By Claire Ortiz Hill
Bargains a rethinking of the principles of 20th-century analytic philosophy. The textual content examines the writings of Gottlob Frege, Bertrand Russell and Willard Quine and considers how the extensionalistic remedy of identification initiated through Frege has served as a hidden constraint on next paintings.
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Extra resources for Rethinking Identity and Metaphysics: On the Foundations of Analytic Philosophy
5). 6), but our analysis will be rather brief, because a full treatment requires the language and techniques of signal processing (ﬁlter theory), which are in general more familiar to electrical engineers than to physicists. Moreover, the 2-D DWT is mostly used in image processing, which is not the main thrust of the present book. 4). We consider 2-D signals of ﬁnite energy, represented by complex-valued, square integrable functions s 2 L2 ðR2 ; d 2 x~Þ. This condition may be relaxed, to allow, for instance, a plane wave or a function.
As a matter of fact, all the approaches that we have mentioned above in the 1-D case have been extended to 2-D: orthonormal bases, biorthogonal bases, wavelet packets, lifting scheme. These topics will be discussed in detail in Chapter 2. Wavelet analysis: a new tool in physics 19 However, the continuous transform may also be extended to 2 (or more) dimensions, with exactly the same properties as in the 1-D case [2, 26]. 10)): Z ~ Sða; ; bÞ $ d 2 x~ " ðaÀ1 rÀ ðx~ À b~ÞÞsðx~Þ; a > 0; 0 < 2; b 2 R2 ; ð1:26Þ where s is the signal and is the analysing wavelet, which is translated by b~, dilated by a and rotated by an angle ðrÀ is the rotation operator).
For all these reasons, signal analysts turn to time-frequency (TF) representations. The idea is that one needs two parameters: one, called a, characterizes the frequency, the other one, b, indicates the position in the signal. This concept of a TF representation is in fact quite old and familiar. The most obvious example is simply a musical score! If one requires in addition the transform to be linear, a general TF transform will take the form: Z1 ð1:1Þ sðxÞ 7 ! Sða; bÞ ¼ ab ðxÞ sðxÞ dx; À1 where s is the signal and ab the analysing function.
Rethinking Identity and Metaphysics: On the Foundations of Analytic Philosophy by Claire Ortiz Hill