《永無休止》

無聲之歌，無言之語，此刻醉臥靜穆中。

等待………

甦醒。

號角已響起，急而又急。

人間世，

一事尚且聽未清，一事早待看分明！

………

復黎明，

萬象交織光與影。

Now Available for Download: Processing

12 Comments

I’m a long-time fan of Processing, a free open source programming language and development environment focused on teaching coding in the context of visual arts. It’s why I’m so excited that the latest version, Processing 3.0.1, now officially supports Raspberry Pi. Just as Sonic Pi lets you make your first sound in just one line of code, Processing lets you draw on screen with just one line of code. It’s that easy to get started. But don’t let that fool you, it’s a very powerful and flexible language and development environment.

Screenshot of Processing development environment

We owe a huge thank you to Gottfried Haider, who did the heavy lifting to get Processing running smoothly on the Raspberry Pi and create a hardware input/output library. That’s right, this version of Processing works with the GPIO pins right out of the box. Gottfried says:

I’m excited about having Processing on the Raspberry Pi and other low-cost desktop machines. In the last few years we’ve seen a shift away from easily accessible environments, towards concepts such as mobile platforms, specialized internet-of-things devices and cloud computing. As someone who got into programming by tinkering around with the open and readily available platforms of the time, I believe it’s important to have initiatives such Raspberry Pi and Processing, to promote software literacy and to encourage a future where computers remain a read/write medium.

───

Overview. A short introduction to the Processing software and projects from the community.

For the past fourteen years, Processing has promoted software literacy, particularly within the visual arts, and visual literacy within technology. Initially created to serve as a software sketchbook and to teach programming fundamentals within a visual context, Processing has also evolved into a development tool for professionals. The Processing software is free and open source, and runs on the Mac, Windows, and GNU/Linux platforms.

Processing continues to be an alternative to proprietary software tools with restrictive and expensive licenses, making it accessible to schools and individual students. Its open source status encourages the community participation and collaboration that is vital to Processing’s growth. Contributors share programs, contribute code, and build libraries, tools, and modes to extend the possibilities of the software. The Processing community has written more than a hundred libraries to facilitate computer vision, data visualization, music composition, networking, 3D file exporting, and programming electronics.

Processing is currently developed primarily in Boston (at Fathom Information Design), Los Angeles (at the UCLA Arts Software Studio), and New York City (at NYU’s ITP).

Education

From the beginning, Processing was designed as a first programming language. It was inspired by earlier languages like BASIC and Logo, as well as our experiences as students and teaching visual arts foundation curricula. The same elements taught in a beginning high school or university computer science class are taught through Processing, but with a different emphasis. Processing is geared toward creating visual, interactive media, so the first programs start with drawing. Students new to programming find it incredibly satisfying to make something appear on their screen within moments of using the software. This motivating curriculum has proved successful for leading design, art, and architecture students into programming and for engaging the wider student body in general computer science classes.

Processing is used in classrooms worldwide, often in art schools and visual arts programs in universities, but it’s also found frequently in high schools, computer science programs, and humanities curricula. Museums such as the Exploratorium in San Francisco use Processing to develop their exhibitions. In a National Science Foundation-sponsored survey, students in a college-level introductory computing course taught with Processing at Bryn Mawr College said they would be twice as likely to take another computer science class as the students in a class with a more traditional curriculum.

The innovations in teaching through Processing have been adapted for the Khan Academy computer science tutorials, offered online for free. The tutorials begin with drawing, using most of the Processing functions for drawing. The Processing approach has also been applied to electronics through the Arduino and Wiring projects. Arduino uses a syntax inspired by that used with Processing, and continues to use a modified version of the Processing programming environment to make it easier for students to learn how to program robots and countless other electronics projects.

Culture

The Processing software is used by thousands of visual designers, artists, and architects to create their works. Projects created with Processing have been featured at the Museum of Modern Art in New York, the Victoria and Albert Museum in London, the Centre Georges Pompidou in Paris, and many other prominent venues. Processing is used to create projected stage designs for dance and music performances; to generate images for music videos and film; to export images for posters, magazines, and books; and to create interactive installations in galleries, in museums, and on the street. Some prominent projects include the House of Cards video for Radiohead, the MIT Media Lab’s generative logo, and the Chronograph projected software mural for the Frank Gehry-designed New World Center in Miami. But the most important thing about Processing and culture is not high-profile results – it’s how the software has engaged a new generation of visual artists to consider programming as an essential part of their creative practice.

………

切莫問，

來還來不及 Processing ？！

部落格訊息

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧傅立葉

2015-11-26 懸鉤子

什麼是『傅立葉變換』？維基百科這麼講︰

Fourier transform

In the first row is the graph of the unit pulse function $f(t)$ and its Fourier transform $\hat{f}(\omega)$ , a function of frequency $\omega$ . Translation (that is, delay) in the time domain goes over to complex phase shifts in the frequency domain. In the second row is shown $g(t)$ , a delayed unit pulse, beside the real and imaginary parts of the Fourier transform. The Fourier transform decomposes a function into eigenfunctions for the group of translations.

The Fourier transform decomposes a function of time (a signal) into the frequencies that make it up, similarly to how a musical chord can be expressed as the amplitude (or loudness) of its constituent notes. The Fourier transform of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is called the frequency domain representation of the original signal. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, the domain of the original function is commonly referred to as the time domain. For many functions of practical interest one can define an operation that reverses this: the inverse Fourier transformation, also called Fourier synthesis, of a frequency domain representation combines the contributions of all the different frequencies to recover the original function of time.

Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency,^{[note 1]} so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain. Concretely, this means that any linear time-invariant system, such as a filter applied to a signal, can be expressed relatively simply as an operation on frequencies.^{[note 2]} After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are “simpler” in one or the other, and has deep connections to almost all areas of modern mathematics.

Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.

───

由於『傅立葉變換』並不是三五篇文本可以詳說，故此介紹點網路上的資源給有興趣的讀者。據聞李家同教授曾寫過一篇簡介

傅葉爾轉換(Fourier Transform)

李家同
暨南國際大學資訊工程系
rctlee@ncnu.edu.tw

，例釋說明這個『轉換』是什麼。

若想更深入的了解，何不上一堂 Stanford 大學的公開課︰

Stanford Engineering Everywhere EE261 – The Fourier Transform and its Applications

author: Brad G. Osgood, Computer Science Department, Stanford University
released under terms of: Creative Commons Attribution Non-Commercial (CC-BY-NC)

The goals for the course are to gain a facility with using the Fourier transform, both specific techniques and general principles, and learning to recognize when, why, and how it is used. Together with a great variety, the subject also has a great coherence, and the hope is students come to appreciate both.

Topics include:

The Fourier transform as a tool for solving physical problems.
Fourier series, the Fourier transform of continuous and discrete signals and its properties.
The Dirac delta, distributions, and generalized transforms.
Convolutions and correlations and applications; probability distributions, sampling theory, filters, and analysis of linear systems.
The discrete Fourier transform and the FFT algorithm.
Multidimensional Fourier transform and use in imaging.
Further applications to optics, crystallography.
Emphasis is on relating the theoretical principles to solving practical engineering and science problems.

Course Homepage: http://see.stanford.edu/see/courseinfo.aspx?coll=84d174c2-d74f-493d-92ae-c3f45c0ee091

Course features at Stanford Engineering Everywhere page:

───

同時認真讀讀 Brad Osgood 教授之課堂筆記耶。

Lecture Notes for

EE 261

The Fourier Transform and its Applications

Prof. Brad Osgood
Electrical Engineering Department
Stanford University

樹莓派、樹莓派之學習、樹莓派之教育

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧摺積

2015-11-25 懸鉤子

即使以相同的『數學方法』，因為談論不同的事，有時還是得腦筋『急轉彎』。這個腦筋『急轉彎』的由來，也許是因為得用『非常不同』的觀點來考察事物。突然發現某種不期而遇之『共通處』。若問笛卡爾如何想出『解析幾何』，使得幾何得以用『座標系』與『座標』來處理？雖說不得而知，但是這套數學『思維方法』構成了牛頓力學的骨幹。就像問著『約瑟夫‧傅立葉』 Joseph Fourier 為什麼會將函數用『正弦級數』來作展開？？難道是受到笛卡爾的啟發！或是牛頓的影響！！可能很難定論。此處只能借著科學史上的一個小故事來暗示那種概念間冥冥的『聯繫』︰

電阻的單位是『歐姆』

傅立葉分析
數學上最美麗的樂章

一八二五年德國物理學家蓋歐格‧西蒙‧歐姆 Georg Simon Ohm 是一位優秀的實驗者，很會設計與製造實驗設備，又具有良好的數學素養和嚴謹的處事態度。他使用伏打電堆為電源，繼而用安裝於扭秤 torsion balance 的磁針來測量電流產生的磁力。因為載流導線的電流所產生的磁場與電流成正比，所以只要測量在載流導線附近的磁針所感受到的磁力，就可以度量電流的大小。他將電流通過不同長度的受測電線；由於長度的不同，電流也就不同。並且基於實驗結果推導出了兩者之間的數學關係式︰

$\Delta I = C \ln\left(1+\frac{x}{a}\right)$

其中， $\Delta I$ 是受測電線造成的電流差值， $C$ 是跟實驗參數有關的係數， $x$ 是受測電線的長度， $a$ 是跟固定長度的載流導線有關的常數。不久之後，歐姆便發覺了這個關係式可能是不正確的。

法國數學家與物理學家約瑟夫‧傅立葉男爵 Joseph Fourier 以所提出的『傅立葉級數』稱名於世，對於任何一個周期為 $T$ 的函數 $f(t)$ ，都可以用『無窮級數』表示為

$f(t) = \sum _{k=-\infty}^{+\infty}a_k\cdot e^{i k 2\pi \frac{t}{T}}$

，式中係數 $a_k$ 可以依下式來計算

$a_k = \frac{1}{T}\int_{T}f(t)\cdot e^{-i k 2\pi \frac{t}{T}}dt$

。此處 $i=\sqrt{-1}$ ，而且 $f_k(t)=e^{i k 2\pi \frac{t}{T}}$ 是基本周期為 $T$ 的三角『正、餘弦』函數，這就是周期訊號 $f(t)$ 的『諧波分析』。當 $k = 0$ 時， $a_0$ 係數稱之為『直流分量』，就是訊號 $f(t)$ 在整個周期的平均值。當 $k = \pm 1$ 時， $a_{\pm 1}$ 係數是『一次諧波』頻率，隨著 $k$ 的值，而有『二次諧波』，『三次諧波』等等。並且將之應用於『熱傳導理論』與『振動理論』；據聞他也是『溫室效應』的發現者。一八二二年傅立葉發表了《熱的解析理論》 Théorie analytique de la chaleur)。依據『牛頓冷卻定律』假設了︰兩相鄰分子的熱流與其間微小的溫度差成正比。

受到傅立葉對熱傳導規律研究的啟發，歐姆認為『電流現象』與『熱傳導』相似，便猜想導線中兩點之間的電流也正比於這兩點間的某種驅動力 ── 即現在所說的『電動勢』 ──。由於當時『伏打電堆』所產生的電流並不穩定，據聞歐姆採納了《物理與化學年鑑》的總編輯約翰‧波根多夫 Johann poggendorff 的建議，改用『熱電偶』為電源。雖然『鉍和銅』所作的『溫差電池』保證了電流的穩定性。然而卻產生了測量『電流太小』的難題。歐姆先是利用『電流』的『熱效應』，想藉著『熱脹冷縮』的方法來測量電流大小，但是量測的結果並『不精確』，後來他把奧斯特『電流磁效應』的發現和『庫侖扭秤』巧妙的結合起來設計了一個『電流扭秤』︰讓導線和連接的磁針平行放置，當導線中通過電流時，磁針的偏轉角與導線中的電流成正比，這就代表了電流的大小。

一八二六年歐姆所發表兩篇重要論文中，建立了『電傳導』的『數學模型』和『表達式』。著名的『歐姆定律』發表於一八二七年的《直流電路的數學研究》 Die galvanische Kette, mathematisch bearbeitet 一書中，提出了『電路分析』中『電流』、『電壓』與『電阻』之間的基本關係

$I = \frac{V}{R}$ 。

─── 引自《【Sonic π】電聲學導引《七》》

或許能說明『正交函數族』之概念，並非一蹴而成的，所謂『正交』也不可講成『畢氏定理』自然的擴張。雖然說這些『象徵』表達概念可能的深層『融會』，提示讀者進入的門徑。究其實，依舊是例說寓言罷了。

就像『振幅調變』 $x(t) \cdot y(t)$ 之『頻譜』，可以用『頻譜摺積』來計算，然而什麼是『摺積』呢？

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated. Convolution is similar to cross-correlation. It has applications that include probability, statistics, computer vision, natural language processing, image and signal processing, engineering, and differential equations.

The convolution can be defined for functions on groups other than Euclidean space. For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 10 at DTFT#Properties.) A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing.

Computing the inverse of the convolution operation is known as deconvolution.

Visual comparison of convolution, cross-correlation and autocorrelation.

摺積

在泛函分析中，捲積、疊積、摺積或旋積，是通過兩個函數f和g生成第三個函數的一種數學算子，表徵函數f與經過翻轉和平移的g的重疊部分的面積。如果將參加摺積的一個函數看作區間的指示函數，摺積還可以被看作是「滑動平均」的推廣。

圖示兩個方形脈衝波的捲積。其中函數”g”首先對 $\tau=0$ 反射，接著平移”t”，成為 $g(t-\tau)$ 。那麼重疊部份的面積就相當於”t”處的捲積，其中橫坐標代表待積變量 $\tau$ 以及新函數 $f\ast g$ 的自變量”t”。

圖示方形脈衝波和指數衰退的脈衝波的捲積（後者可能出現於RC電路中），同樣地重疊部份面積就相當於”t”處的捲積。注意到因為”g”是對稱的，所以在這兩張圖中，反射並不會改變它的形狀。

簡單介紹

摺積是分析數學中一種重要的運算。設： $f(x)$ , $g(x)$ 是 $\mathbb{R}$ 上的兩個可積函數，作積分：

\int_{-\infty}^{\infty} f(\tau) g(x - \tau)\, \mathrm{d}\tau

可以證明，關於幾乎所有的 $x \in (-\infty,\infty)$ ，上述積分是存在的。這樣，隨著 $x$ 的不同取值，這個積分就定義了一個新函數 $h(x)$ ，稱為函數 $f$ 與 $g$ 的摺積，記為 $h(x)=(f*g)(x)$ 。我們可以輕易驗證： $(f * g)(x) = (g * f)(x)$ ，並且 $(f * g)(x)$ 仍為可積函數。這就是說，把摺積代替乘法， $L^1(R^1)$ 空間是一個代數，甚至是巴拿赫代數。雖然這裡為了方便我們假設 $\textstyle f, g\in L^1(\mathbb{R})$ ，不過捲積只是運算符號，理論上並不需要對函數 $f,g$ 有特別的限制，雖然常常要求 $f,g$ 至少是可測函數（measurable function）（如果不是可測函數的話，積分可能根本沒有意義），至於生成的卷積函數性質會在運算之後討論。

摺積與傅立葉變換有著密切的關係。例如兩函數的傅立葉變換的乘積等於它們摺積後的傅立葉變換，利用此一性質，能簡化傅立葉分析中的許多問題。

由摺積得到的函數 $f*g$ 一般要比 $f$ 和 $g$ 都光滑。特別當 $g$ 為具有緊支集的光滑函數， $f$ 為局部可積時，它們的摺積 $f * g$ 也是光滑函數。利用這一性質，對於任意的可積函數 $f$ ，都可以簡單地構造出一列逼近於 $f$ 的光滑函數列 $f_s$ ，這種方法稱為函數的光滑化或正則化。

摺積的概念還可以推廣到數列、測度以及廣義函數上去。

The convolution theorem and its applications

What is a convolution?

One of the most important concepts in Fourier theory, and in crystallography, is that of a convolution. Convolutions arise in many guises, as will be shown below. Because of a mathematical property of the Fourier transform, referred to as the convolution theorem, it is convenient to carry out calculations involving convolutions.

But first we should define what a convolution is. Understanding the concept of a convolution operation is more important than understanding a proof of the convolution theorem, but it may be more difficult!

Mathematically, a convolution is defined as the integral over all space of one function at x times another function at u-x. The integration is taken over the variable x (which may be a 1D or 3D variable), typically from minus infinity to infinity over all the dimensions. So the convolution is a function of a new variable u, as shown in the following equations. The cross in a circle is used to indicate the convolution operation.

convolution equals integral of one function at x times other function at u-x

Note that it doesn’t matter which function you take first, i.e. the convolution operation is commutative. We’ll prove that below, but you should think about this in terms of the illustration below. This illustration shows how you can think about the convolution, as giving a weighted sum of shifted copies of one function: the weights are given by the function value of the second function at the shift vector. The top pair of graphs shows the original functions. The next three pairs of graphs show (on the left) the function g shifted by various values of x and, on the right, that shifted function g multiplied by f at the value of x.

illustration of convolution

The bottom pair of graphs shows, on the left, the superposition of several weighted and shifted copies of g and, on the right, the integral (i.e. the sum of all the weighted, shifted copies of g). You can see that the biggest contribution comes from the copy shifted by 3, i.e. the position of the peak of f.

If one of the functions is unimodal (has one peak), as in this illustration, the other function will be shifted by a vector equivalent to the position of the peak, and smeared out by an amount that depends on how sharp the peak is. But alternatively we could switch the roles of the two functions, and we would see that the bimodal function g has doubled the peaks of the unimodal function f.

───

它是否有所謂傳統『乘法類似物』的耶！！或者『類比推理』只該看成是『發現』的邏輯乎？？

樹莓派、樹莓派之學習、樹莓派之教育

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧周期

2015-11-24 懸鉤子

經過『九點圓』以及『留白』的洗禮，也許我們可以借著

Miller Puckette 之

1.7 Periodic Signals
A signal x[n] is said to repeat at a period τ if

x[n + τ ] = x[n]

for all n. Such a signal would also repeat at periods 2τ and so on; the smallest τ if any at which a signal repeats is called the signal’s period. In discussing periods of digital audio signals, we quickly run into the difficulty of describing signals whose “period” isn’t an integer, so that the equation above doesn’t make sense. For now we’ll effectively ignore this difficulty by supposing that the signal x[n] may somehow be interpolated between the samples so that it’s well defined whether n is an integer or not.

A sinusoid has a period (in samples) of 2π/ω where ω is the angular frequency. More generally, any sum of sinusoids with frequencies 2πk/ω, for integers k, will repeat after 2π/ω samples. Such a sum is called a Fourier Series:

x[n] = a0 + a1 cos (ωn + φ1 ) + a2 cos (2ωn + φ2 ) + · · · + ap cos (pωn + φp )

Moreover, if we make certain technical assumptions (in effect that signals only contain frequencies up to a finite bound), we can represent any periodic signal as such a sum. This is the discrete-time variant of Fourier analysis which will reappear in Chapter 9.

The angular frequencies of the sinusoids above are all integer multiples of ω. They are called the harmonics of ω, which in turn is called the fundamental. In terms of pitch, the harmonics ω, 2ω, . . . are at intervals of 0, 1200, 1902, 2400, 2786, 3102, 3369, 3600, …, cents above the fundamental; this sequence of pitches is sometimes called the harmonic series. The first six of these are all quite close to multiples of 100; in other words, the first six harmonics of a pitch in the Western scale land close to (but not always exactly on) other pitches of the same scale; the third and sixth miss only by 2 cents and the fifth misses by 14.

………

文本，談點『周期』概念容易『誤解』的事。

維基百科的『周期函數』Periodic function 詞條講︰

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function which is not periodic is called aperiodic.

An illustration of a periodic function with period $P.$

Definition

A function f is said to be periodic with period P (P being a nonzero constant) if we have

f(x+P) = f(x) \,\!

for all values of x in the domain. If there exists a least positive^[1] constant P with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P. This definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane.

A function that is not periodic is called aperiodic.

Properties

If a function f is periodic with period P, then for all x in the domain of f and all integers n,

f(x + nP) = f(x).

If f(x) is a function with period P, then f(ax+b), where a is a positive constant, is periodic with period P/|a|. For example, f(x)=sinx has period 2π, therefore sin(5x) will have period 2π/5.

Double-periodic functions

A function whose domain is the complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions. (“Incommensurate” in this context means not real multiples of each other.)

Quotient spaces as domain

In signal processing you encounter the problem, that Fourier series represent periodic functions and that Fourier series satisfy convolution theorems (i.e. convolution of Fourier series corresponds to multiplication of represented periodic function and vice versa), but periodic functions cannot be convolved with the usual definition, since the involved integrals diverge. A possible way out is to define a periodic function on a bounded but periodic domain. To this end you can use the notion of a quotient space:

{\mathbb{R}/\mathbb{Z}} = \{x+\mathbb{Z} : x\in\mathbb{R}\} = \{\{y : y\in\mathbb{R}\land y-x\in\mathbb{Z}\} : x\in\mathbb{R}\}

That is, each element in ${\mathbb{R}/\mathbb{Z}}$ is an equivalence class of real numbers that share the same fractional part. Thus a function like $f : {\mathbb{R}/\mathbb{Z}}\to\mathbb{R}$ is a representation of a 1-periodic function.

也許足以澄清 Miller Puckette 文本中大多數的內容。首先一個以 $P$ 為『周期』的函數，自然也以 $n \cdot P$ 為周期【※ $n$ 非零正整數】，所以實數周期函數才會有最小周期值。

如果 $f(x)$ 以 $P_1$ 為周期， $g(x)$ 以 $P_2$ 為周期，假使 $\frac{P_1}{P_2}$ 的比值是個『有理數』，可以用『最簡分數』表示成 $\frac{n}{m}$ 。也就是說 $m \cdot P_1 \ = \ n \cdot P_2 \ = \ T$

此處的 $T$ 也就是 $f(x) + g(x)$ 的周期。然而周期不必是『有理數』，周期的『比值』當然也未必是『有理數』，因此

兩個周期函數的和，卻未必是個周期函數

！！？？

要是此時重讀《字詞網絡︰ WordNet 《一》索引》系列文本︰

Natural Language Processing with Python
— Analyzing Text with the Natural Language Toolkit

Steven Bird, Ewan Klein, and Edward Loper

一書第二章第五節《 2.5 WordNet 》之『字詞網絡』概念階層片段

※或可參考【譯著】

Figure 2-8. Fragment of WordNet concept hierarchy: Nodes correspond to synsets; edges indicate the hypernym/hyponym relation, i.e., the relation between superordinate and subordinate concepts.

『WordNet』字詞網絡計畫啟始於一九八五年，在普林斯頓大學『認知科學實驗室』由心理學教授『喬治‧A‧米勒』 George Armitage Miller 的指導下建立和維護的英語『詞彙資料庫』 lexical database 字典。因為它包含了多種『字詞』間之『語義關係』，所以別於通常意義下的『字典』。『WordNet』是什麼？也許最好先讀讀『創造者』怎麼說︰

What is WordNet?

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the creators of WordNet and do not necessarily reflect the views of any funding agency or Princeton University.When writing a paper or producing a software application, tool, or interface based on WordNet, it is necessary to properly cite the source. Citation figures are critical to WordNet funding.

About WordNet

WordNet® is a large lexical database of English. Nouns, verbs, adjectives and adverbs are grouped into sets of cognitive synonyms (synsets), each expressing a distinct concept. Synsets are interlinked by means of conceptual-semantic and lexical relations. The resulting network of meaningfully related words and concepts can be navigated with the browser. WordNet is also freely and publicly available for download. WordNet’s structure makes it a useful tool for computational linguistics and natural language processing.

WordNet superficially resembles a thesaurus, in that it groups words together based on their meanings. However, there are some important distinctions. First, WordNet interlinks not just word forms—strings of letters—but specific senses of words. As a result, words that are found in close proximity to one another in the network are semantically disambiguated. Second, WordNet labels the semantic relations among words, whereas the groupings of words in a thesaurus does not follow any explicit pattern other than meaning similarity.

Structure

The main relation among words in WordNet is synonymy, as between the words shut and close or car and automobile. Synonyms–words that denote the same concept and are interchangeable in many contexts–are grouped into unordered sets (synsets). Each of WordNet’s 117 000 synsets is linked to other synsets by means of a small number of “conceptual relations.” Additionally, a synset contains a brief definition (“gloss”) and, in most cases, one or more short sentences illustrating the use of the synset members. Word forms with several distinct meanings are represented in as many distinct synsets. Thus, each form-meaning pair in WordNet is unique.

Relations

The most frequently encoded relation among synsets is the super-subordinate relation (also called hyperonymy, hyponymy or ISA relation). It links more general synsets like {furniture, piece_of_furniture} to increasingly specific ones like {bed} and {bunkbed}. Thus, WordNet states that the category furniture includes bed, which in turn includes bunkbed; conversely, concepts like bed and bunkbed make up the category furniture. All noun hierarchies ultimately go up the root node {entity}. Hyponymy relation is transitive: if an armchair is a kind of chair, and if a chair is a kind of furniture, then an armchair is a kind of furniture. WordNet distinguishes among Types (common nouns) and Instances (specific persons, countries and geographic entities). Thus, armchair is a type of chair, Barack Obama is an instance of a president. Instances are always leaf (terminal) nodes in their hierarchies.

Meronymy, the part-whole relation holds between synsets like {chair} and {back, backrest}, {seat} and {leg}. Parts are inherited from their superordinates: if a chair has legs, then an armchair has legs as well. Parts are not inherited “upward” as they may be characteristic only of specific kinds of things rather than the class as a whole: chairs and kinds of chairs have legs, but not all kinds of furniture have legs.

Verb synsets are arranged into hierarchies as well; verbs towards the bottom of the trees (troponyms) express increasingly specific manners characterizing an event, as in {communicate}-{talk}-{whisper}. The specific manner expressed depends on the semantic field; volume (as in the example above) is just one dimension along which verbs can be elaborated. Others are speed (move-jog-run) or intensity of emotion (like-love-idolize). Verbs describing events that necessarily and unidirectionally entail one another are linked: {buy}-{pay}, {succeed}-{try}, {show}-{see}, etc.

Adjectives are organized in terms of antonymy. Pairs of “direct” antonyms like wet-dry and young-old reflect the strong semantic contract of their members. Each of these polar adjectives in turn is linked to a number of “semantically similar” ones: dry is linked to parched, arid, dessicated and bone-dry and wet to soggy, waterlogged, etc. Semantically similar adjectives are “indirect antonyms” of the contral member of the opposite pole. Relational adjectives (“pertainyms”) point to the nouns they are derived from (criminal-crime).
There are only few adverbs in WordNet (hardly, mostly, really, etc.) as the majority of English adverbs are straightforwardly derived from adjectives via morphological affixation (surprisingly, strangely, etc.)

Cross-POS relations

The majority of the WordNet’s relations connect words from the same part of speech (POS). Thus, WordNet really consists of four sub-nets, one each for nouns, verbs, adjectives and adverbs, with few cross-POS pointers. Cross-POS relations include the “morphosemantic” links that hold among semantically similar words sharing a stem with the same meaning: observe (verb), observant (adjective) observation, observatory (nouns). In many of the noun-verb pairs the semantic role of the noun with respect to the verb has been specified: {sleeper, sleeping_car} is the LOCATION for {sleep} and {painter}is the AGENT of {paint}, while {painting, picture} is its RESULT.

More Information

Fellbaum, Christiane (2005). WordNet and wordnets. In: Brown, Keith et al. (eds.), Encyclopedia of Language and Linguistics, Second Edition, Oxford: Elsevier, 665-670

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是否會有不同之體驗乎？？！！

部落格訊息

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧留白

2015-11-23 懸鉤子

若問為什麼平面上的一個一般三角形可以如下圖表示

，只用著 $a \ , b \ , \ c$ 三個參數？即使在思考過 $a$ 是『底』之『長』， $c$ 是此『底』之『高』， $b$ 是此『高』距與此『底』一端的距離。我們深信這就『確定』了那個三角形。然而若再問︰如果此三角形的三個頂點用更一般的 $A \ (x_0, y_0)$ 、 $B \ (x_1,y_1)$ 、 $C \ (x_2,y_2)$ 來表達，如是分明有六個參數。那麼這兩種『表述』當真是一樣的嗎？設想你在桌面上『移動』一個三角形，從此『位置』此『方位』到達彼『位置』彼『方位』，你會認為這個三角形『改變』了嗎？？假使『直覺』以為『不變』，這個三角形就必得有使之『不變』的『因由』，這個『因由』不必『參照』解析幾何的『座標』而確立。或可說它就是歐式幾何一個三角形的『定義』內涵而已。如此而言，一個『確定』的三角形，可由它的三個『邊長』來『確立』，所以六個參數補之以三個確定之邊長關係，豈非還是三個參數的耶？？

因為這個『歐式幾何』的『留白』，常使人懷疑『解析幾何』簡化『座標系』的『選擇』，到底『圖形』的『自由度』是幾何的了。說難道易，就請讀者思索︰平面上的『 □ 』與『 ○ 』，到底一方一圓需要幾個參數來描述的呢？

從物理上講，那個三角形就是『剛體』 rigid body ，它在『運動』中保持『形狀』的『不變性』。而且不同觀察者間的『座標變換』可以用

Rigid transformation

In mathematics, a rigid transformation (isometry) of a vector space preserves distances between every pair of points.^[1]^[2] Rigid transformations of the plane R², space R³, or real n-dimensional space Rⁿ are termed a Euclidean transformation because they form the basis of Euclidean geometry.^[3]

The rigid transformations include rotations, translations, reflections, or their combination. Sometimes reflections are excluded from the definition of a rigid transformation by imposing that the transformation also preserve the handedness of figures in the Euclidean space (a reflection would not preserve handedness; for instance, it would transform a left hand into a right hand). To avoid ambiguity, this smaller class of transformations is known as proper rigid transformations (informally, also known as roto-translations). In general, any proper rigid transformation can be decomposed as a rotation followed by a translation, while any rigid transformation can be decomposed as an improper rotation followed by a translation (or as a sequence of reflections).

Any object will keep the same shape and size after a proper rigid transformation.

All rigid transformations are examples of affine transformations. The set of all (proper and improper) rigid transformations is a group called the Euclidean group, denoted E(n) for n-dimensional Euclidean spaces. The set of proper rigid transformation is called special Euclidean group, denoted SE(n).

In kinematics, proper rigid transformations in a 3-dimensional Euclidean space, denoted SE(3), are used to represent the linear and angular displacement of rigid bodies. According to Chasles’ theorem, every rigid transformation can be expressed as a screw displacement.

Formal definition

A rigid transformation is formally defined as a transformation that, when acting on any vector v, produces a transformed vector T(v) of the form

T(v) = R v + t

where R^T = R⁻¹ (i.e., R is an orthogonal transformation), and t is a vector giving the translation of the origin.

A proper rigid transformation has, in addition,

det(R) = 1

which means that R does not produce a reflection, and hence it represents a rotation (an orientation-preserving orthogonal transformation). Indeed, when an orthogonal transformation matrix produces a reflection, its determinant is –1.

Distance formula

A measure of distance between points, or metric, is needed in order to confirm that a transformation is rigid. The Euclidean distance formula for Rⁿ is the generalization of the Pythagorean theorem. The formula gives the distance squared between two points X and Y as the sum of the squares of the distances along the coordinate axes, that is

d(\mathbf{X},\mathbf{Y})^2 = (X_1-Y_1)^2 + (X_2-Y_2)^2 + \ldots + (X_n-Y_n)^2 = (\mathbf{X}-\mathbf{Y})\cdot(\mathbf{X}-\mathbf{Y}).

where X=(X₁, X₂, …, X_n) and Y=(Y₁, Y₂, …, Y_n), and the dot denotes the scalar product.

Using this distance formula, a rigid transformation g:Rⁿ→Rⁿ has the property,

d(g(\mathbf{X}), g(\mathbf{Y}))^2 = d(\mathbf{X}, \mathbf{Y})^2.

───

來作轉換的矣！！

如是細思二維空間上的『旋轉』︰

Two dimensions

In two dimensions, to carry out a rotation using matrices the point $(x, y)$ to be rotated (orientation from positive $x$ to $y$ ) is written as a vector, then multiplied by a matrix calculated from the angle, $θ$ :

\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}

where $(x', y')$ are the coordinates of the point that after rotation, and the formulae for $x'$ and $y'$ can be seen to be

\text{[math]}

The vectors $\begin{bmatrix} x \\ y \end{bmatrix}$ and $\begin{bmatrix} x' \\ y' \end{bmatrix}$ have the same magnitude and are separated by an angle $θ$ as expected.

Points on the $R 2$ plane can be also presented as complex numbers: the point $(x, y)$ in the plane is represented by the complex number

z = x + iy

This can be rotated through an angle $θ$ by multiplying it by $e iθ$ , then expanding the product using Euler’s formula as follows:

\text{[math]}

and equating real and imaginary parts gives the same result as a two-dimensional matrix:

\text{[math]}

Since complex numbers form a commutative ring, vector rotations in two dimensions are commutative, unlike in higher dimensions. They have only one degree of freedom, as such rotations are entirely determined by the angle of rotation.^[1]

───

那一方一圓問題的答案，是否不言而喻的哩？？！！

FreeSandal

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧休止♪