菜鸟的菜地: 五月 2011

2011年5月9日星期一

basic staff

1 central slice theory

1.1 expression of projection

$P(r,\theta) = \int_{\infty}^{-\infty} f(x,y)\delta(xcos\theta + ysin\theta - r)dxdy$

1.2 Central slice theory

目标：证明对投影的一维傅里叶变换得到的系数等于对应投影的二维傅里叶变换系数

Goal : prove that the 1D Fourier transform of the Projection == 2D Fourier transform

$P(w) = F(xcos\theta,ysin\theta)$

$P(w) = \int_{\infty}^{-\infty} P(r,\theta) e^{-2 \pi iwr}dr = \int_{\infty}^{-\infty} \int_{\infty}^{-\infty} \int_{\infty}^{-\infty} f(x,y)\delta(xcos\theta + ysin\theta - r)e^{-2 \pi iwr}dxdydr$

$= \int_{\infty}^{-\infty} \int_{\infty}^{-\infty}f(x,y) \int_{\infty}^{-\infty}\delta(xcos\theta + ysin\theta - r)e^{-2 \pi iwr}drdxdy$

$= \int_{\infty}^{-\infty} \int_{\infty}^{-\infty}f(x,y)e^{-2 \pi i(xcos\theta + y sin\theta)w}dxdy$

$\int_{\infty}^{-\infty} \int_{\infty}^{-\infty}f(x,y)e^{-2 \pi i(xu + y v)}|_{u = wcos\theta,v = wsin\theta} dxdy$

$= F(xcos\theta,ysin\theta)$

2 Filtered Back Projection(FBP) for parallel projection

希望通过反变换投影的一维傅里叶变换来重建图像

$f(x,y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} F(u,v) e ^{2\pi i(xu+yv)}dudv$ // inverse 2d Fourier transform

$= \int_{0}^{2\pi} \int_{0}^{\infty} F(w,\theta) e^{2\pi i (xwcos\theta + ywsin\theta)}wdwd\theta$ // from Cartesian coordinate system to polar coordinate system

$= \int_{0}^{\pi} [\int_{-\infty}^{\infty} F(w,\theta) e^{2\pi i (xcos\theta + ysin\theta)w}|w|dw] d\theta$ //变换积分限F(w, /theta) = F(-w,/theta + /pi).

F(w, /theta) 就是投影的一维傅里叶变换 , 所以中括号里的部分就是对经过ramp filter 滤波后的数据进行一维傅里叶逆变换，第二个积分就是back projection。

2011年5月5日星期四

lp norm

The optimization problem :

$||Ax - b||^{2}_{2} + ||x||^{p}_{p}$

we have

$||Dx||^{p}_{p} = \sum (\sqrt{(x_{i,j+1} - x_{i,j})^{2} + (x_{i+1,j} - x_{i,j})^{2}})^{p}$

and

$\frac{\partial ||Dx||^{p}_{p}}{\partial x_{i,j}} = p/2 [ (x_{i,j+1} - x_{i,j})^{2} + (x_{i+1,j} - x_{i,j})^{2} ]^{ p/2 -1} (-2(x_{i,j+1} - x_{i,j}) - 2(x_{i+1,j} - x_{i,j}) ) + p/2 [ (x_{i,j} - x_{i,j-1})^{2} + (x_{i+1,j-1} - x_{i,j-1})^{2} ]^{ p/2 -1} (2x_{i,j}) + p/2 [ (x_{i+1,j+1} - x_{i-1,j})^{2} + (x_{i,j} - x_{i-1,j})^{2} ]^{ p/2 -1} (2(x_{i,j})$

2011年5月2日星期一

Discrete tomography by convex conconve regularization and d.c. programming

1 The paper introduce a regularization term for discrete tomography.

2 The paper use d.c. programing (difference of convex function) to solve the optimization problem. Here the cost function is a convex function minus a convex function, in other words a convex-concave function.

Two things confused me.

1 prime-dual method

The paper did not describe how to derive the iterations.

A referenced paper has to be read

[a d.c. optimization algorithm for solving the trust region subproblem]

2 the gradient for an indicate function

say the indicate function is

$\delta _{C} (x) = \left\{\begin{matrix} 0, & x \in C \\ + \infty & x \notin C \end{matrix}\right.$

The paper only present that if (xi) = 1

$\partial \delta _{C} (x)$ is 1 , if (xi) = 0

$\partial \delta _{C} (x)$ is -1.

what about other case?

so from (31) to (32) is not clear.

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