Mathematical principles in image processing 18--inner product and outer product

Source: Internet
Author: User

Welcome to my Blog column "A detailed explanation of mathematical principles in image Processing"

Full-text catalogs see mathematical Principles in Image Processing (master)

http://blog.csdn.net/baimafujinji/article/details/48467225

A detailed explanation of the mathematical Principles in image processing (part of the link has been posted)

http://blog.csdn.net/baimafujinji/article/details/48751037


Inner product and outer product of 1.3.2


Because cos (Π/2) = 0. Of course, this is also a number of textbooks on the introduction of vector product at the beginning of the most often used to define the way. But it must be made clear that this representation is only a very narrow definition. If we introduce the inner product of vector from this definition, it is actually putting the cart before the horse. Because for high-dimensional vectors, the meaning of the angle is ambiguous. For example, in three-dimensional coordinate space, and then introduce one-dimensional time coordinate to form a four-dimension space, then how to explain the angle between the time vector and the space vector? So the reader must be clear, first of all should be given as this section at the beginning of the definition of the inner product, and then the two-dimensional or three-dimensional space under the definition of the angle. On this basis, we will prove the cosine law.



If according to A. B = |a| | b|cosθ This definition, because 0<=cosθ<=1, apparently Cauchy-Schwartz inequalities are established. But such a way of proving the same also made the mistake of putting the cart before the horse. The Cauchy-Schwartz inequality does not define the dimensions of the vector, in other words it is a vector of any dimension, and the definition of the angle is ambiguous. The right idea should also prove the Cauchy-Schwartz inequality from the very beginning of this section, because there is an inequality relationship, and then we think that there is a coefficient between the product and the vector modulus between 0 and 1, and then we use the cosθ To express this coefficient, and then get a · b = |a| | b|cosθ the expression. Here's a proof of the Cauchy-Schwartz inequality.


Prove:

Similar to the inner product, the outer product of the vector A, B, can also be narrowly defined as


I've sorted out some of the math basics that might be used in image processing, dividing it into 6 chapters (see the link above in the full-text catalog). If you are particularly interested in one of these sections, but it has not yet been published, you can leave a comment below the blog, and I will adjust the order of publication accordingly. However, it is important to pinpoint the chapter markings (for example , 1.3.7) rather than to use a phrase like "5th chapter" or "Wavelet section" in general. Because when I have all the entire chapters published, it may be three months before the time has passed.

In addition, some readers have suggested that it is very desirable to learn the contents of chapter III (mainly because the application of partial differential equations in image processing has been narrated by me in this part of the content). To this end, I specially organized the third chapter of the manuscript to share to the reader. readers who are in need can leave a message at the bottom of the blog to tell me your email address, every 10 email address, I will send a complete third chapter of the document. since CSDN's private message function is not very stable recently, please do not send me a private messages, you may not receive any reply.



Mathematical principles in image processing 18--inner product and outer product

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