(kJKnBmV6QBp6OH~v#T12~BhqkiwHc?WJD~wPMtL1c3wIuqBUAp!i zTwsWEl-OARa>5a%<&=8#o`e>#i)(tGp}-}vj|FD7<8{Q_mxu$rV2)cnVvLH`6IOU& zd&ln;rnqO{g4(CY9Iwr6vd?VC8}sWLT$}-Cz!`7`oPnP+z&l%|yJzUFGvEw318WBK zd))>)oM^^Hy#1}&~xN44cd56n|?84ZKzqaKZO(hA&?01#u@kp20j4yzc{V{ literal 0 HcmV?d00001 diff --git a/README.md b/README.md index 8d8cbdc..c86fdf4 100644 --- a/README.md +++ b/README.md @@ -1,3 +1,67 @@ -# Shear & PSF Reading Group +# Biases in, and corrections to, KSB shear measurements. Viola et al, 2010 Notes and tools derived from reading groups sessions +[arxiv link](https://arxiv.org/abs/1006.2470) + +## Goals of the Paper + +1. Show the theoretical and practical caveats of KSB. +2. Suggest an amelioration of KSB. + +## Main Results and Conclusion + +1. The assumptions in KSB introduce biaises especially for large amplitude +values. +2. Rewriting correctly the Taylor expansion of the ellipticity up to the third +order (see KSB3) gives results with biaises smaller by orders of magnitude. +3. The PSF corrections in KSB are not accurate however they partially cancel +other biaises. + +## Definitions and Equations + +*. Reduced shear, g: (eq. 7) +*. Complex ellipticity, $\chi$: (eq. 10) +*. Shear responsivity $\epsilon$: (eq. 14) +*. KSB (eq. 32) +*. KSB1 (eq. 34) +*. KSBtr (eq.36) +*. KSB3 (eq. 44) +*. Inverse Problem (eq. 50) +*. Circular PSF Formula (eq. 58) +*. Anisotropic PSf Formula (eq. 67) + +> The reduced shear, g, is preferred over the shear responsivity, +$\epsilon$, for its robustness to noise in the data. + +## Assumptions in KSB +1. The reduced shear, g, is constant and the ellipticity average is null, +$\left< \chi \right>=0$ (see eq. 12). +2. Many Taylor expansions like in the reduced shear, g (see eq. 29). +> Taylor expansions gives a polynomial approximation around a certain reference +value. The further from the reference we evaluate the expansion the bigger the +biais. In this paper, since the reference value is zero, the biais will be +greater for large amplitude values. + +3. Assume equality between (eq. 27) and (eq. 28). +4. The PSF has at most slight anisotropies, it is mostly circular. + +## Notations Clarification + +1. The angular coordinates are solid angles so they have two components. These +coordinates can be assimilated to linear coordinates. +2. The star in eq. 12 stands for the conjugate. +3. Rewriting eq.48-50: + i. (eq. 48) $$\chi = f\left( \tilde{g}, \chi^{s}\right)$$ + ii. (eq. 49) $$\chi = h \left( \chi^{obs}\right)$$ + iii. (eq. 50) $$\chi^{obs} = h^{-1}\left[ f \left( \tilde{g}, h\left(\chi^{obs}\right)\right)\right]$$ + + where $\chi^{s}$ is the intrinsic ellipticity of the galaxy, $\chi^{obs}$ is + the ellipticity of the lensed galaxy and $\chi$ is the ellipticity of the + convoluted and lensed galaxy. +4. In (eq. 51): $$\chi^{sh}_{\alpha} = \chi_{\alpha} + \chi^{s}_{\alpha}$$ +5. In (eq. 52): the asterisk, *, in $P^{sm,*}$ is to indicate the PSF. +Explanation: * --> star --> dirac --> PSF --> illuminati + +## Implementation + +[Link](https://github.com/pmelchior/shapelens/blob/master/src/KSB.cc) to Peter Melchior implementation of KSB codes in C. diff --git a/weak_lensing_scheme.key b/weak_lensing_scheme.key new file mode 100755 index 0000000000000000000000000000000000000000..6dc0e61432abef6e6db825467a417c4f37eef4b8 GIT binary patch literal 475436 zcmb@tcUV*1(b@9%uycb@y)KhATXJG*4|T5HeVGka#OS+gcj^eCxV0RO;oaD$<9 zinMo41OV{QagSW~(B9u((!*a|Q(jg{PEkQlT=SuvqWC>ISs8If&3p3V@>-8%A1TO4 zX==(Th|5UJOSpMEU;MdP2Hbe8rK<&?pa1}L$Q$4S1-Snx*!3jh~sVTdupQBYD|qN1dx{zo|}BFOasDn@FiThbbrnV;Iz-1cFSdGjIv3cqG`2kW!nSOHlF z-$>f4Z0sDIHwA@+@7xu+Cnv9Qb?b@%l4^$!e=PfSit zBmd0IqR^O?)wT7%8=JWOgTo{IG2!I&AHFC6l>cFieEtt-{|~+x$$VX+qN1dt`G+rx zOF{oI&PYXlOZqaC##0)5ALiRKZ?3Rte#o!xpyii+hGli|{e6{9;2ug4_YZ6Ta`u16 zSmggF&i7P`BPDnfr`um8K^H){|lG@H_-eGSN;Lo|Aq@P 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Show the theoretical and practical caveats of KSB. -2. Suggest an amelioration of KSB. +2. Suggest improvements to KSB. ## Main Results and Conclusion @@ -48,7 +48,8 @@ greater for large amplitude values. ## Notations Clarification 1. The angular coordinates are solid angles so they have two components. These -coordinates can be assimilated to linear coordinates. +coordinates can be approximated by linear coordinates because the angles are +very small. 2. The star in eq. 12 stands for the conjugate. 3. Rewriting eq.48-50: i. (eq. 48) $$\chi = f\left( \tilde{g}, \chi^{s}\right)$$ From 9f605edd6931a4fc85ea6ea5b3fd63abea938673 Mon Sep 17 00:00:00 2001 From: Fadi Nammour Date: Sun, 7 Feb 2021 21:30:10 +0100 Subject: [PATCH 03/12] Added KSB theory --- shearbook/methods/ksb.ipynb | 34 +++++++++++++++++++++++++++++++--- 1 file changed, 31 insertions(+), 3 deletions(-) diff --git a/shearbook/methods/ksb.ipynb b/shearbook/methods/ksb.ipynb index d3e8a08..5d6436c 100644 --- a/shearbook/methods/ksb.ipynb +++ b/shearbook/methods/ksb.ipynb @@ -11,9 +11,37 @@ "cell_type": "markdown", "metadata": {}, "source": [ + "In this section, we introduce the Kaiser-Squire-Broadhurst method (KSB) to measure galaxies shape using the image of the galaxy surface brightness.\n", + "\n", + "## Theory\n", + "\n", + "KSB estimates the reduced shear $g$ using the polarisation $\\chi$. The exact relation between these quantities is :\n", + "\n", + "$$\n", + "\\chi^s = \\frac{\\chi - 2 g +g^2 \\chi^\\ast}{1 + | g |^2 - \\Re \\left( g \\chi^\\ast \\right) } \\quad,\n", + "$$(g_chi)\n", + "where $\\chi^s$ is the intrinsic (unlensed) polarisation.\n", + "\n", + "Assuming that the average of $\\chi^s$ vanishe, i.e. $\\left< \\chi^s \\right>=0$, and in a certain region $g$ is constant, $g$ is approximated by rotating the coordinate frame such that only one of its component does not vanish:\n", + "\n", + "$$\n", + "g \\approx \\frac{ \\left< \\chi \\right> }{ 2 \\left(1 - \\sigma^2_\\chi\\right) } \n", + " + \\frac{\\left< \\chi \\right>^3}{8} \\frac{1 - 5\\sigma^2_\\chi}{ \\left(1 - \\sigma^2_\\chi\\right)^4} \n", + " + \\mathcal{O} \\left( \\left< \\chi \\right>^5 \\right) \\quad,\n", + "$$\n", + "\n", + "with $\\sigma^2_\\chi$ being the standard deviation of $\\chi^s$ distribution.\n", + "\n", + "\n", + "```{note}\n", + "The polarisation $\\chi$ is used for shear measurement instead of the ellipticity $\\epsilon$ in Weak Lensing because in practice its estimator is less noisy. \n", + "```\n", + "\n", "```{warning}\n", - "In progress!\n", - "```" + "The approximation of $g$ is done by applying a Taylor Expansion of $\\chi$ around 0, i.e. for small value of $\\chi$, on the average of {eq}`g_chi`.\n", + "```\n", + "\n", + "## Standard KSB" ] }, { @@ -40,7 +68,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.5" + "version": "3.8.6" } }, "nbformat": 4, From d0180c26e30487617ba24167d9bc557c705dc527 Mon Sep 17 00:00:00 2001 From: Fadi Nammour Date: Fri, 12 Feb 2021 15:04:04 +0100 Subject: [PATCH 04/12] ksb updates --- shearbook/methods/ksb.ipynb | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/shearbook/methods/ksb.ipynb b/shearbook/methods/ksb.ipynb index 5d6436c..1059296 100644 --- a/shearbook/methods/ksb.ipynb +++ b/shearbook/methods/ksb.ipynb @@ -41,7 +41,9 @@ "The approximation of $g$ is done by applying a Taylor Expansion of $\\chi$ around 0, i.e. for small value of $\\chi$, on the average of {eq}`g_chi`.\n", "```\n", "\n", - "## Standard KSB" + "## Standard KSB\n", + "\n", + "The core idea of KSB is to estimate the polarisation on raw data, i.e. images blurred by a known PSF and contaminated with additive Gaussian noise, using a weighting window on each image then apply corrections directly in the polarisation space. " ] }, { From 5f77793953ecf06a4d9da40009c8c8ed85b51f5b Mon Sep 17 00:00:00 2001 From: Fadi Nammour Date: Mon, 5 Apr 2021 23:54:23 +0200 Subject: [PATCH 05/12] weighted moments first draft --- shearbook/moments/moments-weighted.ipynb | 251 ++++++++++++++++++++++- 1 file changed, 243 insertions(+), 8 deletions(-) diff --git a/shearbook/moments/moments-weighted.ipynb b/shearbook/moments/moments-weighted.ipynb index d6a338e..25eac15 100644 --- a/shearbook/moments/moments-weighted.ipynb +++ b/shearbook/moments/moments-weighted.ipynb @@ -5,24 +5,259 @@ "metadata": {}, "source": [ "(content:weighted)=\n", - "# Weighted Moments" + "# Weighted Moments\n", + "\n", + "Observed galaxy images are corrupted by noise. Straightforward measures of ellipticity on such images lack robustness, see figure below for example." + ] + }, + { + "cell_type": "code", + "execution_count": 43, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Ellipticity of the clean star image : -0.0040+i0.1980\n", + "Ellipticity of the noisy star image : -0.0050+i0.2190\n" + ] + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The noisy star and the clean star images are binary images.\n", + "They differ from each other by one pixel on the bottom right of the noisy star image.\n", + "This pixel induces a difference in ellipticity estimation of : 10.6% between the two images\n" + ] + } + ], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "\n", + "def get_centroid(data):\n", + " \n", + " # Sum flux over x and y individually\n", + " sum_i = np.array([np.sum(data, axis=i) for i in (1, 0)])\n", + " \n", + " # Get range of x and y values\n", + " ranges = np.array([np.arange(i) for i in data.shape])\n", + " \n", + " # Calculate centroids\n", + " cents = np.sum(sum_i * ranges, axis=1) / np.sum(data)\n", + " \n", + " return cents.astype(int)\n", + "\n", + "def get_moments(data):\n", + " \n", + " centroid = get_centroid(data)\n", + " ranges = np.array([np.arange(i) for i in data.shape])\n", + " \n", + " x = np.outer(ranges[0] - centroid[0], np.ones(data.shape[1]))\n", + " y = np.outer(np.ones(data.shape[0]), ranges[1] - centroid[1])\n", + " \n", + " q = np.array([np.sum(data * xi * xj) for xi in (x, y) for xj in (x, y)])\n", + " q = (q / np.sum(data)).reshape(2, 2).astype('complex')\n", + " \n", + " return q\n", + "\n", + "def get_ellipticity(data, method='chi'):\n", + " \n", + " # Calculate moments\n", + " q = get_moments(data)\n", + " \n", + " # Calculate the image size.\n", + " r2 = q[0, 0] + q[1, 1]\n", + "\n", + " # Calculate the numerator\n", + " num = (q[0, 0] - q[1, 1] + 2 * np.complex(0, q[0, 1]))\n", + " \n", + " # Calculate the denominator\n", + " den = r2\n", + " \n", + " if method == 'epsilon':\n", + " den += 2 * np.sqrt(q[0, 0] * q[1, 1] - q[0, 1] ** 2)\n", + " \n", + " # Calculate the ellipticity/polarisation\n", + " ellip = num / den\n", + "\n", + " return np.around([ellip.real, ellip.imag], 3)\n", + "\n", + "#load toy image and generate noisy image\n", + "star_clean = np.load('star.npy')\n", + "n_row,n_col = star_clean.shape\n", + "star_noisy = np.copy(star_clean)\n", + "noise_coord = (n_row-3,n_col-3)\n", + "star_noisy[noise_coord] = 1\n", + "\n", + "#estimate ellipticities\n", + "e_clean = get_ellipticity(star_clean)\n", + "e_noisy = get_ellipticity(star_noisy)\n", + "print('Ellipticity of the clean star image : {:.4f}+i{:.4f}'.format(*e_clean))\n", + "print('Ellipticity of the noisy star image : {:.4f}+i{:.4f}'.format(*e_noisy))\n", + "\n", + "#compute deviation\n", + "e_clean_norm = np.linalg.norm(e_clean)\n", + "e_noisy_norm = np.linalg.norm(e_noisy)\n", + "deviation = np.abs((e_noisy_norm-e_clean_norm)/e_clean_norm)*100\n", + "\n", + "#show image\n", + "cmap = 'bone'\n", + "plt.figure(3,figsize=(10,7))\n", + "plt.subplot(121)\n", + "plt.imshow(star_clean,cmap=cmap)\n", + "plt.axis('off')\n", + "plt.title('Clean star image')\n", + "plt.subplot(122)\n", + "plt.imshow(star_noisy,cmap=cmap)\n", + "plt.axis('off')\n", + "plt.title('Noisy star image')\n", + "circle = plt.Circle(noise_coord, 2, color='r', lw = 3, fill=False)\n", + "plt.gcf().gca().add_artist(circle)\n", + "plt.tight_layout()\n", + "plt.show()\n", + "print('The noisy star and the clean star images are binary images.')\n", + "print('They differ from each other by one pixel on the bottom right of the noisy star image.')\n", + "print('This pixel induces a difference in ellipticity estimation of : {:.1f}% between the two images'.format(deviation))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "```{warning}\n", - "In progress!\n", - "```" + "One way to have more robust measurements is to reduce the weight of the pixels containing only noise in the image before computing the moments. For this purpose we use a weighting window $W$ to define the weighted quadrupole moments of an image as follows:\n", + "\n", + "$$\n", + " Q_{w,xy} = \\frac{\\sum_{x,y} W(x,y) I(x,y) (x - \\bar{x})(y - \\bar{y})}{\\sum_{x,y} W(x,y) I(x,y)}\n", + "$$ (weighted_moments)\n", + "\n", + "## Weight dependent measures\n", + "\n", + "In equation {eq}`weighted_moments` the output varies with $W$ which means that the resulting shape estimation also varies depending on $W$ see example below. Usually the window is chosen to be the best fit of a 2D Gaussian on the galaxy image." + ] + }, + { + "cell_type": "code", + "execution_count": 44, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "def draw_gaussian(e1, e2, sigma, x_c, y_c, shape):\n", + " \n", + " # compute centered grid\n", + " ranges = np.array([np.arange(i) for i in shape])\n", + " x = np.outer(ranges[0] - x_c, np.ones(shape[1]))\n", + " y = np.outer(np.ones(shape[0]),ranges[1] - y_c)\n", + " \n", + " # shift it to match centroid\n", + " x1 = (1-e1/2)*x - e2/2*y\n", + " y1 = (1+e1/2)*y - e2/2*x\n", + " \n", + " # compute elliptical gaussian\n", + " return np.exp(-(x1 ** 2 + y1 ** 2) / (2 * sigma))\n", + "\n", + "w50 = draw_gaussian(0,0,50,48,58, (96,96))\n", + "w80 = draw_gaussian(0,0,80,48,58, (96,96))\n", + "\n", + "plt.figure(4,figsize=(10,7))\n", + "plt.subplot(121)\n", + "plt.imshow(w50,cmap=cmap)\n", + "plt.axis('off')\n", + "plt.title('Small Gaussian Window')\n", + "plt.subplot(122)\n", + "plt.imshow(w40*star_noisy,cmap=cmap)\n", + "plt.axis('off')\n", + "plt.title('Windowed Star Image')\n", + "plt.tight_layout()\n", + "plt.show()\n", + "\n", + "plt.figure(5,figsize=(10,7))\n", + "plt.subplot(121)\n", + "plt.imshow(w80,cmap=cmap)\n", + "plt.axis('off')\n", + "plt.title('Big Gaussian Window')\n", + "plt.subplot(122)\n", + "plt.imshow(w80*star_noisy,cmap=cmap)\n", + "plt.axis('off')\n", + "plt.title('Windowed Star Image')\n", + "plt.tight_layout()\n", + "plt.show()" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 48, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Ellipticity of the star image with a small window: -0.0080+i0.0810\n", + "Ellipticity of the star image with a large window: -0.0070+i0.1150\n", + "The choice of the window induces a difference in ellipticity estimation of : 29.4% between the two images\n" + ] + } + ], + "source": [ + "#estimate ellipticities\n", + "e_w0 = get_ellipticity(w50*star_noisy)\n", + "e_w80 = get_ellipticity(w80*star_noisy)\n", + "print('Ellipticity of the star image with a small window: {:.4f}+i{:.4f}'.format(*e_w50))\n", + "print('Ellipticity of the star image with a large window: {:.4f}+i{:.4f}'.format(*e_w80))\n", + "\n", + "#compute deviation\n", + "e_w50_norm = np.linalg.norm(e_w50)\n", + "e_w80_norm = np.linalg.norm(e_w80)\n", + "deviation_w = np.abs((e_w50_norm-e_w80_norm)/e_w80_norm)*100\n", + "print('The choice of the window induces a difference in ellipticity estimation of : {:.1f}% between the two images'.format(deviation_w))" + ] + }, + { + "cell_type": "markdown", "metadata": {}, - "outputs": [], - "source": [] + "source": [ + "The bias induced by the weighting window can be corrected, for more details see [MOMENT BASED METHOD] section." + ] } ], "metadata": { @@ -41,7 +276,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.5" + "version": "3.8.6" } }, "nbformat": 4, From e8c0b009ad05adbf37ac0fa3c60765beadd631c7 Mon Sep 17 00:00:00 2001 From: Fadi Nammour Date: Tue, 6 Apr 2021 18:14:07 +0200 Subject: [PATCH 06/12] content added to weighted moments and KSB --- shearbook/methods/ksb.ipynb | 70 ++++++++++++++++++--- shearbook/methods/weak_lensing_scheme.jpeg | Bin 0 -> 162460 bytes shearbook/moments/moments-basic.ipynb | 2 +- shearbook/moments/star.npy | Bin 0 -> 9344 bytes 4 files changed, 61 insertions(+), 11 deletions(-) create mode 100644 shearbook/methods/weak_lensing_scheme.jpeg create mode 100644 shearbook/moments/star.npy diff --git a/shearbook/methods/ksb.ipynb b/shearbook/methods/ksb.ipynb index 1059296..abac31d 100644 --- a/shearbook/methods/ksb.ipynb +++ b/shearbook/methods/ksb.ipynb @@ -4,14 +4,63 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "# KSB" + "# KSB\n", + "\n", + "In this section, we introduce the Kaiser-Squire-Broadhurst method (KSB) to measure galaxies shape using the image of the galaxy surface brightness." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "In this section, we introduce the Kaiser-Squire-Broadhurst method (KSB) to measure galaxies shape using the image of the galaxy surface brightness.\n", + "## Reminder : Reduced Shear\n", + "\n", + "A quick note on how the reduced shear is defined in Weak Lensing.\n", + "\n", + "![Weak Lensing Scheme](weak_lensing_scheme.jpeg)\n", + "\n", + "In the scheme above, let $\\Sigma$ be the surface mass density of the Dark Matter (L) then its lensing potential is given by:\n", + "$$\n", + "\\Psi(\\theta) = \\frac{4G}{c^2}\n", + " \\frac{D_{l}D_{s}}{D_{ls}}\n", + " \\int \\Sigma{\\left(\\theta'\\right)} \\ln \\left\\|\n", + " \\theta - \\theta' \n", + " \\right\\| \n", + " d^2\\theta' \\quad,\n", + "$$ (lensing)\n", + "where $G$ and $c$ are the gravitational constant and the speed of light in vacuum and $D_{l}$, $D_{s}$ and $D_{l,s}$ are the angular diameter distances between the observer and the lens (OL), the observer and the source (OS) and the lens and the source (LS), respectively.\n", + "\n", + "The deflection of the light rays is approximated well enough by the angle\n", + "$$\n", + "\\alpha(\\theta) = \\nabla \\Psi(\\theta) \\quad.\n", + "$$\n", + "This angle is related to the angular positions of the source $\\beta$ to the image $\\theta$ on the sky by the lens equation:\n", + "$$\n", + "\\beta = \\theta - \\alpha(\\theta) \\quad.\n", + "$$ (lens)\n", + "Under weak lensing assumptions (lens mapping almost constant along the solid angle of the soucre), the lens mapping is locally linear and the Jacobian matrix of the image distorsion is:\n", + "$$\n", + "A \\equiv \\frac{\\partial \\beta}{\\partial \\theta} \n", + " = \\left( \\delta_{ij} - \\frac{\\partial^2 \\Psi(\\theta)}{\\partial \\theta_{i} \\partial \\theta_{j}} \\right)\n", + " = \\begin{pmatrix}\n", + " 1-\\kappa -\\gamma_1 & -\\gamma_2 \\\\\n", + " -\\gamma_2 & 1-\\kappa+\\gamma_! \n", + " \\end{pmatrix}\n", + "$$ (jacobian_lensing)\n", + "with the convergence\n", + "$$\n", + "\\kappa (\\theta) = \\frac{1}{2}\\left(\\Psi_{11} + \\Psi_{22}\\right)\n", + "$$\n", + "and the two components\n", + "$$\n", + "\\gamma_1 = \\frac{1}{2} \\left(\\Psi_{11} - \\Psi_{22}\\right), \\quad \\gamma_2 = \\Psi_{12}\n", + "$$\n", + "of the complex shear $\\gamma = \\gamma_1 + \\boldsymbol{i} \\gamma_2$.\n", + "\n", + "In the KSB method, the reduced shear is the measure of image distorsions and is defined as:\n", + "$$\n", + "g = \\frac{\\gamma}{1-\\kappa}\n", + "$$\n", "\n", "## Theory\n", "\n", @@ -43,15 +92,16 @@ "\n", "## Standard KSB\n", "\n", - "The core idea of KSB is to estimate the polarisation on raw data, i.e. images blurred by a known PSF and contaminated with additive Gaussian noise, using a weighting window on each image then apply corrections directly in the polarisation space. " + "The core idea of KSB is to estimate the polarisation on raw data, i.e. images blurred by a known PSF and contaminated with additive Gaussian noise, using a weighting window on each image then apply corrections directly in the polarisation space.\n", + "\n", + "In the following, the PSF effect and the weighting window effect are handled separately.\n", + "\n", + "### Weighting window\n", + "\n", + "$$\n", + "\\chi_\\alpha - \\chi^s_\\alpha = g^\\beta P^{sh}_{\\alpha \\beta}\n", + "$$" ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [] } ], "metadata": { diff --git a/shearbook/methods/weak_lensing_scheme.jpeg b/shearbook/methods/weak_lensing_scheme.jpeg new file mode 100644 index 0000000000000000000000000000000000000000..b39539953f8485e7929e2b09f84045b3ef267034 GIT binary patch literal 162460 zcmd?Qc{E#Z*Eb%WC~eg|m9#}oMa{Fc6ct6yQ%PH6sCf#4mZIjGTU4rsAch)3%`-(& zb5T>wb3zdk@k_t=b3f0$e(U$g{rui%t@pd$_hcn!pX;2nud{cqYY(4mpR=E5^MLDM 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?6~Ljw^M0M;IYE}!;Ab#4Dfb{fZcLbdv#G-)uN?Xht>uV?195!Ntp zU`}>@(c|KYEn}Y7YG&q@#Wcu^XW@g;BN8zoaVS#9Ce9<1<)OC z_0q)?qlf!!+X172nV*N$C~2f@-8mh331$)XJXSAwuKdZpK+f{Yp5ueIH%0xn-S-5? zJb0mMlQzFFoc|yG&8P2~lck`Qqls~O4vJ^-G6)MFNRvQrFfCUbc{;K#G?eXZ=L$ou zm=w>9pj=&PMG8B1vHUE3!g2kbvn9oryhsjMEopK<>f= z&o9R!g|_2_ZE*^I#|e#x0XZ&ASx4_z|8=F%zaBj5zjd~M8D{Jx! z41D>J9am3a!2k@v01UtY48Q;kzyJ*J8t{1W!s89_(#3d#49^&lA>oqAfSq@)XoVNo>4jm7(Aol6kyQ4G?)XN@FM*u zwk$_}TY@vbIe-A)6hMNn4z%>D>B-~LH32?H4$q literal 0 HcmV?d00001 From 7b10eb6e0678a0c9840b28073f3579f8889e79ab Mon Sep 17 00:00:00 2001 From: Fadi Nammour Date: Tue, 6 Apr 2021 23:10:01 +0200 Subject: [PATCH 07/12] moment formula modification + added content to ksb --- shearbook/methods/ksb.ipynb | 40 ++++++++++++++++++++++++ shearbook/moments/moments-basic.ipynb | 4 +-- shearbook/moments/moments-weighted.ipynb | 16 ++++++++-- 3 files changed, 55 insertions(+), 5 deletions(-) diff --git a/shearbook/methods/ksb.ipynb b/shearbook/methods/ksb.ipynb index abac31d..5203cf2 100644 --- a/shearbook/methods/ksb.ipynb +++ b/shearbook/methods/ksb.ipynb @@ -97,11 +97,51 @@ "In the following, the PSF effect and the weighting window effect are handled separately.\n", "\n", "### Weighting window\n", + "Here the polarisation $\\chi$ is estimated using the weighted moments $Q_w$ and $\\chi^s$ is the unaccessible polarisation of the source observed by a perfect telescope (no PSF effect) and without any noise.\n", "\n", + "By introducing the shear polarisability matrix, $P^{sh}$, a first order approximation in terms of $g$ gives :\n", "$$\n", "\\chi_\\alpha - \\chi^s_\\alpha = g^\\beta P^{sh}_{\\alpha \\beta}\n", + "$$ (chichi)\n", + "\n", + "with $\\alpha, \\beta \\in \\{1,2\\}$ the indices of the components and\n", + "$$\n", + "P^{sh}_{\\alpha \\beta} = - 2 \\frac{\\chi_{\\alpha}L_{\\beta}}{Q_{w,xx}+Q_{w,yy}} \n", + " - 2 \\chi_{\\alpha}\\chi_{\\beta}\n", + " + 2 \\frac{B_{\\alpha \\beta}}{Q_{w,xx}+Q_{w,yy}}\n", + " + 2 \\delta_{\\alpha \\beta}\n", + "$$\n", + "such that $\\delta_{\\alpha \\beta}$ is the Kronecker delta and\n", + "$$\n", + "L_\\beta = \\frac{1}{\\sigma^2} \\sum_{x,y} I(x,y) W'\\left(\\frac{(x,y)}{\\sigma}\\right) \\eta_\\beta\n", + "$$\n", + "\n", + "$$\n", + "B_{\\alpha \\beta} = \\frac{1}{\\sigma^2} \\sum_{x,y} I(x,y) W'\\left(\\frac{(x,y)}{\\sigma}\\right) \\eta_\\beta \\eta_\\beta\n", + "$$\n", + "\n", + "where $W$ is the weighting window function and $\\sigma$ its scaling factor along with\n", + "\n", + "$$\n", + "\\eta_1 = x^2 - y^2\n", + "$$\n", + "and\n", + "$$\n", + "\\eta_2 = 2xy.\n", + "$$\n", + "\n", + "By averaing equation {eq}`chichi`, $\\left<\\chi^s_\\alpha\\right> = 0$ (no intrinsic alignment assumption), $\\left = g_\\alpha$ (weak lensing assumption) and\n", + "$$\n", + "\\left< \\chi^s_\\alpha \\right> = \n", "$$" ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] } ], "metadata": { diff --git a/shearbook/moments/moments-basic.ipynb b/shearbook/moments/moments-basic.ipynb index 078f325..c25594a 100644 --- a/shearbook/moments/moments-basic.ipynb +++ b/shearbook/moments/moments-basic.ipynb @@ -221,9 +221,9 @@ "Finally, we can define the second moment or quadrupole moments of an image as follows.\n", "\n", "$$\n", - " Q_{xy} = \\frac{\\sum_{x,y} I(x,y)(x - \\bar{x})(y - \\bar{y})}{\\sum_{x,y} I(x,y)}\n", + " Q_{k_1 k_2} = \\frac{\\sum_{x,y} I(x,y)(k_1 - \\bar{k_1})(k_2 - \\bar{k_2})}{\\sum_{x,y} I(x,y)}\n", "$$ (moments)\n", - "\n", + "where $k_1,k_2 \\in \\{x,y\\}$, the quadrupoles are $Q_{xx},Q_{xy}$and $Q_{yy}$ (with $Q_{xy}=Q_{yx}$).\n", "Equation {eq}`moments` is used to determine object shapes. In the following cell we a simple implementation of this equation." ] }, diff --git a/shearbook/moments/moments-weighted.ipynb b/shearbook/moments/moments-weighted.ipynb index 25eac15..4f6d1d2 100644 --- a/shearbook/moments/moments-weighted.ipynb +++ b/shearbook/moments/moments-weighted.ipynb @@ -142,12 +142,22 @@ "One way to have more robust measurements is to reduce the weight of the pixels containing only noise in the image before computing the moments. For this purpose we use a weighting window $W$ to define the weighted quadrupole moments of an image as follows:\n", "\n", "$$\n", - " Q_{w,xy} = \\frac{\\sum_{x,y} W(x,y) I(x,y) (x - \\bar{x})(y - \\bar{y})}{\\sum_{x,y} W(x,y) I(x,y)}\n", + " Q_{w,k_1 k_2} = \\frac{\\sum_{x,y} W(x,y) I(x,y) (k_1 - \\bar{k_1})(k_2 - \\bar{k_2})}{\\sum_{x,y} W(x,y) I(x,y)}\n", "$$ (weighted_moments)\n", - "\n", + "with $k_1,k_2 \\in \\{x,y\\}$.\n", "## Weight dependent measures\n", "\n", - "In equation {eq}`weighted_moments` the output varies with $W$ which means that the resulting shape estimation also varies depending on $W$ see example below. Usually the window is chosen to be the best fit of a 2D Gaussian on the galaxy image." + "In equation {eq}`weighted_moments` the output varies with $W$ which means that the resulting shape estimation also varies depending on $W$ see example below. Usually the window is chosen to be the best fit of a 2D Gaussian on the galaxy image. The expression of 2D isotropic Gaussian window centered in 0 is :\n", + "$$\n", + "W(p) = \\frac{1}{2\\pi} \\exp\\left(-\\frac{\\|p\\|^2}{2}\\right)\n", + "$$\n", + "with $p=(x,y)$. Once the centroid, $\\bar{p}=\\left(\\bar{x},\\bar{y}\\right)$ and the scaling factor $\\sigma$ are defined for a specific image, the window is translated and rescaled as :\n", + "$$\n", + "W\\left(\\frac{p-\\bar{p}}{\\sigma}\\right) \\quad.\n", + "$$\n", + "```{note}\n", + "In the case of isotropic windows, only the norm of the point considered is required and the value of the window can be noted as $W(\\|p\\|^2)$. \n", + "```" ] }, { From 8f39a68376d4fd8b7caf8f83d5de42dac6299245 Mon Sep 17 00:00:00 2001 From: Fadi Nammour Date: Thu, 8 Apr 2021 00:41:54 +0200 Subject: [PATCH 08/12] added content to ksb --- shearbook/methods/ksb.ipynb | 85 +++++++++++++++++++++++++++++++------ 1 file changed, 72 insertions(+), 13 deletions(-) diff --git a/shearbook/methods/ksb.ipynb b/shearbook/methods/ksb.ipynb index 5203cf2..eb1a7a3 100644 --- a/shearbook/methods/ksb.ipynb +++ b/shearbook/methods/ksb.ipynb @@ -92,26 +92,31 @@ "\n", "## Standard KSB\n", "\n", - "The core idea of KSB is to estimate the polarisation on raw data, i.e. images blurred by a known PSF and contaminated with additive Gaussian noise, using a weighting window on each image then apply corrections directly in the polarisation space.\n", + "The core idea of KSB is to deduce the reduced shear by estimating the polarisation on raw data, i.e. images blurred by a known PSF and contaminated with additive Gaussian noise, and using a weighting window on each image then apply corrections directly in the polarisation space.\n", "\n", - "In the following, the PSF effect and the weighting window effect are handled separately.\n", - "\n", - "### Weighting window\n", + "In the following, the PSF effect and the weighting window effect are handled separately." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Handling the Weighting Window\n", "Here the polarisation $\\chi$ is estimated using the weighted moments $Q_w$ and $\\chi^s$ is the unaccessible polarisation of the source observed by a perfect telescope (no PSF effect) and without any noise.\n", "\n", "By introducing the shear polarisability matrix, $P^{sh}$, a first order approximation in terms of $g$ gives :\n", "$$\n", - "\\chi_\\alpha - \\chi^s_\\alpha = g^\\beta P^{sh}_{\\alpha \\beta}\n", + "\\chi - \\chi^s = P^{sh} g\n", "$$ (chichi)\n", "\n", - "with $\\alpha, \\beta \\in \\{1,2\\}$ the indices of the components and\n", + "with\n", "$$\n", "P^{sh}_{\\alpha \\beta} = - 2 \\frac{\\chi_{\\alpha}L_{\\beta}}{Q_{w,xx}+Q_{w,yy}} \n", " - 2 \\chi_{\\alpha}\\chi_{\\beta}\n", " + 2 \\frac{B_{\\alpha \\beta}}{Q_{w,xx}+Q_{w,yy}}\n", " + 2 \\delta_{\\alpha \\beta}\n", "$$\n", - "such that $\\delta_{\\alpha \\beta}$ is the Kronecker delta and\n", + "such that $\\alpha, \\beta \\in \\{1,2\\}$ the indices of the components and $\\delta_{\\alpha \\beta}$ is the Kronecker delta and\n", "$$\n", "L_\\beta = \\frac{1}{\\sigma^2} \\sum_{x,y} I(x,y) W'\\left(\\frac{(x,y)}{\\sigma}\\right) \\eta_\\beta\n", "$$\n", @@ -130,18 +135,72 @@ "\\eta_2 = 2xy.\n", "$$\n", "\n", - "By averaing equation {eq}`chichi`, $\\left<\\chi^s_\\alpha\\right> = 0$ (no intrinsic alignment assumption), $\\left = g_\\alpha$ (weak lensing assumption) and\n", + "By averaing equation {eq}`chichi`, $\\left<\\chi^s\\right> = 0$ (no intrinsic alignment assumption), $\\left = g$ (weak lensing assumption) and by multplying both sides by $\\left^{-1}$, we obtain:\n", + "$$\n", + "g = \\left^{-1}\\left< \\chi \\right> \n", + "$$ (gpx)\n", + "Equation {eq}`gpx` applies when the all the sources have locally constant morphology and size. In practice, this does not hold because the sources are affected by the PSF which adds instability to the morphology and size. Commonly in cosmic-shear, the averages are interchanged:\n", + "$$\n", + "\\left< \\tilde{g} \\right> = \\left< \\left(P^{sh}\\right)^{-1} \\chi \\right> \n", + "$$ (gpx1)\n", + "with the assumption that $\\left< \\left( P^{sh}\\right)^{-1} \\chi^s \\right> = 0$ which does not hold in reality because $P^{sh}$ depends on $\\chi$.\n", + "\n", + "Nevertheless, $\\tilde{g}$ is the solution to {eq}`chichi` with $\\chi^s=0$. This corresponds to the shear of a circular source. The tru value of $g$ is obtained by averaging $\\tilde{g}$." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Handling the PSF Effect\n", + "\n", + "In KSB the PSF is supposed to be almost istropic and its effects on the source polarisation are linearly modelled by:\n", + "$$\n", + "\\chi = \\chi^{obs} + \\chi^{g}\n", + "$$\n", + "where $\\chi$ is the polarisation that would have been observed without the PSF effect (it is the same $\\chi$ from the section above), $\\chi^{obs}$ is the polarisation observed by the telescope and $\\chi^g$ contains the effects of the PSF which expression is:\n", "$$\n", - "\\left< \\chi^s_\\alpha \\right> = \n", + "\\chi^g = P^{sm} \\left( P^{sm,*}\\right)^{-1} \\chi^{sh,*}\n", + "$$\n", + "$P^{sm}$ is the smear polarizability matrix and it accounts for the effects of the isotrpic part of the PSF, $P^{sm,*}$ and $\\chi^{sh,*}$ are respectively the smear polarizability and the polarization of a star and are used to account for the anisotropies of the PSF that are supposed to be small. The star is supposed to give an empirical estimation of the PSF near the source. The expression of the smear polarizabilty is:\n", + "$$\n", + "P^{sm}_{\\alpha \\beta}=\\frac{1}{Q_{xx}+Q_{yy}}\\left[ \n", + " \\left( M + 2\\frac{Q'_{xx}+Q'_{yy}}{\\sigma^2}\n", + " \\right) \\delta_{\\alpha \\beta}\n", + " + G_{\\alpha \\beta}\n", + " -\\chi_\\alpha \\left( 2 F_\\beta + L'_\\beta \\right)\n", + " \\right]\n", + "$$\n", + "where $L'_\\alpha$ is equivalent to $L_\\alpha$ calculated with the second order derivative of the weighting window function, $Q'_{k,k}$ (with $k \\in {x,y}$) is equivalent to the moments weighted by the first order derivative of the weighting window function and\n", + "$$\n", + "M = \\sum_{x,y} I(x,y) W \\left( \\frac{\\left(x,y\\right)}{\\sigma} \\right) \\quad,\n", + "$$\n", + "\n", + "$$\n", + "F_\\alpha = \\frac{1}{\\sigma^2}\\sum_{x,y} I(x,y) W' \\left( \\frac{\\left(x,y\\right)}{\\sigma} \\right) \\eta_\\alpha \\quad,\n", + "$$\n", + "\n", + "$$\n", + "G_{\\alpha \\beta} = \\frac{1}{\\sigma^4} \\sum_{x,y} I(x,y) W'' \\left( \\frac{\\left(x,y\\right)}{\\sigma} \\right) \\eta_\\alpha \\eta_\\beta \\quad.\n", "$$" ] }, { - "cell_type": "code", - "execution_count": null, + "cell_type": "markdown", "metadata": {}, - "outputs": [], - "source": [] + "source": [ + "### KSB in practice\n", + "\n", + "The reduced shear is estimated by removing the PSF effect then solving equation {eq}`gpx1`.\n", + "\n", + "The original KSB had an incorrect approximation of $g$. Many ameliorations were suggested, namely [VIOLA PAPER] which pushed the approximations up to the third order.\n", + "\n", + "First order approximation and limited Taylor expansion induce a large bias for the points that are distant from the [SEE TAYLOR SERIES NOTEBOOK]. This means that for high values of $g$ the error is more important than for smaller values.\n", + "\n", + "Also KSB works under the hypothesis that the PSF has small anisotropies which is not always the case, specially in Radio Interferometry for example.\n", + "\n", + "Finally, the correction of the weighting window effect is approximative which adds another bias to the estimation of $g$ that depends on the choice of the window parameters (e.g. $\\sigma$)." + ] } ], "metadata": { From 96a6b132796694f88197e7df4f699540fb7687e3 Mon Sep 17 00:00:00 2001 From: Fadi Nammour Date: Thu, 8 Apr 2021 00:47:56 +0200 Subject: [PATCH 09/12] ksb equation spacing adjusted --- shearbook/methods/ksb.ipynb | 29 +++++++++++++++++++++++++++++ 1 file changed, 29 insertions(+) diff --git a/shearbook/methods/ksb.ipynb b/shearbook/methods/ksb.ipynb index eb1a7a3..06122be 100644 --- a/shearbook/methods/ksb.ipynb +++ b/shearbook/methods/ksb.ipynb @@ -20,6 +20,7 @@ "![Weak Lensing Scheme](weak_lensing_scheme.jpeg)\n", "\n", "In the scheme above, let $\\Sigma$ be the surface mass density of the Dark Matter (L) then its lensing potential is given by:\n", + "\n", "$$\n", "\\Psi(\\theta) = \\frac{4G}{c^2}\n", " \\frac{D_{l}D_{s}}{D_{ls}}\n", @@ -28,17 +29,23 @@ " \\right\\| \n", " d^2\\theta' \\quad,\n", "$$ (lensing)\n", + "\n", "where $G$ and $c$ are the gravitational constant and the speed of light in vacuum and $D_{l}$, $D_{s}$ and $D_{l,s}$ are the angular diameter distances between the observer and the lens (OL), the observer and the source (OS) and the lens and the source (LS), respectively.\n", "\n", "The deflection of the light rays is approximated well enough by the angle\n", + "\n", "$$\n", "\\alpha(\\theta) = \\nabla \\Psi(\\theta) \\quad.\n", "$$\n", + "\n", "This angle is related to the angular positions of the source $\\beta$ to the image $\\theta$ on the sky by the lens equation:\n", + "\n", "$$\n", "\\beta = \\theta - \\alpha(\\theta) \\quad.\n", "$$ (lens)\n", + "\n", "Under weak lensing assumptions (lens mapping almost constant along the solid angle of the soucre), the lens mapping is locally linear and the Jacobian matrix of the image distorsion is:\n", + "\n", "$$\n", "A \\equiv \\frac{\\partial \\beta}{\\partial \\theta} \n", " = \\left( \\delta_{ij} - \\frac{\\partial^2 \\Psi(\\theta)}{\\partial \\theta_{i} \\partial \\theta_{j}} \\right)\n", @@ -47,17 +54,22 @@ " -\\gamma_2 & 1-\\kappa+\\gamma_! \n", " \\end{pmatrix}\n", "$$ (jacobian_lensing)\n", + "\n", "with the convergence\n", + "\n", "$$\n", "\\kappa (\\theta) = \\frac{1}{2}\\left(\\Psi_{11} + \\Psi_{22}\\right)\n", "$$\n", "and the two components\n", + "\n", "$$\n", "\\gamma_1 = \\frac{1}{2} \\left(\\Psi_{11} - \\Psi_{22}\\right), \\quad \\gamma_2 = \\Psi_{12}\n", "$$\n", + "\n", "of the complex shear $\\gamma = \\gamma_1 + \\boldsymbol{i} \\gamma_2$.\n", "\n", "In the KSB method, the reduced shear is the measure of image distorsions and is defined as:\n", + "\n", "$$\n", "g = \\frac{\\gamma}{1-\\kappa}\n", "$$\n", @@ -105,18 +117,22 @@ "Here the polarisation $\\chi$ is estimated using the weighted moments $Q_w$ and $\\chi^s$ is the unaccessible polarisation of the source observed by a perfect telescope (no PSF effect) and without any noise.\n", "\n", "By introducing the shear polarisability matrix, $P^{sh}$, a first order approximation in terms of $g$ gives :\n", + "\n", "$$\n", "\\chi - \\chi^s = P^{sh} g\n", "$$ (chichi)\n", "\n", "with\n", + "\n", "$$\n", "P^{sh}_{\\alpha \\beta} = - 2 \\frac{\\chi_{\\alpha}L_{\\beta}}{Q_{w,xx}+Q_{w,yy}} \n", " - 2 \\chi_{\\alpha}\\chi_{\\beta}\n", " + 2 \\frac{B_{\\alpha \\beta}}{Q_{w,xx}+Q_{w,yy}}\n", " + 2 \\delta_{\\alpha \\beta}\n", "$$\n", + "\n", "such that $\\alpha, \\beta \\in \\{1,2\\}$ the indices of the components and $\\delta_{\\alpha \\beta}$ is the Kronecker delta and\n", + "\n", "$$\n", "L_\\beta = \\frac{1}{\\sigma^2} \\sum_{x,y} I(x,y) W'\\left(\\frac{(x,y)}{\\sigma}\\right) \\eta_\\beta\n", "$$\n", @@ -130,19 +146,25 @@ "$$\n", "\\eta_1 = x^2 - y^2\n", "$$\n", + "\n", "and\n", + "\n", "$$\n", "\\eta_2 = 2xy.\n", "$$\n", "\n", "By averaing equation {eq}`chichi`, $\\left<\\chi^s\\right> = 0$ (no intrinsic alignment assumption), $\\left = g$ (weak lensing assumption) and by multplying both sides by $\\left^{-1}$, we obtain:\n", + "\n", "$$\n", "g = \\left^{-1}\\left< \\chi \\right> \n", "$$ (gpx)\n", + "\n", "Equation {eq}`gpx` applies when the all the sources have locally constant morphology and size. In practice, this does not hold because the sources are affected by the PSF which adds instability to the morphology and size. Commonly in cosmic-shear, the averages are interchanged:\n", + "\n", "$$\n", "\\left< \\tilde{g} \\right> = \\left< \\left(P^{sh}\\right)^{-1} \\chi \\right> \n", "$$ (gpx1)\n", + "\n", "with the assumption that $\\left< \\left( P^{sh}\\right)^{-1} \\chi^s \\right> = 0$ which does not hold in reality because $P^{sh}$ depends on $\\chi$.\n", "\n", "Nevertheless, $\\tilde{g}$ is the solution to {eq}`chichi` with $\\chi^s=0$. This corresponds to the shear of a circular source. The tru value of $g$ is obtained by averaging $\\tilde{g}$." @@ -155,14 +177,19 @@ "### Handling the PSF Effect\n", "\n", "In KSB the PSF is supposed to be almost istropic and its effects on the source polarisation are linearly modelled by:\n", + "\n", "$$\n", "\\chi = \\chi^{obs} + \\chi^{g}\n", "$$\n", + "\n", "where $\\chi$ is the polarisation that would have been observed without the PSF effect (it is the same $\\chi$ from the section above), $\\chi^{obs}$ is the polarisation observed by the telescope and $\\chi^g$ contains the effects of the PSF which expression is:\n", + "\n", "$$\n", "\\chi^g = P^{sm} \\left( P^{sm,*}\\right)^{-1} \\chi^{sh,*}\n", "$$\n", + "\n", "$P^{sm}$ is the smear polarizability matrix and it accounts for the effects of the isotrpic part of the PSF, $P^{sm,*}$ and $\\chi^{sh,*}$ are respectively the smear polarizability and the polarization of a star and are used to account for the anisotropies of the PSF that are supposed to be small. The star is supposed to give an empirical estimation of the PSF near the source. The expression of the smear polarizabilty is:\n", + "\n", "$$\n", "P^{sm}_{\\alpha \\beta}=\\frac{1}{Q_{xx}+Q_{yy}}\\left[ \n", " \\left( M + 2\\frac{Q'_{xx}+Q'_{yy}}{\\sigma^2}\n", @@ -171,7 +198,9 @@ " -\\chi_\\alpha \\left( 2 F_\\beta + L'_\\beta \\right)\n", " \\right]\n", "$$\n", + "\n", "where $L'_\\alpha$ is equivalent to $L_\\alpha$ calculated with the second order derivative of the weighting window function, $Q'_{k,k}$ (with $k \\in {x,y}$) is equivalent to the moments weighted by the first order derivative of the weighting window function and\n", + "\n", "$$\n", "M = \\sum_{x,y} I(x,y) W \\left( \\frac{\\left(x,y\\right)}{\\sigma} \\right) \\quad,\n", "$$\n", From d853cb35f394f7bc419d9a536aa084f4f91c4b08 Mon Sep 17 00:00:00 2001 From: Fadi Nammour Date: Sat, 1 May 2021 18:39:10 +0200 Subject: [PATCH 10/12] ksb prereview modifications done --- notebooks/taylor_series.ipynb | 219 ++++++++++++++++++ shearbook/_toc.yml | 5 + shearbook/appendix/.DS_Store | Bin 0 -> 6148 bytes shearbook/appendix/taylor_series.ipynb | 201 ++++++++++++++++ shearbook/intro/weak_lensing.ipynb | 98 ++++++++ .../weak_lensing_scheme.jpeg | Bin shearbook/methods/ksb.ipynb | 95 +++----- shearbook/methods/shear_estimation.png | Bin 0 -> 178011 bytes 8 files changed, 552 insertions(+), 66 deletions(-) create mode 100755 notebooks/taylor_series.ipynb create mode 100644 shearbook/appendix/.DS_Store create mode 100644 shearbook/appendix/taylor_series.ipynb create mode 100644 shearbook/intro/weak_lensing.ipynb rename shearbook/{methods => intro}/weak_lensing_scheme.jpeg (100%) create mode 100644 shearbook/methods/shear_estimation.png diff --git a/notebooks/taylor_series.ipynb b/notebooks/taylor_series.ipynb new file mode 100755 index 0000000..8f9a155 --- /dev/null +++ b/notebooks/taylor_series.ipynb @@ -0,0 +1,219 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Taylor Series for Approximations\n", + "\n", + "Taylor series are commonly used in physics to approximate functions making them easier to handle specially when solving equations. In this notebook we give a visual example on how it works and the biases that it introduces.\n", + "\n", + "## Theoretical Formula\n", + "\n", + "Consider a function $f$ that is $n$ times differentiable in a point $a$. Then by Taylor's theorem, for any point $x$ in the domain of f, we have the Taylor expansion about the point $a$ is defined as:\n", + "\\begin{equation}\n", + "f(x) = f(a) + \\sum_{k=1}^n \\frac{f^{k}(a)}{k!}(x-a)^k + o\\left((x-a)^n\\right) \\quad,\n", + "\\end{equation}\n", + "where $f^{(k)}$ is the derivative of order $k$ of $f$. Usually, we consider $a=0$ which gives:\n", + "\\begin{equation}\n", + "f(x) = f(0) + \\sum_{k=1}^n \\frac{f^{k}(0)}{k!}(x)^k + o\\left((x)^n\\right) \\quad.\n", + "\\end{equation}\n", + "\n", + "For example, the exponential, $e$ is infinitely differentiable with $e^{(k)}=e$ and $e^0=1$. This gives us the following Taylor expansion:\n", + "\\begin{equation}\n", + "e(x) = 1 + \\sum_{k=1}^\\infty \\frac{x^k}{k!} \\quad.\n", + "\\end{equation}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Visualising Taylor Expansion Approximation and its Bias\n", + "\n", + "Let us see visually how the Taylor expansion approximatees a given function. We start by defining our function below, for example we will consider the exponential function, $e$ again up to order 3." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "code_folding": [ + 0 + ] + }, + "outputs": [], + "source": [ + "#### FOLDED CELL\n", + "%matplotlib inline\n", + "import matplotlib.pyplot as plt\n", + "from IPython.display import Markdown as md\n", + "from sympy import Symbol, series, lambdify, latex\n", + "from sympy.functions import *\n", + "from ipywidgets import interactive_output\n", + "import ipywidgets as widgets\n", + "from sympy.parsing.sympy_parser import parse_expr\n", + "import numpy as np\n", + "\n", + "x = Symbol('x')" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "order = 3\n", + "func = exp(x)" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "code_folding": [ + 0 + ] + }, + "outputs": [], + "source": [ + "#### FOLDED CELL\n", + "taylor_exp = series(func,x,n=order+1)\n", + "approx = lambdify(x, sum(taylor_exp.args[:-1]), \"numpy\")\n", + "func_np = lambdify(x, func, \"numpy\")\n", + "latex_func = '$'+latex(func)+'$'\n", + "latex_taylor = '\\\\begin{equation} '+latex(taylor_exp)+' \\end{equation}'" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "variables": { + " latex_func ": "$e^{x}$", + "latex_taylor": "\\begin{equation} 1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + O\\left(x^{4}\\right) \\end{equation}" + } + }, + "source": [ + "The Taylor expansion of {{ latex_func }} is :\n", + "{{latex_taylor}}\n", + "\n", + "Now let's plot the function and its expansion while considering a point, noted $p$, to study the biais that we introduce when we approximate the function by its expansion:" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "code_folding": [ + 0 + ] + }, + "outputs": [ + { + "data": { + "application/vnd.jupyter.widget-view+json": { + "model_id": "5f5d027e2c284af4869d9e9c6ea083f9", + "version_major": 2, + "version_minor": 0 + }, + "text/plain": [ + "Output(layout=Layout(height='650px'))" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "application/vnd.jupyter.widget-view+json": { + "model_id": "d8b970e851ef4e688880bc1db4a46238", + "version_major": 2, + "version_minor": 0 + }, + "text/plain": [ + "HBox(children=(FloatSlider(value=3.0, description='Point Absciss', max=4.0, min=-4.0, step=0.2), IntSlider(val…" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "#### FOLDED CELL\n", + "order = widgets.IntSlider(min=0,max=20,step=1,value=3,description='Order')\n", + "x_min = -4\n", + "x_max = 4\n", + "x1 = widgets.FloatSlider(min=x_min,max=x_max,value=3,step=0.2,description='Point Absciss')\n", + "func = widgets.Text('exp(x)',description='Function')\n", + "text_offset = np.array([-0.15,2.])\n", + "\n", + "\n", + "ui = widgets.HBox([x1, order, func])\n", + "\n", + "def f(order=widgets.IntSlider(min=1,max=10,step=1,value=3)\n", + " ,x1=1.5\n", + " ,func='exp(x)'):\n", + " func_sp = parse_expr(func)\n", + " taylor_exp = series(func_sp,x,n=order+1)\n", + " approx = lambdify(x, sum(taylor_exp.args[:-1]), \"numpy\")\n", + " func_np = lambdify(x, func_sp, \"numpy\")\n", + " n_points = 1000\n", + " x_array = np.linspace(x_min,x_max,n_points)\n", + " approx_array = np.array([approx(z) for z in x_array])\n", + " func_array = np.array([func_np(z) for z in x_array])\n", + " func_x1 = func_np(x1)\n", + " approx_x1 = approx(x1)\n", + " plt.figure(42,figsize=(10,10))\n", + " plt.plot(x_array,approx_array,color='blue',label='Taylor Expansion')\n", + " plt.plot(x_array,func_array,color='green',label=func)\n", + " plt.plot(0,approx(0),color='black',marker='o')\n", + " plt.annotate(r'(0,0)',[0,approx(0)],xytext=text_offset)\n", + " plt.plot([x1,x1]\n", + " ,[-np.max(np.abs([np.min(func_array),np.max(func_array)])),min(approx_x1, func_x1)]\n", + " ,'--',color='black',marker='x')\n", + " plt.plot([x1,x1],[approx_x1, func_x1],'r--',marker='x')\n", + " plt.annotate(r'$p_{approx}$',[x1,approx(x1)],xytext=[x1,approx(x1)]-text_offset)\n", + " plt.annotate(r'$p$',[x1,func_np(x1)],xytext=[x1,func_np(x1)]-text_offset)\n", + " plt.xlim([x_min,x_max])\n", + " plt.ylim(-np.max(np.abs([np.min(func_array),np.max(func_array)]))\n", + " ,np.max(np.abs([np.min(func_array),np.max(func_array)])))\n", + " plt.legend()\n", + " plt.show()\n", + " print('Approximation bias : {}'.format(func_x1-approx_x1))\n", + " return None\n", + "interactive_plot = widgets.interactive_output(f, {'order': order, 'x1': x1, 'func': func})\n", + "interactive_plot.layout.height = '650px'\n", + "display(interactive_plot, ui)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Notice that the further $p$ gets away from the point of the expansion (in that case $0$), the higher the approximation bias gets. Samely, the lower the order of approximation is, the higher the approximation bias gets." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.6" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/shearbook/_toc.yml b/shearbook/_toc.yml index aa5aba1..1afa5e6 100644 --- a/shearbook/_toc.yml +++ b/shearbook/_toc.yml @@ -5,6 +5,7 @@ chapters: - file: intro/about - file: intro/run_code + - file: intro/weak_lensing - part: Shape Measurement chapters: @@ -35,6 +36,10 @@ sections: - file: calibration/calibration-ref +- part: Appendix + chapters: + - file: appendix/taylor_series + - part: Contributing chapters: - file: contrib/contributors diff --git a/shearbook/appendix/.DS_Store b/shearbook/appendix/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..dc020f17468bc0c34e805a13781172636d930b9e GIT binary patch literal 6148 zcmeHKF;2rk5S)b+k!T{(P+m!e#1E_}DX4h>V4#R(;fO#%=PqyIl`y+|AS??h6$P4+ z_U3MH_s*We>jhxT*N01B0bovZ#@>f1eBW(?qv|m-jc4?D#tn9OBp(LPJ;e*|@P=-b zzu_r6TwsGO_ny#Rb^U$Y^(`~}ao=g@9|6t;DIf);fE17dQs6KJy!I0FlVpujKnh5K zZw37OP-xCg~V64D#E*D<^*R)^y|8bI5Qa}p)D+O$}x?U~$O0~95 yUe0T6q21EHz}N`q5T%$LrI-sZ#b-a_iqE;qo$K5v0IhDDVl!lpjI> literal 0 HcmV?d00001 diff --git a/shearbook/appendix/taylor_series.ipynb b/shearbook/appendix/taylor_series.ipynb new file mode 100644 index 0000000..751775f --- /dev/null +++ b/shearbook/appendix/taylor_series.ipynb @@ -0,0 +1,201 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Taylor Series for Approximations\n", + "\n", + "Taylor series are commonly used in physics to approximate functions making them easier to handle specially when solving equations. In this notebook we give a visual example on how it works and the biases that it introduces.\n", + "\n", + "## Theoretical Formula\n", + "\n", + "Consider a function $f$ that is $n$ times differentiable in a point $a$. Then by Taylor's theorem, for any point $x$ in the domain of f, we have the Taylor expansion about the point $a$ is defined as:\n", + "\\begin{equation}\n", + "f(x) = f(a) + \\sum_{k=1}^n \\frac{f^{k}(a)}{k!}(x-a)^k + o\\left((x-a)^n\\right) \\quad,\n", + "\\end{equation}\n", + "where $f^{(k)}$ is the derivative of order $k$ of $f$. Usually, we consider $a=0$ which gives:\n", + "\\begin{equation}\n", + "f(x) = f(0) + \\sum_{k=1}^n \\frac{f^{k}(0)}{k!}(x)^k + o\\left((x)^n\\right) \\quad.\n", + "\\end{equation}\n", + "\n", + "For example, the exponential, $e$ is infinitely differentiable with $e^{(k)}=e$ and $e^0=1$. This gives us the following Taylor expansion:\n", + "\\begin{equation}\n", + "e(x) = 1 + \\sum_{k=1}^\\infty \\frac{x^k}{k!} \\quad.\n", + "\\end{equation}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Visualising Taylor Expansion Approximation and its Bias\n", + "\n", + "Let us see visually how the Taylor expansion approximatees a given function. We start by defining our function below, for example we will consider the exponential function, $e$ again up to order 3." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "code_folding": [ + 0 + ] + }, + "outputs": [], + "source": [ + "#### FOLDED CELL\n", + "%matplotlib inline\n", + "import matplotlib.pyplot as plt\n", + "from IPython.display import Markdown as md\n", + "from sympy import Symbol, series, lambdify, latex\n", + "from sympy.functions import *\n", + "from ipywidgets import interactive_output\n", + "import ipywidgets as widgets\n", + "from sympy.parsing.sympy_parser import parse_expr\n", + "import numpy as np\n", + "\n", + "x = Symbol('x')" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "order = 3\n", + "func = exp(x)" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "code_folding": [ + 0 + ] + }, + "outputs": [], + "source": [ + "#### FOLDED CELL\n", + "taylor_exp = series(func,x,n=order+1)\n", + "approx = lambdify(x, sum(taylor_exp.args[:-1]), \"numpy\")\n", + "func_np = lambdify(x, func, \"numpy\")\n", + "latex_func = '$'+latex(func)+'$'\n", + "latex_taylor = '\\\\begin{equation} '+latex(taylor_exp)+' \\end{equation}'" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "variables": { + " latex_func ": "$e^{x}$", + "latex_taylor": "\\begin{equation} 1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + O\\left(x^{4}\\right) \\end{equation}" + } + }, + "source": [ + "The Taylor expansion of {{ latex_func }} is :\n", + "{{latex_taylor}}\n", + "\n", + "Now let's plot the function and its expansion while considering a point, noted $p$, to study the biais that we introduce when we approximate the function by its expansion:" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "code_folding": [ + 0 + ] + }, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Approximation order : 3\n", + "Approximation bias : 7.085536923187668\n" + ] + } + ], + "source": [ + "#### FOLDED CELL\n", + "text_offset = np.array([-0.15,2.])\n", + "\n", + "func='exp(x)'\n", + "order = 3\n", + "x1 = 3\n", + "x_min, x_max = -4,4\n", + "func_sp = parse_expr(func)\n", + "taylor_exp = series(func_sp,x,n=order+1)\n", + "approx = lambdify(x, sum(taylor_exp.args[:-1]), \"numpy\")\n", + "func_np = lambdify(x, func_sp, \"numpy\")\n", + "n_points = 1000\n", + "x_array = np.linspace(x_min,x_max,n_points)\n", + "approx_array = np.array([approx(z) for z in x_array])\n", + "func_array = np.array([func_np(z) for z in x_array])\n", + "func_x1 = func_np(x1)\n", + "approx_x1 = approx(x1)\n", + "plt.figure(42,figsize=(10,10))\n", + "plt.plot(x_array,approx_array,color='blue',label='Taylor Expansion')\n", + "plt.plot(x_array,func_array,color='green',label=func)\n", + "plt.plot(0,approx(0),color='black',marker='o')\n", + "plt.annotate(r'(0,0)',[0,approx(0)],xytext=text_offset)\n", + "plt.plot([x1,x1]\n", + " ,[-np.max(np.abs([np.min(func_array),np.max(func_array)])),min(approx_x1, func_x1)]\n", + " ,'--',color='black',marker='x')\n", + "plt.plot([x1,x1],[approx_x1, func_x1],'r--',marker='x')\n", + "plt.annotate(r'$p_{approx}$',[x1,approx(x1)],xytext=[x1,approx(x1)]-text_offset)\n", + "plt.annotate(r'$p$',[x1,func_np(x1)],xytext=[x1,func_np(x1)]-text_offset)\n", + "plt.xlim([x_min,x_max])\n", + "plt.ylim(-np.max(np.abs([np.min(func_array),np.max(func_array)]))\n", + " ,np.max(np.abs([np.min(func_array),np.max(func_array)])))\n", + "plt.legend()\n", + "plt.show()\n", + "print('Approximation order : {}'.format(order))\n", + "print('Approximation bias : {}'.format(func_x1-approx_x1))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Notice that the further $p$ gets away from the point of the expansion (in that case $0$), the higher the approximation bias gets. Samely, the lower the order of approximation is, the higher the approximation bias gets." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.6" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/shearbook/intro/weak_lensing.ipynb b/shearbook/intro/weak_lensing.ipynb new file mode 100644 index 0000000..63ce796 --- /dev/null +++ b/shearbook/intro/weak_lensing.ipynb @@ -0,0 +1,98 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Weak Lensing\n", + "\n", + "A quick note on how the reduced shear is defined in Weak Lensing.\n", + "\n", + "![Weak Lensing Scheme](weak_lensing_scheme.jpeg)\n", + "\n", + "In the scheme above, let $\\Sigma$ be the surface mass density of the Dark Matter (L) then its lensing potential is given by:\n", + "\n", + "$$\n", + "\\Psi(\\theta) = \\frac{4G}{c^2}\n", + " \\frac{D_{l}D_{s}}{D_{ls}}\n", + " \\int \\Sigma{\\left(\\theta'\\right)} \\ln \\left\\|\n", + " \\theta - \\theta' \n", + " \\right\\| \n", + " d^2\\theta' \\quad,\n", + "$$ (lensing)\n", + "\n", + "where $G$ and $c$ are the gravitational constant and the speed of light in vacuum and $D_{l}$, $D_{s}$ and $D_{l,s}$ are the angular diameter distances between the observer and the lens (OL), the observer and the source (OS) and the lens and the source (LS), respectively.\n", + "\n", + "The deflection of the light rays is approximated well enough by the angle\n", + "\n", + "$$\n", + "\\alpha(\\theta) = \\nabla \\Psi(\\theta) \\quad.\n", + "$$\n", + "\n", + "This angle is related to the angular positions of the source $\\beta$ to the image $\\theta$ on the sky by the lens equation:\n", + "\n", + "$$\n", + "\\beta = \\theta - \\alpha(\\theta) \\quad.\n", + "$$ (lens)\n", + "\n", + "Under weak lensing assumptions (lens mapping almost constant along the solid angle of the soucre), the lens mapping is locally linear and the Jacobian matrix of the image distorsion is:\n", + "\n", + "$$\n", + "A \\equiv \\frac{\\partial \\beta}{\\partial \\theta} \n", + " = \\left( \\delta_{ij} - \\frac{\\partial^2 \\Psi(\\theta)}{\\partial \\theta_{i} \\partial \\theta_{j}} \\right)\n", + " = \\begin{pmatrix}\n", + " 1-\\kappa -\\gamma_1 & -\\gamma_2 \\\\\n", + " -\\gamma_2 & 1-\\kappa+\\gamma_! \n", + " \\end{pmatrix}\n", + "$$ (jacobian_lensing)\n", + "\n", + "with the convergence\n", + "\n", + "$$\n", + "\\kappa (\\theta) = \\frac{1}{2}\\left(\\Psi_{11} + \\Psi_{22}\\right)\n", + "$$\n", + "and the two components\n", + "\n", + "$$\n", + "\\gamma_1 = \\frac{1}{2} \\left(\\Psi_{11} - \\Psi_{22}\\right), \\quad \\gamma_2 = \\Psi_{12}\n", + "$$\n", + "\n", + "of the complex shear $\\gamma = \\gamma_1 + \\boldsymbol{i} \\gamma_2$.\n", + "\n", + "In the KSB method, the reduced shear is the measure of image distorsions and is defined as:\n", + "\n", + "$$\n", + "g = \\frac{\\gamma}{1-\\kappa}\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.6" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/shearbook/methods/weak_lensing_scheme.jpeg b/shearbook/intro/weak_lensing_scheme.jpeg similarity index 100% rename from shearbook/methods/weak_lensing_scheme.jpeg rename to shearbook/intro/weak_lensing_scheme.jpeg diff --git a/shearbook/methods/ksb.ipynb b/shearbook/methods/ksb.ipynb index 06122be..3effc04 100644 --- a/shearbook/methods/ksb.ipynb +++ b/shearbook/methods/ksb.ipynb @@ -13,69 +13,6 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## Reminder : Reduced Shear\n", - "\n", - "A quick note on how the reduced shear is defined in Weak Lensing.\n", - "\n", - "![Weak Lensing Scheme](weak_lensing_scheme.jpeg)\n", - "\n", - "In the scheme above, let $\\Sigma$ be the surface mass density of the Dark Matter (L) then its lensing potential is given by:\n", - "\n", - "$$\n", - "\\Psi(\\theta) = \\frac{4G}{c^2}\n", - " \\frac{D_{l}D_{s}}{D_{ls}}\n", - " \\int \\Sigma{\\left(\\theta'\\right)} \\ln \\left\\|\n", - " \\theta - \\theta' \n", - " \\right\\| \n", - " d^2\\theta' \\quad,\n", - "$$ (lensing)\n", - "\n", - "where $G$ and $c$ are the gravitational constant and the speed of light in vacuum and $D_{l}$, $D_{s}$ and $D_{l,s}$ are the angular diameter distances between the observer and the lens (OL), the observer and the source (OS) and the lens and the source (LS), respectively.\n", - "\n", - "The deflection of the light rays is approximated well enough by the angle\n", - "\n", - "$$\n", - "\\alpha(\\theta) = \\nabla \\Psi(\\theta) \\quad.\n", - "$$\n", - "\n", - "This angle is related to the angular positions of the source $\\beta$ to the image $\\theta$ on the sky by the lens equation:\n", - "\n", - "$$\n", - "\\beta = \\theta - \\alpha(\\theta) \\quad.\n", - "$$ (lens)\n", - "\n", - "Under weak lensing assumptions (lens mapping almost constant along the solid angle of the soucre), the lens mapping is locally linear and the Jacobian matrix of the image distorsion is:\n", - "\n", - "$$\n", - "A \\equiv \\frac{\\partial \\beta}{\\partial \\theta} \n", - " = \\left( \\delta_{ij} - \\frac{\\partial^2 \\Psi(\\theta)}{\\partial \\theta_{i} \\partial \\theta_{j}} \\right)\n", - " = \\begin{pmatrix}\n", - " 1-\\kappa -\\gamma_1 & -\\gamma_2 \\\\\n", - " -\\gamma_2 & 1-\\kappa+\\gamma_! \n", - " \\end{pmatrix}\n", - "$$ (jacobian_lensing)\n", - "\n", - "with the convergence\n", - "\n", - "$$\n", - "\\kappa (\\theta) = \\frac{1}{2}\\left(\\Psi_{11} + \\Psi_{22}\\right)\n", - "$$\n", - "and the two components\n", - "\n", - "$$\n", - "\\gamma_1 = \\frac{1}{2} \\left(\\Psi_{11} - \\Psi_{22}\\right), \\quad \\gamma_2 = \\Psi_{12}\n", - "$$\n", - "\n", - "of the complex shear $\\gamma = \\gamma_1 + \\boldsymbol{i} \\gamma_2$.\n", - "\n", - "In the KSB method, the reduced shear is the measure of image distorsions and is defined as:\n", - "\n", - "$$\n", - "g = \\frac{\\gamma}{1-\\kappa}\n", - "$$\n", - "\n", - "## Theory\n", - "\n", "KSB estimates the reduced shear $g$ using the polarisation $\\chi$. The exact relation between these quantities is :\n", "\n", "$$\n", @@ -222,13 +159,39 @@ "\n", "The reduced shear is estimated by removing the PSF effect then solving equation {eq}`gpx1`.\n", "\n", - "The original KSB had an incorrect approximation of $g$. Many ameliorations were suggested, namely [VIOLA PAPER] which pushed the approximations up to the third order.\n", + "The original KSB had an incorrect approximation of $g$. Many ameliorations were suggested, namely [VIOLA PAPER] which pushed the approximations up to the third order. Alternative methods to KSB exist, for example there are HSM and REGAUSS which is used in HSC [give references and define acronyms].\n", "\n", - "First order approximation and limited Taylor expansion induce a large bias for the points that are distant from the [SEE TAYLOR SERIES NOTEBOOK]. This means that for high values of $g$ the error is more important than for smaller values.\n", + "First order approximation and limited Taylor expansion induce a large bias for the points that are distant from the [SEE TAYLOR SERIES NOTEBOOK]. This means that for high values of $g$ the error is more important than for smaller values, as seen in the figure below.\n", + "\n", + "\n", + "```{figure} shear_estimation.png\n", + "---\n", + "width: 650px\n", + "name: ksb_shear_est\n", + "---\n", + "Shear estimation for unconvolved and almost noiseless galaxies with Sersic profile. KSB (red solidline) goes to third order approximations which are incorrect, KSB1 (violet dotted) is a first order approximation, KSBtr (green long-dashed line) is a third order approximation that is not mathematically justified and KSB3 (blue short-dashed line) is the correct third order approximation. Source [VIOLA PAPER]\n", + "```\n", "\n", "Also KSB works under the hypothesis that the PSF has small anisotropies which is not always the case, specially in Radio Interferometry for example.\n", "\n", - "Finally, the correction of the weighting window effect is approximative which adds another bias to the estimation of $g$ that depends on the choice of the window parameters (e.g. $\\sigma$)." + "Furthermore, the correction of the weighting window effect is approximative which adds another bias to the estimation of $g$ that depends on the choice of the window parameters (e.g. $\\sigma$).\n", + "\n", + "Finally in the following the following code snippet, an example of how to use the GalSim KSB implementation to estimate the shear of a galaxy contained in `image` which was convolved by `psf` (assuming that both `image` and `psf` are numpy arrays):\n", + "\n", + "```python\n", + "import galsim\n", + "from galsim import Image\n", + "import galsim.hsm\n", + "\n", + "#set scale\n", + "scale = 0.1\n", + "#create a galsim version of the data\n", + "image_galsim = Image(image,scale=scale)\n", + "psf_galsim = Image(psf,scale=scale)\n", + "#estimate the moments of the observation image\n", + "ell=galsim.hsm.EstimateShear(image_galsim\n", + " ,psf_galsim,shear_est='KSB')\n", + "```" ] } ], diff --git a/shearbook/methods/shear_estimation.png b/shearbook/methods/shear_estimation.png new file mode 100644 index 0000000000000000000000000000000000000000..30bc9300ed6507ab6cab912bf7e25ff81c3f024b GIT binary patch literal 178011 zcmeFZcRbbo|38i>8D*2K$jFSWWEDd8-XoGt_6nVl%tNxXv-i%HjF7zv*_-Um_xU+yU(?vMLpK98TgtT;9XDFzY}61Jp-s3H;)dI=H|+79|f z@S9PqhH4}vEJ;%l5qU`w5h{6G>!+p`Mo35!e&K4D)RkI^;!)_}@6O+Tsd2i96sH-l9OJu;62f*XgjV!b5nWnpvBWqMvS{K3Y2@gZNU4#{ zO?ITV?$d;=dA^cYwoPKKyK5(&gaR=6&PzyszPw6Q3kxFElt|TY*}@5tgm-*gSEA7r zY2&d9VOVtNh01=#10R(A?hJf*WEA$c9_CDp^n?^n-+ocJ3T;`#=fvW3iaWl6nx8N; zvam0f4US$TzAzcuX2!|M(j%<37b5e-{3GYzzL3q&ZWj3#RuYAWd}_2crLZ169|=BZ 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a/shearbook/_bibliography/z_moments_methods.bib b/shearbook/_bibliography/z_moments_methods.bib index 25d362d..98e9ec2 100644 --- a/shearbook/_bibliography/z_moments_methods.bib +++ b/shearbook/_bibliography/z_moments_methods.bib @@ -18,3 +18,38 @@ @ARTICLE{2011MNRAS.410.2156V adsurl = {https://ui.adsabs.harvard.edu/abs/2011MNRAS.410.2156V}, adsnote = {Provided by the SAO/NASA Astrophysics Data System} } + +@ARTICLE{1995ApJ.KSB, + author = {{Kaiser}, Nick and {Squires}, Gordon and {Broadhurst}, Tom}, + title = "{A Method for Weak Lensing Observations}", + journal = {\apj}, + keywords = {COSMOLOGY: OBSERVATIONS, COSMOLOGY: DARK MATTER, GALAXIES: FORMATION, COSMOLOGY: GRAVITATIONAL LENSING, COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE, Astrophysics}, + year = 1995, + month = aug, + volume = {449}, + pages = {460}, + doi = {10.1086/176071}, +archivePrefix = {arXiv}, + eprint = {astro-ph/9411005}, + primaryClass = {astro-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/1995ApJ...449..460K}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{2003MNRAS.343..459H, + author = {{Hirata}, Christopher and {Seljak}, Uro{\v{s}}}, + title = "{Shear calibration biases in weak-lensing surveys}", + journal = {\mnras}, + keywords = {gravitational lensing, methods: data analysis, Astrophysics}, + year = 2003, + month = aug, + volume = {343}, + number = {2}, + pages = {459-480}, + doi = {10.1046/j.1365-8711.2003.06683.x}, +archivePrefix = {arXiv}, + eprint = {astro-ph/0301054}, + primaryClass = {astro-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2003MNRAS.343..459H}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} \ No newline at end of file diff --git a/shearbook/methods/ksb.ipynb b/shearbook/methods/ksb.ipynb index 3effc04..2fb1779 100644 --- a/shearbook/methods/ksb.ipynb +++ b/shearbook/methods/ksb.ipynb @@ -159,9 +159,9 @@ "\n", "The reduced shear is estimated by removing the PSF effect then solving equation {eq}`gpx1`.\n", "\n", - "The original KSB had an incorrect approximation of $g$. Many ameliorations were suggested, namely [VIOLA PAPER] which pushed the approximations up to the third order. Alternative methods to KSB exist, for example there are HSM and REGAUSS which is used in HSC [give references and define acronyms].\n", + "The original KSB {cite}`1995ApJ.KSB` had an incorrect approximation of $g$. Many ameliorations were suggested, namely from {cite}`2011MNRAS.410.2156V` which pushed the approximations up to the third order. Alternative methods to KSB exist, for example there is the re-Gaussianization approach which is used in HSC {cite}`2003MNRAS.343..459H`.\n", "\n", - "First order approximation and limited Taylor expansion induce a large bias for the points that are distant from the [SEE TAYLOR SERIES NOTEBOOK]. This means that for high values of $g$ the error is more important than for smaller values, as seen in the figure below.\n", + "First order approximation and limited Taylor expansion induce a large bias for the points that are distant from the expansion point, for more details see the [appendix on Taylor series](../appendix/taylor_series.ipynb). This means that for high values of $g$ the error is more important than for smaller values, as seen in the figure below.\n", "\n", "\n", "```{figure} shear_estimation.png\n", @@ -169,7 +169,7 @@ "width: 650px\n", "name: ksb_shear_est\n", "---\n", - "Shear estimation for unconvolved and almost noiseless galaxies with Sersic profile. KSB (red solidline) goes to third order approximations which are incorrect, KSB1 (violet dotted) is a first order approximation, KSBtr (green long-dashed line) is a third order approximation that is not mathematically justified and KSB3 (blue short-dashed line) is the correct third order approximation. Source [VIOLA PAPER]\n", + "Shear estimation for unconvolved and almost noiseless galaxies with Sersic profile. KSB (red solidline) goes to third order approximations which are incorrect, KSB1 (violet dotted) is a first order approximation, KSBtr (green long-dashed line) is a third order approximation that is not mathematically justified and KSB3 (blue short-dashed line) is the correct third order approximation. Source {cite}`2011MNRAS.410.2156V`.\n", "```\n", "\n", "Also KSB works under the hypothesis that the PSF has small anisotropies which is not always the case, specially in Radio Interferometry for example.\n", @@ -183,7 +183,7 @@ "from galsim import Image\n", "import galsim.hsm\n", "\n", - "#set scale\n", + "#set pixel scale\n", "scale = 0.1\n", "#create a galsim version of the data\n", "image_galsim = Image(image,scale=scale)\n", From f9001595de886dfddaeac93a3b7a48f14134274f Mon Sep 17 00:00:00 2001 From: Fadi Nammour Date: Fri, 7 May 2021 15:28:15 +0200 Subject: [PATCH 12/12] ksb files clean-up --- shearbook/_toc.yml | 2 - shearbook/methods/ksb-variants.ipynb | 48 ------------------ shearbook/methods/ksb.ipynb | 2 +- shearbook/methods/viola2011.md | 73 ---------------------------- 4 files changed, 1 insertion(+), 124 deletions(-) delete mode 100644 shearbook/methods/ksb-variants.ipynb delete mode 100644 shearbook/methods/viola2011.md diff --git a/shearbook/_toc.yml b/shearbook/_toc.yml index 1afa5e6..90d4cc1 100644 --- a/shearbook/_toc.yml +++ b/shearbook/_toc.yml @@ -18,8 +18,6 @@ - file: methods/moments-methods sections: - file: methods/ksb - - file: methods/ksb-variants - - file: methods/viola2011 - file: methods/moments-methods-ref - file: model/model-intro sections: diff --git a/shearbook/methods/ksb-variants.ipynb b/shearbook/methods/ksb-variants.ipynb deleted file mode 100644 index 7800f7e..0000000 --- a/shearbook/methods/ksb-variants.ipynb +++ /dev/null @@ -1,48 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# KSB Variants" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "```{warning}\n", - "In progress!\n", - "```" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/shearbook/methods/ksb.ipynb b/shearbook/methods/ksb.ipynb index 2fb1779..90e3dcc 100644 --- a/shearbook/methods/ksb.ipynb +++ b/shearbook/methods/ksb.ipynb @@ -176,7 +176,7 @@ "\n", "Furthermore, the correction of the weighting window effect is approximative which adds another bias to the estimation of $g$ that depends on the choice of the window parameters (e.g. $\\sigma$).\n", "\n", - "Finally in the following the following code snippet, an example of how to use the GalSim KSB implementation to estimate the shear of a galaxy contained in `image` which was convolved by `psf` (assuming that both `image` and `psf` are numpy arrays):\n", + "Finally, a C++ implementation of KSB by Peter Melchior can be found on this [GitHub repository](https://github.com/pmelchior/shapelens/blob/master/src/KSB.cc). And in the following code snippet, an example of how to use the GalSim KSB implementation to estimate the shear of a galaxy contained in `image` which was convolved by `psf` (assuming that both `image` and `psf` are numpy arrays):\n", "\n", "```python\n", "import galsim\n", diff --git a/shearbook/methods/viola2011.md b/shearbook/methods/viola2011.md deleted file mode 100644 index 7554207..0000000 --- a/shearbook/methods/viola2011.md +++ /dev/null @@ -1,73 +0,0 @@ -# Biases in, and corrections to, KSB shear measurements. Viola et al, 2010 - -```{warning} -In progress! -``` - -Notes and tools derived from reading groups sessions - -{cite}`2011MNRAS.410.2156V` - -## Goals of the Paper - -1. Show the theoretical and practical caveats of KSB. -2. Suggest improvements to KSB. - -## Main Results and Conclusion - -1. The assumptions in KSB introduce biaises especially for large amplitude -values. -2. Rewriting correctly the Taylor expansion of the ellipticity up to the third -order (see KSB3) gives results with biaises smaller by orders of magnitude. -3. The PSF corrections in KSB are not accurate however they partially cancel -other biaises. - -## Definitions and Equations - -*. Reduced shear, g: (eq. 7) -*. Complex ellipticity, $\chi$: (eq. 10) -*. Shear responsivity $\epsilon$: (eq. 14) -*. KSB (eq. 32) -*. KSB1 (eq. 34) -*. KSBtr (eq.36) -*. KSB3 (eq. 44) -*. Inverse Problem (eq. 50) -*. Circular PSF Formula (eq. 58) -*. Anisotropic PSf Formula (eq. 67) - -> The reduced shear, g, is preferred over the shear responsivity, -$\epsilon$, for its robustness to noise in the data. - -## Assumptions in KSB -1. The reduced shear, g, is constant and the ellipticity average is null, -$\left< \chi \right>=0$ (see eq. 12). -2. Many Taylor expansions like in the reduced shear, g (see eq. 29). -> Taylor expansions gives a polynomial approximation around a certain reference -value. The further from the reference we evaluate the expansion the bigger the -biais. In this paper, since the reference value is zero, the biais will be -greater for large amplitude values. - -3. Assume equality between (eq. 27) and (eq. 28). -4. The PSF has at most slight anisotropies, it is mostly circular. - -## Notations Clarification - -1. The angular coordinates are solid angles so they have two components. These -coordinates can be approximated by linear coordinates because the angles are -very small. -2. The star in eq. 12 stands for the conjugate. -3. Rewriting eq.48-50: - i. (eq. 48) $$\chi = f\left( \tilde{g}, \chi^{s}\right)$$ - ii. (eq. 49) $$\chi = h \left( \chi^{obs}\right)$$ - iii. (eq. 50) $$\chi^{obs} = h^{-1}\left[ f \left( \tilde{g}, h\left(\chi^{obs}\right)\right)\right]$$ - - where $\chi^{s}$ is the intrinsic ellipticity of the galaxy, $\chi^{obs}$ is - the ellipticity of the lensed galaxy and $\chi$ is the ellipticity of the - convoluted and lensed galaxy. -4. In (eq. 51): $$\chi^{sh}_{\alpha} = \chi_{\alpha} + \chi^{s}_{\alpha}$$ -5. In (eq. 52): the asterisk, *, in $P^{sm,*}$ is to indicate the PSF. -Explanation: * --> star --> dirac --> PSF --> illuminati - -## Implementation - -[Link](https://github.com/pmelchior/shapelens/blob/master/src/KSB.cc) to Peter Melchior implementation of KSB codes in C.