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{1487} 
ref: 0
tags: adaptive optics sensorless retina fluorescence imaging optimization zernicke polynomials
date: 11152019 02:51 gmt
revision:0
[head]


PMID26819812 Wavefront sensorless adaptive optics fluorescence biomicroscope for in vivo retinal imaging in mice
 
{1341} 
ref: 0
tags: image registration optimization camera calibration sewing machine
date: 07152016 05:04 gmt
revision:20
[19] [18] [17] [16] [15] [14] [head]


Recently I was tasked with converting from image coordinates to real world coordinates from stereoscopic cameras mounted to the endeffector of a robot. The end goal was to let the user (me!) click on points in the image, and have the robot record that position & ultimately move to it. The overall strategy is to get a set of points in both image and RW coordinates, then fit some sort of model to the measured data. I began by printing out a grid of (hopefully evenlyspaced and perpendicular) lines via a laserprinter; spacing was ~1.1 mm. This grid was manually aligned to the axes of robot motion by moving the robot along one axis & checking that the lines did not jog. The images were modeled as a grating with quadratic phase in $u,v$ texture coordinates: $p_h(u,v) = sin((a_h u/1000 + b_h v/1000 + c_h)v + d_h u + e_h v + f_h) + 0.97$ (1) $p_v(u,v) = sin((a_v u/1000 + b_v v/1000 + c_v)u + d_v u + e_v v + f_v) + 0.97$ (2) $I(u,v) = 16 p_h p_v / ( \sqrt{ 2 + 16 p_h^2 + 16 p_v^2})$ (3) The 1000 was used to make the parameter search distribution more spherical; $c_h,c_v$ were bias terms to seed the solver; 0.97 was a dutycycle term fit by inspection to the image data; (3) is a modified sigmoid. $I$ was then optimized over the parameters using a GPUaccelerated (CUDA) nonlinear stochastic optimization: $(a_h,b_h,d_h,e_h,f_h  a_v,b_v,d_v,e_v,f_v) = Argmin \sum_u \sum_v (I(u,v)  Img(u,v))^2$ (4) Optimization was carried out by drawing parameters from a normal distribution with a diagonal covariance matrix, set by inspection, and mean iteratively set to the best solution; horizontal and vertical optimization steps were separable and carried out independently. The equation (4) was sampled 18k times, and equation (3) 34 billion times per frame. Hence the need for GPU acceleration. This yielded a set of 10 parameters (again, $c_h$ and $c_v$ were bias terms and kept constant) which modeled the data (e.g. grid lines) for each of the two cameras. This process was repeated every 0.1 mm from 0  20 mm height (z) from the target grid, resulting in a sampled function for each of the parameters, e.g. $a_h(z)$ . This required 13 trillion evaluations of equation (3). Now, the task was to use this model to generate the forward and reverse transform from image to world coordinates; I approached this by generating a data set of the grid intersections in both image and world coordinates. To start this process, the known image origin $u_{origin}_{z=0},v_{origin}_{z=0}$ was used to find the corresponding roots of the periodic axillary functions $p_h,p_v$ : $\frac{3 \pi}{ 2} + 2 \pi n_h = a_h u v/1000 + b_h v^2/1000 + (c_h + e_h)v + d_h u + f_h$ (5) $\frac{3 \pi}{ 2} + 2 \pi n_h = a_v u^2/1000 + b_v u v/1000 + (c_v + d_v)u + e_v v + f_v$ (6) Or .. $n_h = round( (a_h u v/1000 + b_h v^2/1000 + (c_h + e_h)v + d_h u + f_h  \frac{3 \pi}{ 2} ) / (2 \pi )$ (7) $n_v = round( (a_v u^2/1000 + b_v u v/1000 + (c_v + d_v)u + e_v v + f_v  \frac{3 \pi}{ 2} ) / (2 \pi)$ (8) From this, we get variables $n_{h,origin}_{z=0} and n_{v,origin}_{z=0}$ which are the offsets to align the sine functions $p_h,p_v$ with the physical origin. Now, the reverse (world to image) transform was needed, for which a twostage newton scheme was used to solve equations (7) and (8) for $u,v$ . Note that this is an equation of phase, not image intensity  otherwise this direct method would not work! First, the equations were linearized with three steps of (911) to get in the right ballpark: $u_0 = 640, v_0 = 360$ $n_h = n_{h,origin}_{z} + [30 .. 30] , n_v = n_{v,origin}_{z} + [20 .. 20]$ (9) $B_i = {\left[ \begin{matrix} \frac{3 \pi}{ 2} + 2 \pi n_h  a_h u_i v_i / 1000  b_h v_i^2  f_h \\ \frac{3 \pi}{ 2} + 2 \pi n_v  a_v u_i^2 / 1000  b_v u_i v_i  f_v \end{matrix} \right]}$ (10) $A_i = {\left[ \begin{matrix} d_h && c_h + e_h \\ c_v + d_v && e_v \end{matrix} \right]}$ and ${\left[ \begin{matrix} u_{i+1} \\ v_{i+1} \end{matrix} \right]} = mldivide(A_i,B_i)$ (11) where mldivide is the Matlab operator. Then three steps with the full Jacobian were made to attain accuracy: $J_i = {\left[ \begin{matrix} a_h v_i / 1000 + d_h && a_h u_i / 1000 + 2 b_h v_i / 1000 + c_h + e_h \\ 2 a_v u_i / 1000 + b_v v_i / 1000 + c_v + d_v && b_v u_i / 1000 + e_v \end{matrix} \right]}$ (12) $K_i = {\left[ \begin{matrix} a_h u_i v_i/1000 + b_h v_i^2/1000 + (c_h+e_h) v_i + d_h u_i + f_h  \frac{3 \pi}{ 2}  2 \pi n_h \\ a_v u_i^2/1000 + b_v u_i v_i/1000 + (c_v+d_v) u_i + e_v v + f_v  \frac{3 \pi}{ 2}  2 \pi n_v \end{matrix} \right]}$ (13) ${\left[ \begin{matrix} u_{i+1} \\ v_{i+1} \end{matrix} \right]} = {\left[ \begin{matrix} u_i \\ v_i \end{matrix} \right]}  J^{1}_i K_i$ (14) Solutions $(u,v)$ were verified by plugging back into equations (7) and (8) & verifying $n_h, n_v$ were the same. Inconsistent solutions were discarded; solutions outside the image space $[0, 1280),[0, 720)$ were also discarded. The process (10)  (14) was repeated to tile the image space with gird intersections, as indicated in (9), and this was repeated for all $z$ in $(0 .. 0.1 .. 20)$ , resulting in a large (74k points) dataset of $(u,v,n_h,n_v,z)$ , which was converted to full realworld coordinates based on the measured spacing of the grid lines, $(u,v,x,y,z)$ . Between individual z steps, $n_{h,origin} n_{v,origin}$ was reestimated to minimize (for a current $z'$ ): $(u_{origin}_{z' + 0.1}  u_{origin}_{z' + 0.1})^2 + (v_{origin}_{z' + 0.1} + v_{origin}_{z'})^2$ (15) with gridsearch, and the method of equations (914). This was required as the stochastic method used to find original image model parameters was agnostic to phase, and so phase (via parameter $f_{}$ ) could jump between individual $z$ measurements (the origin did not move much between successive measurements, hence (15) fixed the jumps.) To this dataset, a model was fit: ${\left[ \begin{matrix} u \\ v \end{matrix} \right]} = A {\left[ \begin{matrix} 1 && x && y && z && x'^2 && y'^2 && \prime z'^2 && w^2 && x' y' && x' z' && y' z' && x' w && y' w && z' w \end{matrix} \right]}$ (16) Where $x' = \frac{x}{ 10}$ , $y' = \frac{y}{ 10}$ , $z' = \frac{z}{ 10}$ , and $w = \frac{ 20}{20  z}$ . $w$ was introduced as an axillary variable to assist in perspective mapping, ala computer graphics. Likewise, $x,y,z$ were scaled so the quadratic nonlinearity better matched the data. The model (16) was fit using regular linear regression over all rows of the validated dataset. This resulted in a second set of coefficients $A$ for a model of world coordinates to image coordinates; again, the model was inverted using Newton's method (Jacobian omitted here!). These coefficients, one set per camera, were then integrated into the C++ program for displaying video, and the inverse mapping (using closedform matrix inversion) was used to convert mouse clicks to realworld coordinates for robot motor control. Even with the relatively poor wideFOV cameras employed, the method is accurate to $\pm 50\mu m$ , and precise to $\pm 120\mu m$ .  
{1339}  
* Watch the [http://homes.cs.washington.edu/~todorov/index.php?video=MordatchSIGGRAPH12&paper=Mordatch,%20SIGGRAPH%202012 movies! Discovery of complex behaviors through contactinvariant optimization]
 
{1212} 
ref: Nordhausen1994.02
tags: Utah array electrodes optimization
date: 08142014 01:24 gmt
revision:2
[1] [0] [head]


PMID8180807[0] Optimizing recording capabilities of the Utah Intracortical Electrode Array.
____References____
 
{54}  
!:
 
{968} 
ref: Bassett2009.07
tags: Weinberger congnitive efficiency beta band neuroimagaing EEG task performance optimization network size effort
date: 12282011 20:39 gmt
revision:1
[0] [head]


PMID19564605[0] Cognitive fitness of costefficient brain functional networks.
____References____
 
{821} 
ref: work0
tags: differential evolution function optimization
date: 07092010 14:46 gmt
revision:3
[2] [1] [0] [head]


Differential evolution (DE) is an optimization method, somewhat like NeidlerMead or simulated annealing (SA). Much like genetic algorithms, it utilizes a population of solutions and selection to explore and optimize the objective function. However, it instead of perturbing vectors randomly or greedily descending the objective function gradient, it uses the difference between individual population vectors to update hypothetical solutions. See below for an illustration. At my rather cursory reading, this serves to adapt the distribution of hypothetical solutions (or population of solutions, to use the evolutionary term) to the structure of the underlying function to be optimized. Judging from images/821_1.pdf Price and Storn (the inventors), DE works in situations where simulated annealing (which I am using presently, in the robot vision system) fails, and is applicable to higherdimensional problems than simplex methods or SA. The paper tests DE on 100 dimensional problems, and it is able to solve these with on the order of 50k function evaluations. Furthermore, they show that it finds function extrema quicker than stochastic differential equations (SDE, alas from 85) which uses the gradient of the function to be optimized. I'm surprised that this method slipped under my radar for so long  why hasn't anyone mentioned this? Is it because it has no proofs of convergence? has it more recently been superseded? (the paper is from 1997). Yet, I'm pleased because it means that there are also many other algorithms equally clever and novel (and simple?), out their in the literature or waiting to be discovered.  
{652} 
ref: notes0
tags: policy gradient reinforcement learning aibo walk optimization
date: 12092008 17:46 gmt
revision:0
[head]


Policy Gradient Reinforcement Learning for Fast Quadrupedal Locomotion
 
{409} 
ref: bookmark0
tags: optimization function search matlab linear nonlinear programming
date: 08092007 02:21 gmt
revision:0
[head]


http://www.mat.univie.ac.at/~neum/ very nice collection of links!!  
{141} 
ref: learning0
tags: motor control primitives nonlinear feedback systems optimization
date: 002007 0:0
revision:0
[head]


http://hardm.ath.cx:88/pdf/Schaal2003_LearningMotor.pdf not in pubmed. 