The four vector norms that play signi cant roles in the compressedsensing framework arethe $\iota_0$ ,the$\iota _1$, $\iota_2$, and$\iota_\infty$norms, denoted by$\|x\|_0$,$\|x\|_1$,$\|x\|_2$and$\|x\|_\infty$respectively. Given a vector $x\in R^m$. $\|x\|_0$ is equal to the number of the non-zero elements in the vector $x$. $\|x\|_1=\sum_{i=1}^{m}|x_i|$. $\|x\|_2=\sqrt{x_1^2+x_2^2+...+x_m^2}$. $\|x\|_\infty=\max_i |x_i|$. The vector norm$\|x\|_p$for$p=1, 2, 3, ...$is defined as $\|x\|_p=(\sum_{i=1}^m |x_i|^p )^\frac{1}{p}.$ The$p$-norm of vector$x$is implemented asNorm , with the 2 -norm being returned byNorm . These norms have natural generalizations to matrices, inheriting many appealing properties from the vector case. In particular, there is a parallel duality structure. For two rectangular matrices $X\in R ^{m \times n}$, $X\in R ^{m \times n}$ . Define the inner product as $X,Y:= \sum_{i=1}^m \sum_{j=1}^n X_{ij}Y_{ij}= \sqrt{Tr(X^T Y)}.$ The norm associated with this inner product is called the Frobenius (or Hilbert-Schmidt) norm $\|X\|_F$ . The Frobenius norm is also equal to the Euclidean, or $\iota_2$, norm of the vector of singular values, i.e., $\|X\|_F:= \sqrt{X,X}=\sqrt{X^T X}=({\sum_{i=1}^r \delta_i^2})^{\frac{1}{2}}.$ The operator norm (or induced 2 -norm) of a matrix is equal to its largest singular value (i.e., the $\iota_1$ norm of the singular values): $\|X\|:= \delta_ 1(X).$ The nuclear norm of a matrix is equal to the sum of its singular values, i.e., $\|X\|_*:= \sum_{i=1}^r {\delta_i(X)}$, and is alternatively known by several other names including the Schatten 1 -norm, the Ky Fan r-norm, and the trace class norm. Since the singular values are all positive, the nuclear norm is equal to the $\iota_1$ norm of the vector of singular values. These three norms are related by the following inequalities which hold for any matrix $X$ of rank at most $r$: $\|X\| \leq \|X\|_F \leq \|X\|_{*} \leq \sqrt{r}\|X\|_F \leq r\|X\|.$ Table 1: A dictionary relating the concepts of cardinality and rank minimization . parsimony concept cardinality rank Hilbert Space norm Euclidean Frobenius sparsity inducing norm $\iota_1$ nuclear dual norm $\iota_{\infty}$ operator norm additivity disjoint support orthogonal row and column spaces convex optimization linear programming semide nite programming The rank of a matrix $X$ is equals to the number of the singular values, and the singular values are all non-zeros. The $\iota_0$ norm of a vector is the number of the non-zero elements in the vector. In compressed sensing (in sparse representation), the original objective function is: $\min \|x\|_0$ s.t. $Ax=b$. In low rank representation, the original objective function is: $\min rank(Z)$ s.t. $X=XZ$. I'm curious to the relation ship between sparse representation and low rank representation. References: Weisstein, Eric W. "Vector Norm." From MathWorld --A Wolfram Web Resource. http://mathworld.wolfram.com/VectorNorm.html . Recht, Benjamin, Maryam Fazel, and Pablo A. Parrilo. "Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization." SIAM review 52, no. 3 (2010): 471-501. Apendix: Matlab code: from http://www.mathworks.cn/cn/help/matlab/ref/norm.html Vector and matrix norms Syntax n = norm(X,2) n = norm(X) n = norm(X,1) n = norm(X,Inf) n = norm(X,'fro') n = norm(v,p) n = norm(v,Inf) n = norm(v,-Inf) Description Thenormfunction calculates several different types of matrix and vector norms. If the input is a vector or a matrix: n = norm(X,2) returns the 2-norm ofX. n = norm(X) is the same asn = norm(X,2). n = norm(X,1) returns the 1-norm ofX. n = norm(X,Inf) returns the infinity norm ofX. n = norm(X,'fro') returns the Frobenius norm ofX. In addition, when the input is a vectorv: n = norm(v,p) returns the p -norm ofv. The p -norm issum(abs(v).^p)^(1/p). n = norm(v,Inf) returns the largest element ofabs(v). n = norm(v,-Inf) returns the smallest element ofabs(v).
在使用latex做论文时，很多期刊要求使用eps格式或者pdf格式的矢量图形（vector figure），因为矢量图与位图（bitmap）相比，图形放大和缩小都不会失真。本文介绍下使用visio 2010简单制作pdf格式的一般示意图形的矢量图格式： Step One : 使用visio画出示意图，将visio文件的版面设置成适应图形区域，即：在“desgin”菜单中选择“size”，选择“Fit to Drawing”。 然后另存为后缀格式为“Scalable Vector Graphics”. Step Two : 使用visio打开保存好的“Scalable Vector Graphics”格式的文件，再另存为后缀为“PDF”格式。 Step Three : Latex源文件中将“PDF”格式的图形嵌入，方法如下： \usepackage{graphicx} %%%%%%%%near the top of the LaTeX file, just after the documentclass command. \includegraphics {myfig.pdf} （注意：这里latex中需要使用“pdflatex” or “TeXShop”来进行tex源文件编译！）
On the Derivation of Vector Radiative Transfer Equation for Polarized Radiative Transport in Graded Index Media J.M. Zhao(jmzhao@hit.edu.cn), J.Y. Tan(tanjy.hit.edu.cn), L.H. Liu(lhliu@hit.edu.cn) ABSTRACT: Light transport in graded index media follows a curved trajectory determined by the Fermat’s principle. Besides the effect of variation of the refractive index on the transport of radiative intensity, the curved ray trajectory will induce geometrical effects on the transport of polarization ellipse. This paper presents a complete derivation of vector radiative transfer equation for polarized radiation transport in absorption, emission and scattering graded index media. The derivation is based on the analysis of the conserved quantities for polarized light transport along curved trajectory and a novel approach. The obtained transfer equation can be considered as a generalization of the classic vector radiative transfer equation that is only valid for uniform refractive index media. Several variant forms of the transport equation are also presented, which include the form for Stokes parameters defined with a fixed reference and the Eulerian forms in the ray coordinate and in several common orthogonal coordinate systems. This paper has been accepted by JQSRT. doi:10.1016/j.jqsrt.2011.11.002 The preprint can be download here or from the preprint server athttp://arxiv.org/abs/1110.5134 (or by click arXiv:1110.5134v1 )
MDSM Communication Vol.2-No.12 Dec. 26, 1995 WHAT CAN WE DISCOVER FROM 1,2,3 TO 2,3,4? - An Introduction to vegetation dynamic analysis- Revised June 3, 1997 for home page Let’s consider a situation where the vegetation is recovering from a fire event, such as in Yellowstone National Park . The first year, say 1994, the data collected were 1, 2, 3, and the second year, 1995, were 2, 3, 4, respectively. These measurements represent the abundance of trees, shrubs, and grasses. The data will look like this: Tree Shrub Grass (Total) 1994 1 2 3 (6) 1995 2 3 4 (9) What can we discover from 1-2-3 to 2-3-4? 1). All of us can see that the vegetation changed from 1994 to 1995. 2). Most of us will agree that the vegetation has increased, but different species may increase at different speeds. This may be considered a temporal dynamic of vegetation. 3). Some of us would analyze the data and found out that: Trees increased 2/1=200%, Shrubs increased 3/2=150%, but Grasses increased 4/3=133%, and The Total increased 9/6=150%, respectively. The data will look like this: Tree Shrub Grass (Total) 1994 1 2 3 (6) 1995 2 3 4 (9) Increasing ratio 200% 150% 133% (150%) 4A). However, there may not be many of us who agree that the grasses are relatively decreasing, while all the figures appear to be increasing. To discover the instantaneous changing trends of the vegetation in 1995, we use the total increasing ratio to adjust each single increasing ratio and name this Trend Values, which is short for Multivariate Instantaneous Successional Trends for 1995(Bai, 1996) . The total increasing ratio is 9/6=150%. After using this Total ratio to adjust the ratio of each species, the Trend Values for each species are: T(t)=200%/150%=1.33, T(s)=150%/150%=1.0, and T(g)=133%/150%=0.89, respectively. The above analysis is expressed in the following table: Tree Shrub Grass 1994 1 2 3 (6) 1995 2 3 4 (9) Increasing rate 200% 150% 133% (150%) Trend Value 1.33 1.00 0.89 The Trend Value is the ratio of increment of species adjusted by that of the total. The Trend Value for each species can also be obtained by another means: 4B):The p ropor tions of the tree, shrub, and grass for 1994 were: 1/6=17%, 2/6=33%, and 3/6=50%, respectively. The p r o po rtions for 1995 were: 2/9=22%, 3/9=33%, and 4/9=44% respectively. The increasing ratio, named trends, for the three species are: T(t)=22/17=1.33 T(s)=33/33=1.00, and T(g)=44/50=0.89, respectively. The above trend analysis is shown in the following table: Tree Tree% Shrub Shrub% Grass Grass% Total 1994 1 17 2 33 3 50 6 1995 2 22 3 33 4 45 9 Trend Values 1.33 1 0.89 The percent is the ratio of the value of the species over total. And the Trend Value is the ratio of percent of 1995 over 1994. 5). We can sort the species by their Trend Values: Tree 1.33 Shrub 1.00 Grass 0.89 We see t he trend for trees is increasing, as the Trend Value of the trees is greater than one; while the trend for grasses is decreasing, as the Trend Value of grasses is smaller than one. 6). Without the concept of multivariate instantaneous trend, it is very difficult to convince people that in ten years this vegetation will be dominated by trees , 1.33^10=17 , and most of the grasses will be gone , 0.89^10=0.31 , provided the trend remains the same in the next ten years. The above trend analysis and prediction is shown in the following table: Tree Shrub Grass Total 1994 1 2 3 6 1995 2 3 4 9 Increasing rate 200% 150% 133% 150% Trend Value 1.33 1 0.89 ( T -Value) ^10 17.31 1 0.31 7). Conclusion: We can defin e the adjusted increasing percentage as an vegetation successional trend or instantaneous trend for species i at time k, in a simplified form: T(i,k)=Y'(i,k)/ . Y' is the relative abundance expressed in a percentage form: Y'(i)=Y(i)/{Total }, i=1,2,3. After the trend analysis we discovered that Trees are increasing the most, the Shrubs are stable, but Grasses are decreasing. Furthermore, we may make a best projection based on existing information that vegetation will be dominated by trees in ten years, instead of the present grass community. 8). Discussion: In vegetation dynamic analysis, we assume that the change of a species can only be discovered by comparison of the present over the past of the species, unless it is proved otherwise. In other words, it is our presumption that all species in vegetation are mathematically independent as every species is important and should be kept in the temporal dynamics (successiional trend)analysis. The instantaneous trend is different from the long term trend. It's value is a function of time, instead of a constan t . This makes the instantaneous trend analysis more flexible. The relation of the instantaneous trend and long-term trend will be discussed in another essay (MDSM Communication 4-2, 1997). 9). FURTHER DISCUSSION: First, w e apply our discussion to two-dimensional space (2-space) . Please recall two ancient great Mathematicians: Shang Gao (?) from China and Pythagoras (BC 500-580) from Greece. They discovered the relationship between the vector length and it's components in two dimensional space: 3^2+4^2=5^2. In other words, in 2-space, 3+4=5, instead of 7. While in 2-space, vector sum is interpreted as the vector length, the percentage is interpreted as cosine values: 3/5=0.6, and 4/5=0.8. The two cosine values determine the direction of the two-component-vector ( 2-vector ) . When the cosine values of 0.6 and 0.8 change, the 2-vector will change it's direction on the plane, and vise versa. This can further be extended to m ulti -dimensional space ( m-space ) : vector length of Y(i), L =sqrt{sum } , i=1,2,..m. In our case, 3-vector length of (1,2,3)=sqrt(14)=3.74, and 3-vector length of (2,3,4)=sqrt(29)=5.39. We can use these values to replace the one dimensional sum and begin our discussion all over again. This time, our discussion is in three dimensional space and the trees, shrubs, and grasses are represented by three axes. We can imagine the 3-vector representing the vegetation rotating in 3-space, from (1/3.74, 2/3.74, 3/3.74) of 1994 to (2/5.39, 3/5.39, 4/5.39) of 1995. This supplies a new tool that we may use to investigate the rotation of a n multi-component vector in multi-dimensional space and monitor the vegetation changes over time. SUMMARY OF 9) Using a vector sum instead of a scalar sum in m-space, the change in vegetation can be seen as the m- vector rotating in m-space. Thus, people can monitor the vegetation dynamics by tracing the movement of the m- vector on the unit hypersphere. This new method of performing Multivariate-Instantaneous-Trend analysis and system monitoring was named the Multi-Dimensional Sphere Model (MDSM). In the MDSM, an observation is expressed as a point or an m-vector in m-space. A community is the centroid m-vector of ' n ' observations, where the ' n ' is the observation number. The state of the vegetation is then a standardized (normalized) centroid m-vector, i.e., the center of the projections of ' n ' observations on the unit hypersphere. In the MDSM, the distance expresses the quantity while the direction expresses the quality. In other words, the distance of a vector relates to the production of vegetation, while direction of the vector contains the composition information of the vegetation. MDSM considers that vegetation a =(1,2,3) equals to vegetation a' =(10, 20, 30), as the two are in the same direction (coliner) . However, vegetation a =(1, 2, 3) is different from vegetation b =(3, 2, 1), although the two have the same vector length (norm). Furthermore, the trend may be paralleled as the slope of the trace, or the tangent vector onto the hypersphere, which indicates the reasons that have caused the changes. T. Jay Bai, Ph.D. MDSM Research P.O.Box 272628 Fort Collins, CO 80527 USA 970/495-9716, 970/581-0253 P.S. All the vectors should have been expressed in bolded case. The MDSM supposes the trend remains the same in neighborhood, and makes a prediction for next time interval: P(k+1)=Y(k)*T(k) and this presumption of exponential in neighborhood brings in prediction error. In the new development of MDSM, the prediction was adjusted and corrected with actually sampled data (D) from the next time interval (Jameson, 1986) and (Bai, 1996): E=P(1-alpha)+D*alpha, and R=sinE-P. Where, the E, P, D, and R are M-vectors of expectation, prediction, sampled data, and error, respectively. In the above example, as the 1995 data w as (2, 3, 4), and the trend was (1.33, 1, 0.89), so the prediction of 1996 based on the existing information is P96=(2.66, 3, 3.56). If the sampled data were 3, 4, 5 for 1996, then the expectation for the true value of 1996 would be E=P+D=(2.83, 3.5, 4.28), set alph=0.5, and the prediction error would be R=(0.17, 0.5, 0.72). This prediction error is less than the differences from the two observations: 96-95=(1, 1, 1). The entire procedure of vegetation dynamic analysis is shown in the following table: Tree Shrub Grass Total 1994 1 2 3 6 1995 2 3 4 9 Increasing Rate 200% 150% 133% 150% Trend Value-95 1.33 1.00 0.89 Projection-96 2.66 3 3.56 Observation-96 3 4 5 12 Expectation-96 2.83 3.5 4.28 Error 0.17 0.5 0.72 Interested readers please come to Aug. 14, 1996 session 155 of ESA meeting. -End- Dear friends, The above is a short essay discussing multivariate-instantaneous-trend analysis. Since it was posted on MDSM, SinoEco, and a local net last December, I received quite a few comments. Based on the se comments, I revised it and posted it here, hop ing to get more comments. T. Jay Bai If you are interested in MDSM research and discussion, please sign on: MDSM@gpsrv1.gpsr.colostate.edu or contact me at : JBAI@LAMAR.COLOSTATE.EDU ， now should be updated: mdsm95bai@yahoo.com
In mathematics a Vector field is a construction in vector calculus which associated a vector to every point in a subset of Euclidean space. Vector fields are often used in pysics to model, for example, the speed and direction of a moving fluid thoughout space, or the strength and direction of some force, such as the magnitic or gravitational force, as it changes from point to point.
J.M. Zhao , L. H. Liu, P. -f. Hsu, J. Y. Tan . Spectral Element Method for Vector Radiative Transfer Equation. Journal of Quantitative Spectroscopy and Radiative Transfer . Feb. 2010, 111(3): 433-446. download Abstract: A spectral element method (SEM) is developed to solve polarized radiative transfer in multidimensional participating medium. The angular discretization is based on the discrete-ordinates approach, and the spatial discretization is conducted by spectral element approach. Chebyshev polynomial is used to build basis function on each element. Four various test problems are taken as examples to verify the performance of the SEM. The effectiveness of the SEM is demonstrated. The h and the p convergence characteristics of the SEM are studied. The convergence rate of p-refinement follows the exponential decay trend and is superior to that of h-refinement. The accuracy and efficiency of the higher order approximation in the SEM is well demonstrated for the solution of the VRTE. The predicted angular distribution of brightness temperature and Stokes vector by the SEM agree very well with the benchmark solutions in references. Numerical results show that the SEM is accurate, flexible and effective to solve multidimensional polarized radiative transfer problems.
标签: Vector
