Resource Garage

Friday, October 11, 2013

Hardware Related

Sensor
ADNS3080 Optical Flow Sensor， Link， Optical Mouse based optical flow sensor. Choosen in paper

An Autonomous UAV with an Optical Flow Sensor for Positioning and Navigation

Tuesday, September 10, 2013

Thursday, August 15, 2013

Image stiching in OpenCV

Reference to http://ramsrigoutham.com/2012/11/22/panorama-image-stitching-in-opencv/

Main steps of the code:

Load two images;
Convert to gray scale;
Using SURF detector to find SURF descriptor in both images;
matching the SURF descriptor using FLANN Matcher;
Postprocessing matches to find good matches
Using RANSAC to estimate the Homography matrix using the matched SURF descriptors;
Warping the images based on the homography matrix

The image shows the definition of Homography which transforms 2d Planar point to the image plane.

The following image is shows the initial image and the matched SURF features of the two. Homography is derived from the left image to the right image.

Stiching image result.

Source Code

Tuesday, August 6, 2013

Resource to learn optimization

http://www.quora.com/Mathematical-Optimization/What-are-some-good-resources-to-learn-about-optimization#

Theory behind MPC

MPC is based on iterative, finite horizon optimization of a plant model. At time

t

the current plant state is sampled and a cost minimizing control strategy is computed (via a numerical minimization algorithm) for a relatively short time horizon in the future: $[t,t+T]$ . Specifically, an online or on-the-fly calculation is used to explore state trajectories that emanate from the current state and find (via the solution of Euler-Lagrange equations) a cost-minimizing control strategy until time $t+T$ . Only the first step of the control strategy is implemented, then the plant state is sampled again and the calculations are repeated starting from the current state, yielding a new control and new predicted state path. The prediction horizon keeps being shifted forward and for this reason MPC is also called receding horizon control.

Sunday, May 26, 2013

Hessian Matrix and Jacobian Matrix

Hessian Matrix

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a function. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants".

Given the real-valued function

$f(x_1, x_2, \dots, x_n),\,\!$

if all second partial derivatives of f exist and are continuous over the domain of the function, then the Hessian matrix of f is

$H(f)_{ij}(\mathbf x) = D_i D_j f(\mathbf x)\,\!$

where x = (x₁, x₂, ..., x_n) and D_i is the differentiation operator with respect to the ith argument. Thus

$H(f) = \begin{bmatrix} \dfrac{\partial^2 f}{\partial x_1^2} & \dfrac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_1\,\partial x_n} \\[2.2ex] \dfrac{\partial^2 f}{\partial x_2\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_2^2} & \cdots & \dfrac{\partial^2 f}{\partial x_2\,\partial x_n} \\[2.2ex] \vdots & \vdots & \ddots & \vdots \\[2.2ex] \dfrac{\partial^2 f}{\partial x_n\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_n^2} \end{bmatrix}.$

Because f is often clear from context, $H(f)(\mathbf x)$ is frequently abbreviated to $H(\mathbf x)$ .

The Hessian matrix is related to the Jacobian matrix by $H(f)(\mathbf x)$ = $J(\nabla \! f)(\mathbf x)$ .

The determinant of the above matrix is also sometimes referred to as the Hessian.^[1]

Hessian matrices are used in large-scale optimization problems within Newton-type methods because they are the coefficient of the quadratic term of a local Taylor expansion of a function. That is,

$y=f(\mathbf{x}+\Delta\mathbf{x})\approx f(\mathbf{x}) + J(\mathbf{x})\Delta \mathbf{x} +\frac{1}{2} \Delta\mathbf{x}^\mathrm{T} H(\mathbf{x}) \Delta\mathbf{x}$

where J is the Jacobian matrix, which is a vector (the gradient) for scalar-valued functions. The full Hessian matrix can be difficult to compute in practice; in such situations, quasi-Newton algorithms have been developed that use approximations to the Hessian. The best-known quasi-Newton algorithm is the BFGS algorithm

Jacobian Matrix

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. Specifically, suppose $F : \mathbb{R}^n \rightarrow \mathbb{R}^m$ is a function (which takes as input real n-tuples and produces as output real m-tuples). Such a function is given by m real-valued component functions, $F_1(x_1,\ldots,x_n),\ldots,F_m(x_1,\ldots,x_n)$ . The partial derivatives of all these functions with respect to the variables $x_1,\ldots,x_n$ (if they exist) can be organized in an m-by-n matrix, the Jacobian matrix $J$ of $F$ , as follows:

$J=\begin{bmatrix} \dfrac{\partial F_1}{\partial x_1} & \cdots & \dfrac{\partial F_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial F_m}{\partial x_1} & \cdots & \dfrac{\partial F_m}{\partial x_n} \end{bmatrix}.$

Friday, May 3, 2013

Implementation of KalmanFilter in Opencv

Two Functions used: gemm , solve
gemm: Performs generalized matrix multiplication.

C++: void gemm(InputArray src1, InputArray src2, double alpha, InputArray src3, double gamma, OutputArray dst, intflags=0 )
The function performs generalized matrix multiplication similar to the gemm functions in BLAS level 3. For example, gemm(src1,src2, alpha, src3, beta, dst, GEMM_1_T + GEMM_3_T) corresponds to

$\texttt{dst} = \texttt{alpha} \cdot \texttt{src1} ^T \cdot \texttt{src2} + \texttt{beta} \cdot \texttt{src3} ^T$

solve: Solves one or more linear systems or least-squares problems.

C++: bool solve(InputArray src1, InputArray src2, OutputArray dst, int flags=DECOMP_LU)

solution (matrix inversion) method.

DECOMP_LU Gaussian elimination with optimal pivot element chosen.

DECOMP_CHOLESKY Cholesky $LL^T$ factorization; the matrix src1 must be symmetrical and positively defined.

DECOMP_EIG eigenvalue decomposition; the matrix src1 must be symmetrical.

DECOMP_SVD singular value decomposition (SVD) method; the system can be over-defined and/or the matrix src1 can be singular.

DECOMP_QR QR factorization; the system can be over-defined and/or the matrix src1 can be singular.

DECOMP_NORMAL while all the previous flags are mutually exclusive, this flag can be used together with any of the previous; it means that the normal equations $\texttt{src1}^T\cdot\texttt{src1}\cdot\texttt{dst}=\texttt{src1}^T\texttt{src2}$ are solved instead of the original system $\texttt{src1}\cdot\texttt{dst}=\texttt{src2}$ .

The function solve solves a linear system or least-squares problem (the latter is possible with SVD or QR methods, or by specifying the flag DECOMP_NORMAL ):

$\texttt{dst} = \arg \min _X \| \texttt{src1} \cdot \texttt{X} - \texttt{src2} \|$