From ee1d2242eb69067f56c7580515e596c585ec6881 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Gi=C3=B9=20Marcer?= Date: Wed, 20 May 2020 18:45:27 +0200 Subject: [PATCH] images: neatly renamed all the images --- .../{landau-histo.pdf => 1-landau-histo.pdf} | Bin .../{landau-kde.pdf => 1-landau-kde.pdf} | Bin .../{landau-small.pdf => 1-landau-small.pdf} | Bin notes/images/{landau.pdf => 1-landau.pdf} | Bin .../{gamma-area.png => 2-gamma-area.png} | Bin .../images/{histo-i-F.pdf => 3-histo-i-F.pdf} | Bin .../images/{histo-i-u.pdf => 3-histo-i-u.pdf} | Bin .../images/{histo-p-F.pdf => 3-histo-p-F.pdf} | Bin .../images/{histo-p-u.pdf => 3-histo-p-u.pdf} | Bin notes/images/{dip.pdf => 4-dip.pdf} | Bin notes/images/{expected.pdf => 4-expected.pdf} | Bin notes/images/{fit.pdf => 4-fit.pdf} | Bin notes/images/{MC_MC.pdf => 5-MC_MC.pdf} | Bin notes/images/{MC_MC_MI.pdf => 5-MC_MC_MI.pdf} | Bin notes/images/{MC_MI_VE.pdf => 5-MC_MI_VE.pdf} | Bin ...md-hist-0.005.pdf => 6-emd-hist-0.005.pdf} | Bin ...{emd-hist-0.01.pdf => 6-emd-hist-0.01.pdf} | Bin ...{emd-hist-0.05.pdf => 6-emd-hist-0.05.pdf} | Bin .../{emd-hist-0.pdf => 6-emd-hist-0.pdf} | Bin ...ounds-0.005.pdf => 6-emd-rounds-0.005.pdf} | Bin ...-rounds-0.01.pdf => 6-emd-rounds-0.01.pdf} | Bin ...-rounds-0.05.pdf => 6-emd-rounds-0.05.pdf} | Bin .../{emd-rounds-0.pdf => 6-emd-rounds-0.pdf} | Bin notes/images/{original.pdf => 6-original.pdf} | Bin notes/images/{smoothed.pdf => 6-smoothed.pdf} | Bin .../{fisher-plane.pdf => 7-fisher-plane.pdf} | Bin .../{fisher-proj.pdf => 7-fisher-proj.pdf} | Bin notes/images/{fisher.png => 7-fisher.png} | Bin .../{percep-plane.pdf => 7-percep-plane.pdf} | Bin .../{percep-proj.pdf => 7-percep-proj.pdf} | Bin notes/images/{points.pdf => 7-points.pdf} | Bin notes/images/fraun-conv-0.05.pdf | Bin 16946 -> 0 bytes notes/images/fraun-conv-0.5.pdf | Bin 16999 -> 0 bytes notes/images/fraun-conv-1.pdf | Bin 16668 -> 0 bytes notes/images/fraun-fft-0.05.pdf | Bin 17587 -> 0 bytes notes/images/fraun-fft-0.5.pdf | Bin 17583 -> 0 bytes notes/images/fraun-fft-1.pdf | Bin 17215 -> 0 bytes notes/images/fraun-noise-fft.pdf | Bin 16696 -> 0 bytes notes/images/fraun-noise-rl.pdf | Bin 16648 -> 0 bytes notes/images/fraun-original.pdf | Bin 15537 -> 0 bytes notes/images/fraun-rl-0.05.pdf | Bin 17817 -> 0 bytes notes/images/fraun-rl-0.5.pdf | Bin 17786 -> 0 bytes notes/images/fraun-rl-1.pdf | Bin 17407 -> 0 bytes notes/images/landau-comp.pdf | Bin 9738 -> 0 bytes notes/images/nome.pdf | Bin 12040 -> 0 bytes notes/images/plot.pdf | Bin 12979 -> 0 bytes notes/images/prova.pdf | Bin 12040 -> 0 bytes notes/sections/1.md | 8 +- notes/sections/2.md | 2 +- notes/sections/3.md | 8 +- notes/sections/4.md | 6 +- notes/sections/5.md | 6 +- notes/sections/6.md | 171 ++++++++++-------- notes/sections/7.md | 12 +- 54 files changed, 114 insertions(+), 99 deletions(-) rename notes/images/{landau-histo.pdf => 1-landau-histo.pdf} (100%) rename notes/images/{landau-kde.pdf => 1-landau-kde.pdf} (100%) rename notes/images/{landau-small.pdf => 1-landau-small.pdf} (100%) rename notes/images/{landau.pdf => 1-landau.pdf} (100%) rename notes/images/{gamma-area.png => 2-gamma-area.png} (100%) rename notes/images/{histo-i-F.pdf => 3-histo-i-F.pdf} (100%) rename notes/images/{histo-i-u.pdf => 3-histo-i-u.pdf} (100%) rename notes/images/{histo-p-F.pdf => 3-histo-p-F.pdf} (100%) rename notes/images/{histo-p-u.pdf => 3-histo-p-u.pdf} (100%) rename notes/images/{dip.pdf => 4-dip.pdf} (100%) rename notes/images/{expected.pdf => 4-expected.pdf} (100%) rename notes/images/{fit.pdf => 4-fit.pdf} (100%) rename notes/images/{MC_MC.pdf => 5-MC_MC.pdf} (100%) rename notes/images/{MC_MC_MI.pdf => 5-MC_MC_MI.pdf} (100%) rename notes/images/{MC_MI_VE.pdf => 5-MC_MI_VE.pdf} (100%) rename notes/images/{emd-hist-0.005.pdf => 6-emd-hist-0.005.pdf} (100%) rename notes/images/{emd-hist-0.01.pdf => 6-emd-hist-0.01.pdf} (100%) rename notes/images/{emd-hist-0.05.pdf => 6-emd-hist-0.05.pdf} 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\log(t) -xt} \sin (\pi t) $$ -![Landau distribution.](images/landau-small.pdf){width=50%} +![Landau distribution.](images/1-landau-small.pdf){width=50%} The GNU Scientific Library (GSL) provides a number of functions for generating random variates following tens of probability distributions. Thus, the function @@ -20,7 +20,7 @@ an histogram and plotted with matplotlib. The result is shown in @fig:landau. ![Example of N = 10'000 points generated with the `gsl_ran_landau()` function and plotted in a 100-bins histogram ranging from -20 to -80.](images/landau-histo.pdf){#fig:landau} +80.](images/1-landau-histo.pdf){#fig:landau} ## Randomness testing of the generated sample @@ -102,7 +102,7 @@ PDF, very few parameters can be easily checked: mode, median and full width at half maximum (FWHM). ![Landau distribution with emphatized mode $m_e$ and - FWHM = ($x_+ - x_-$).](images/landau.pdf) + FWHM = ($x_+ - x_-$).](images/1-landau.pdf) \begin{figure} \hypertarget{fig:parameters}{% @@ -304,7 +304,7 @@ $$ ![Example of a Moyal distribution density obtained by the KDE method. The rug plot shows the original sample used in the reconstruction. The 0.6 factor - compensate for the otherwise peak height reduction.](images/landau-kde.pdf) + compensate for the otherwise peak height reduction.](images/1-landau-kde.pdf) On the other hand, obtaining a good estimate of the FWHM from a sample is much more difficult. In principle, it could be measured by binning the data and diff --git a/notes/sections/2.md b/notes/sections/2.md index d218f49..3432030 100644 --- a/notes/sections/2.md +++ b/notes/sections/2.md @@ -26,7 +26,7 @@ numbers, packed in series, partial sums or infinite products. Thus, the efficiency of the methods lies on how quickly they converge to their limit. ![The area of the blue region converges to the Euler–Mascheroni -constant.](images/gamma-area.png){#fig:gamma width=7cm} +constant.](images/2-gamma-area.png){#fig:gamma width=7cm} ## Computing the constant diff --git a/notes/sections/3.md b/notes/sections/3.md index 1a74ffe..16cca17 100644 --- a/notes/sections/3.md +++ b/notes/sections/3.md @@ -45,14 +45,14 @@ easily generated by the GSL function `gsl_rng_uniform()`.
![Uniformly distributed points with $\theta$ evenly distributed between - 0 and $\pi$.](images/histo-i-u.pdf){width=50%} + 0 and $\pi$.](images/3-histo-i-u.pdf){width=50%} ![Points uniformly distributed on a spherical - surface.](images/histo-p-u.pdf){width=50%} + surface.](images/3-histo-p-u.pdf){width=50%} ![Sample generated according to $F$ with $\theta$ evenly distributed between - 0 and $\pi$.](images/histo-i-F.pdf){width=50%} + 0 and $\pi$.](images/3-histo-i-F.pdf){width=50%} ![Sample generated according to $F$ with $\theta$ properly - distributed.](images/histo-p-F.pdf){width=50%} + distributed.](images/3-histo-p-F.pdf){width=50%} Examples of samples. On the left, points with $\theta$ evenly distributed between 0 and $\pi$; on the right, points with $\theta$ properly distributed. diff --git a/notes/sections/4.md b/notes/sections/4.md index 6c5f081..e0281a5 100644 --- a/notes/sections/4.md +++ b/notes/sections/4.md @@ -150,7 +150,7 @@ $$ {#eq:dip} Namely: ![Plot of the expected distribution with - $P_{\text{max}} = 10$.](images/expected.pdf){#fig:plot} + $P_{\text{max}} = 10$.](images/4-expected.pdf){#fig:plot} ## Monte Carlo simulation @@ -194,7 +194,7 @@ following. At first $S_j = 0 \wedge \text{num}_j = 0 \, \forall \, j$, then: For $P_{\text{max}} = 10$ and $n = 50$, the following result was obtained: -![Sampled points histogram.](images/dip.pdf) +![Sampled points histogram.](images/4-dip.pdf) In order to check whether the expected distribution (@eq:dip) properly matches the produced histogram, a chi-squared minimization was applied. Being a simple @@ -232,4 +232,4 @@ distribution. In @fig:fit, the fit function superimposed on the histogram is shown. ![Fitted sampled data. $P^{\text{oss}}_{\text{max}} = 10.005 - \pm 0.018$, $\chi_r^2 = 0.071$.](images/fit.pdf){#fig:fit} + \pm 0.018$, $\chi_r^2 = 0.071$.](images/4-fit.pdf){#fig:fit} diff --git a/notes/sections/5.md b/notes/sections/5.md index 27d78d4..d5fdbe4 100644 --- a/notes/sections/5.md +++ b/notes/sections/5.md @@ -77,7 +77,7 @@ given. ![Estimated values of $I$ obatined by Plain MC technique with different number of function calls; logarithmic scale; errorbars showing their estimated uncertainties. As can be seen, the process does a sort o seesaw -around the correct value.](images/MC_MC.pdf){#fig:MC} +around the correct value.](images/5-MC_MC.pdf){#fig:MC} --------------------------------------------------------------------------- calls $I^{\text{oss}}$ $\sigma$ diff @@ -220,7 +220,7 @@ to give an overall result and an estimate of its error [@sayah19]. ![Estimations $I^{\text{oss}}$ of the integral $I$ obtained for the three implemented method for different values of function calls. Errorbars - showing their estimated uncertainties.](images/MC_MC_MI.pdf){#fig:MI} + showing their estimated uncertainties.](images/5-MC_MC_MI.pdf){#fig:MI} Results for this particular sample are shown in black in @fig:MI and some of them are listed in @tbl:MI. Except for the first very little number of calls, @@ -400,7 +400,7 @@ calls, the difference is smaller than \SI{1e-10}{}. ![Only the most accurate results are shown in order to stress the differences between VEGAS (in gray) and MISER (in black) methods - results.](images/MC_MI_VE.pdf){#fig:MI_VE} + results.](images/5-MC_MI_VE.pdf){#fig:MI_VE} In conclusion, between a plain Monte Carlo technique, stratified sampling and importance sampling, the last one turned out to be the most powerful mean to diff --git a/notes/sections/6.md b/notes/sections/6.md index b0a174f..91d1cc5 100644 --- a/notes/sections/6.md +++ b/notes/sections/6.md @@ -87,13 +87,14 @@ The so obtained sample was binned and stored in a histogram with a customizable number $n$ of bins (default set $n = 150$) ranging from $\theta = 0$ to $\theta = \pi/2$ bacause of the system symmetry. In @fig:original an example is shown. -![Example of intensity histogram.](images/original.pdf){#fig:original} +![Example of intensity histogram.](images/6-original.pdf){#fig:original} ## Gaussian convolution {#sec:convolution} -The sample has then to be smeared with a Gaussian function with the aim to -recover the original sample afterwards, implementing a deconvolution routine. +In order to simulate the instrumentation response, the sample was then to be +smeared with a Gaussian function with the aim to recover the original sample +afterwards, implementing a deconvolution routine. For this purpose, a 'kernel' histogram with an even number $m$ of bins and the same bin width of the previous one, but a smaller number of them ($m \sim 6\% \, n$), was created according to a Gaussian distribution with mean $\mu$, @@ -102,10 +103,10 @@ descussed lately. Then, the original histogram was convolved with the kernel in order to obtain the smeared signal. As an example, the result obtained for $\sigma = \Delta \theta$, where $\Delta \theta$ is the bin width, is shown in @fig:convolved. -As expected, the smeared signal looks smoother with respect to the original -one. +The smeared signal looks smoother with respect to the original one: the higher +$\sigma$, the greater the smoothness. -![Convolved signal.](images/smoothed.pdf){#fig:convolved} +![Convolved signal.](images/6-smoothed.pdf){#fig:convolved} The convolution was implemented as follow. Consider the definition of convolution of two functions $f(x)$ and $g(x)$: @@ -328,16 +329,25 @@ function'. Consider the problem of estimating the frequeny distribution $f(\xi)$ of a variable $\xi$ when the available measure is a sample {$x_i$} of points -dronwn not by $f(x)$ but by an other function $\phi(x)$ such that: +dronwn not by $f(x)$ but by another function $\phi(x)$ such that: $$ \phi(x) = \int d\xi \, f(\xi) P(x | \xi) -$$ +$$ {#eq:conv} where $P(x | \xi) \, d\xi$ is the probability (presumed known) that $x$ falls -in the interval $(x, x + dx)$ when $\xi = \xi$. An example of this problem is -precisely that of correcting an observed distribution $\phi(x)$ for the effect -of observational errors, which are represented by the function $P (x | \xi)$, -called point spread function. +in the interval $(x, x + dx)$ when $\xi = \xi$. If the so-colled point spread +function $P(x | \xi)$ follows a normal distribution with variance $\sigma$, +namely: +$$ + P(x | \xi) = \frac{1}{\sqrt{2 \pi} \sigma} + \exp \left( - \frac{(x - \xi)^2}{2 \sigma^2} \right) +$$ + +then, @eq:conv becomes a convolution and finding $f(\xi)$ turns out to be a +deconvolution. +An example of this problem is precisely that of correcting an observed +distribution $\phi(x)$ for the effect of observational errors, which are +represented by the function $P (x | \xi)$. Let $Q(\xi | x) d\xi$ be the probability that $\xi$ comes from the interval $(\xi, \xi + d\xi)$ when the measured quantity is $x = x$. The probability that @@ -382,22 +392,18 @@ $$ P(x | \xi) $$ {#eq:solution} -If the spread function $P(x | \xi)$ follows a normal distribution with variance -$\sigma$, namely: -$$ - P(x | \xi) = \frac{1}{\sqrt{2 \pi} \sigma} - \exp \left( - \frac{(x - \xi)^2}{2 \sigma^2} \right) -$$ - -then, @eq:solution can be rewritten in terms of convolutions: +When the spread function $P(x | \xi)$ is Gaussian, @eq:solution can be +rewritten in terms of convolutions: $$ f^{t + 1} = f^{t}\left( \frac{\phi}{{f^{t}} * P} * P^{\star} \right) $$ + where $P^{\star}$ is the flipped point spread function [@lucy74]. -In this special case, the Gaussian kernel stands for the point spread function -and, dealing with discrete values, the division and multiplication are element -wise and the convolution is to be carried out as described in @sec:convolution. +In this special case, the Gaussian kernel which was convolved with the original +histogram stands for the point spread function and, dealing with discrete +values, the division and multiplication are element wise and the convolution is +to be carried out as described in @sec:convolution. When implemented, this method results in an easy step-wise routine: - create a flipped copy of the kernel; @@ -405,53 +411,16 @@ When implemented, this method results in an easy step-wise routine: - compute the convolutions, the product and the division at each step; - proceed until a given number of reiterations is achieved. -In this case, the zero-order was set $f(\xi) = 0.5 \, \forall \, \xi$. +In this case, the zero-order was set $f(\xi) = 0.5 \, \forall \, \xi$. Different +number of iterations where tested. Results are discussed in +@sec:conv_results. -## Results comparison {#sec:conv_Results} +## The earth mover's distance -In [@fig:results1; @fig:results2; @fig:results3] the results obtained for three -different $\sigma$s are shown. The tested values are $\Delta \theta$, $0.5 \, -\Delta \theta$ and $0.05 \, \Delta \theta$, where $\Delta \theta$ is the bin -width of the original histogram, which is the one previously introduced in -@fig:original. In each figure, the convolved signal is shown above, the -histogram deconvolved with the FFT method is in the middle and the one -deconvolved with RL is located below. - -As can be seen, increasing the value of $\sigma$ implies a stronger smoothing of -the curve. The FFT deconvolution process seems not to be affected by $\sigma$ -amplitude changes: it always gives the same outcome, which is exactly the -original signal. In fact, the FFT is the analitical result of the deconvolution. -In the real world, it is unpratical, since signals are inevitably blurred by -noise. -The same can't be said about the RL deconvolution, which, on the other hand, -looks heavily influenced by the variance magnitude: the greater $\sigma$, the -worse the deconvolved result. In fact, given the same number of steps, the -deconvolved signal is always the same 'distance' far form the convolved one: -if it very smooth, the deconvolved signal is very smooth too and if the -convolved is less smooth, it is less smooth too. - -The original signal is shown below for convenience. - - - -It was also implemented the possibility to add a Poisson noise to the -convolved histogram to check weather the deconvolution is affected or not by -this kind of interference. It was took as an example the case with $\sigma = -\Delta \theta$. In @fig:poisson the results are shown for both methods when a -Poisson noise with mean $\mu = 50$ is employed. -In both cases, the addition of the noise seems to partially affect the -deconvolution. When the FFT method is applied, it adds little spikes nearly -everywhere on the curve and it is particularly evident on the edges, where the -expected data are very small. On the other hand, the Richardson-Lucy routine is -less affected by this further complication. - -In order to quantify the similarity of a deconvolution outcome with the original -signal, a null hypotesis test was made up. -Likewise in @sec:Landau, the original sample was treated as a population from -which other samples of the same size were sampled with replacements. For each -new sample, the earth mover's distance with respect to the original signal was -computed. +With the aim of comparing the two deconvolution methods, the similarity of a +deconvolved outcome with the original signal was quantified using the earth +mover's distance. In statistics, the earth mover's distance (EMD) is the measure of distance between two probability distributions [@cock41]. Informally, the distributions @@ -460,22 +429,36 @@ a region and the EMD is the minimum cost of turning one pile into the other, where the cost is the amount of dirt moved times the distance by which it is moved. It is valid only if the two distributions have the same integral, that is if the two piles have the same amount of dirt. -Computing the EMD is based on a solution to the well-known transportation -problem, which can be formalized as follows. +Computing the EMD is based on a solution to the transportation problem, which +can be formalized as follows. -Consider two vectors: +Consider two vectors $P$ and $Q$ which represent the two probability +distributions whose EMD has to be measured: $$ - P = \{ (p_1, w_{p1}) \dots (p_n, w_{pm}) \} \et + P = \{ (p_1, w_{p1}) \dots (p_m, w_{pm}) \} \et Q = \{ (q_1, w_{q1}) \dots (q_n, w_{qn}) \} $$ -where $p_i$ and $q_i$ are the 'values' and $w_{pi}$ and $w_{qi}$ are their -weights. The entries $d_{ij}$ of the ground distance matrix $D_{ij}$ are -defined as the distances between $p_i$ and $q_j$. -The aim is to find the flow $F =$ {$f_{ij}$}, where $f_{ij}$ is the flow -between $p_i$ and $p_j$ (which would be the quantity of moved dirt), which -minimizes the cost $W$: +L'istogramma P deve essere distrutto in modo tale da ottenere l'istogramma Q, +che in partenza è vuoto ma so che vorrò avere w_qj in ogni bin che sta alla +posizione qj. +- sposto solo da P a Q +- sposto non più di ogni ingresso di P +- ottengo non più di ogni ingreddo di Q +- sposto tutto quello che posso: o ottengo tutto Q o ho finito P + +e non devono venire uguali, quindi! + + + + +where $p_i$ and $q_i$ are the 'values' (that is, the location of the dirt) and +$w_{pi}$ and $w_{qi}$ are the 'weights' (that is, the quantity of dirt). A +ground distance matrix $D_{ij}$ is defined such as its entries $d_{ij}$ are the +distances between $p_i$ and $q_j$. The aim is to find the flow matrix $F_{ij}$, +where each entry $f_{ij}$ is the flow from $p_i$ to $q_j$ (which would be +the quantity of moved dirt), which minimizes the cost $W$: $$ W (P, Q, F) = \sum_{i = 1}^m \sum_{j = 1}^n f_{ij} d_{ij} @@ -486,7 +469,7 @@ with the constraints: \begin{align*} &f_{ij} \ge 0 \hspace{15pt} &1 \le i \le m \wedge 1 \le j \le n \\ &\sum_{j = 1}^n f_{ij} \le w_{pi} &1 \le i \le m \\ - &\sum_{j = 1}^m f_{ij} \le w_{qj} &1 \le j \le n + &\sum_{i = 1}^m f_{ij} \le w_{qj} &1 \le j \le n \end{align*} $$ \sum_{j = 1}^n f_{ij} \sum_{j = 1}^m f_{ij} \le w_{qj} @@ -555,3 +538,35 @@ the original one cannot be disporoved if its distance from the original signal is grater than \textcolor{red}{value}. \textcolor{red}{counts} + + +## Results comparison {#sec:conv_results} + +As can be seen, increasing the value of $\sigma$ implies a stronger smoothing of +the curve. The FFT deconvolution process seems not to be affected by $\sigma$ +amplitude changes: it always gives the same outcome, which is exactly the +original signal. In fact, the FFT is the analitical result of the deconvolution. +In the real world, it is unpratical, since signals are inevitably blurred by +noise. +The same can't be said about the RL deconvolution, which, on the other hand, +looks heavily influenced by the variance magnitude: the greater $\sigma$, the +worse the deconvolved result. In fact, given the same number of steps, the +deconvolved signal is always the same 'distance' far form the convolved one: +if it very smooth, the deconvolved signal is very smooth too and if the +convolved is less smooth, it is less smooth too. + +The original signal is shown below for convenience. + + + +It was also implemented the possibility to add a Poisson noise to the +convolved histogram to check weather the deconvolution is affected or not by +this kind of interference. It was took as an example the case with $\sigma = +\Delta \theta$. In @fig:poisson the results are shown for both methods when a +Poisson noise with mean $\mu = 50$ is employed. +In both cases, the addition of the noise seems to partially affect the +deconvolution. When the FFT method is applied, it adds little spikes nearly +everywhere on the curve and it is particularly evident on the edges, where the +expected data are very small. On the other hand, the Richardson-Lucy routine is +less affected by this further complication. + diff --git a/notes/sections/7.md b/notes/sections/7.md index 7d0b3ba..e37604c 100644 --- a/notes/sections/7.md +++ b/notes/sections/7.md @@ -40,7 +40,7 @@ An example of the two samples is shown in @fig:points. ![Example of points sorted according to two Gaussian with the given parameters. Noise points in pink and signal points -in yellow.](images/points.pdf){#fig:points} +in yellow.](images/7-points.pdf){#fig:points} Assuming not to know how the points were generated, a model of classification must then be implemented in order to assign each point to the right class @@ -96,7 +96,7 @@ maximization, it can be found that $w \propto (m_2 − m_1)$. resulting from projection onto the line joining the class means: note that there is considerable overlap in the projected space. The right plot shows the corresponding projection based on the Fisher linear discriminant, showing the -greatly improved classes separation.](images/fisher.png){#fig:overlap} +greatly improved classes separation.](images/7-fisher.png){#fig:overlap} There is still a problem with this approach, however, as illustrated in @fig:overlap: the two classes are well separated in the original 2D space but @@ -195,9 +195,9 @@ The projection of the points was accomplished by the use of the function this case were the weight vector and the position of the point to be projected.
-![View from above of the samples.](images/fisher-plane.pdf){height=5.7cm} +![View from above of the samples.](images/7-fisher-plane.pdf){height=5.7cm} ![Gaussian of the samples on the projection - line.](images/fisher-proj.pdf){height=5.7cm} + line.](images/7-fisher-proj.pdf){height=5.7cm} Aerial and lateral views of the projection direction, in blue, and the cut, in red. @@ -265,9 +265,9 @@ $$ $$
-![View from above of the samples.](images/percep-plane.pdf){height=5.7cm} +![View from above of the samples.](images/7-percep-plane.pdf){height=5.7cm} ![Gaussian of the samples on the projection - line.](images/percep-proj.pdf){height=5.7cm} + line.](images/7-percep-proj.pdf){height=5.7cm} Aerial and lateral views of the projection direction, in blue, and the cut, in red.