AWCGAIT2015, 27 Mar. 2015: Overview of Quaternion and Clifford Fourier Transforms, E. Hitzer

Transcript

today I want to talk about uh Quan and Clifford F Transformations and I'll try to give an overview and at the beginning because I'm from a Christian University I say something Christian so here is a quation from Christ um about world peace peace I leave with you my peace I give you I do not give you as the world gives do not let your hearts be troubled and do not uh be afraid and um I want to thank my wife my children and my parents and a number of people who cooperated with me and um Professor Palm who successfully organized this conference together with his colleagues and Professor t bner with whom we couldn't be here because he was the supervisor of Professor palm and he also introduced me to Professor pal and uh some time ago I gave a presentation in hoshima and this was not the reason why I wrote this license but uh I understand that part of my research can also be used in the wrong way and I hope it never happens but therefore I wrote this license uh for it so it should be only for peaceful purposes and um kind of uh correct purposes this is a short overview of what I talking about today um about uh does you do you have a laser poter uh does anybody have a laser okay then I'll just try to read off the slides uh so CTIC algebra and calculus and um here number two will be overview uh of Transformations and then a special frequent type the two-sided uh Transformations number the next one is one-sided P Transformations and uh to give you a motivation for using the cliford geometric Algebra I give you here some uh idea uh maybe everybody knows complex numbers and if you make a product of two complex numbers you get this combination X1 y1 - X2 Y2 + I X1 Y2 + X2 y1 and if you rotate it in the complex plane with a phas factor e to the^ I Theta the product x * y changes it rotates by twice the angle and for many applications uh this is not what you want you don't want to express your data in a way that it always changes when you rotate uh the coordinates so but if you make another product xar that's the complex conjugate times y then the result is invariant and it has two components the real and imaginary component and the real component you see is the inner product of the vectors X and Y and the imaginary component is the area of the parallelogram spanned by the vectors X and Y so these uh was a geometrically very meaningful and if you yeah the next point I want to uh talk about is quarians which is not yet taught in most universities but for certain applications in uh aerospace engineering and virtual reality it's a very frequently used tool and you have three imaginary units i j k each of them Squares to minus one and the product of two gives the third there are four numbers the real number QR and then the are three imaginary components you have like this complex conjugate a contan conjugate a norm a scalar part and then the inner product as well that's like a product of four dimensions and if you take two imaginary uh cians that's great um so if you take um two pure K with only imaginary parts then you as well the cian conjugant X over bar * Y and you get again something which is in variant under rotation and the first part is the inner product in three dimensions and um the second part here is the outer product of vectors or the cross product of vectors in uh three dimensions and uh this is the area again of the parallelogram span by the two vectors and what you give it a direction of AR to it and the general ouo product has been studied in 1844 by grasman he is uh German High School Master teacher and he invented a full calculus of extension and then the inner product and the outer product here have been unified by Clifford in 1878 into Clifford Aras and uh that's here the CER product of two vectors it has a Scala part and the spor part and you simply add them like you add a real and complex number like a real an imaginary uh number U being added okay works well and now here in three dimensions you have three vectors E1 E2 and E3 they made three dimensional Vector space but if you look in the C algebra you get the scalar one the three vectors and then you get three area by vectors like the sides of of a cube and you get the volume of the unit cube E1 * E2 * E3 together with orientation as well and I call this I because if you compute according to the rules of the algebra I squ it's also minus one and under the product the Scala and the B vectors they are a some algebra so they C under multiplication and they isomorphic to so you find as soal even sub algebra the cians in as part of the C algebra of three dimensional space and as cion are used for rotation operations also simply this even sub algebra here indicated in red does all the rotations in threedimensional space so any element of the subalgebra here makes a rotation and it's very efficient description of rotation compared to rotation matrices and now in general in a algebra you have a scalar part then a vector part A B Vector part and so on until the highest dimensional kind of volume or hyper volume part this part is also called Scala and there's something principle reverse which is the same like conjugation of planium conjugation and if you apply it and then take the Scala part of the geometric product then you get simply the norm the product of all the coefficients the sum over the product of all the coefficients and this you can use to define the modulus of a Clifford number or a multiv vector and the modulus generalizes Notions of magnitude or lengths of a vector Notions of area volume and and Hyper volume so it's all becoming just one General Norm of mod this notion then I've told you that in this case here um the square of the TR Vector is minus one and um here too in general um if you have q = Z so all the vectors have positive square then the square of the unit SCA the product of all vectors is minus one for the total Dimension being two or three 6 or 7 10 or 11 and so on so there's some important elements which have a negative square always and geometrically a blade B um that is um a product of or horal vectors um describes the vector space so an example is the plade B E1 W E2 or E1 * E2 and this is equivalent to the vector space um using vectors in the direction E1 and E2 and you get a dual blade which is the blade multiplied or divided by the pseudo Scala the example is here E1 E2 inverses of Sol scal is E3 and that's the vector space perpendicular to the blade so if you have the blade then the direction perpendicular you can easily compute as well and scalar I is Central if the dimension of the underlying Vector space is odd Central means I is like a imaginary unit it commutes with every sync so you take a multi vector and you can multiply from the left or the right it's the same result and you can build it up from knowing this algebra like you have real functions and complex functions we can have um functions with pan values and here also functions with General geometric algebra values and so there are the Scala function coefficients and then the basis elements and we do a linear combination and you get a general function and then we take two functions and the second uh is um conjugated with a principle reverse to the product and this is an inner product under the integral and it becomes a norm if you take the Scala part of um product of a function with itself under the inte and um now know already a few elements with square to minus one like the pseudo Scala in two and three dimensions of a Idan Vector space but in general there are many more elements which have a square of minus one or you can say the are SARS of minus one they in general in a cop algebra there's a continuous manifold of half the dimension of the algebra of elements which have a negative square square ofus one and so for a number of different algebras this one here and here so ukian space and here this is isomorphic to pans as well it corresponds to All Pure pans and here there's a mixed signature one vector has a positive square the other has a negative square that's this case here uh you find okay and here now I give you it's a bit overwhelming um overview of some of the glyph Fu transforms which have been studied uh so far and you you basically start with an element from Clifford algebra with squares to minus one and you can go from here and for example make wavelets or F Transformations and then a whole variety because the algebra is noncommutative so if you put a factor to the left or the right if you use one or two factors it all makes a difference and therefore there's such a variety of Transformations and uh in the uh from now on I I cannot explain all of them just a few I will give some explanations about so um this construction here is very General and illustrates illustrates the principle you have in the classical transform a factor e to the power ofus I X from position and Omega for frequency and you replace the I by a general S ofus one from the algebra and the whole expression minus i x Omega becomes a function- s x Omega and this s here squares uh also has a negative square and so in general you have a signal function which maps from the vector space into the Clif algebra and then you put factors on the right and factors on the left and they are uh these exponential factors here and the product of these factors and that's the way you can kind of generally mathematically study these transformations and now a more concrete example from these algebras here and that is um um you Tak the signal function and the signal function maps from the vector space RN into C algebra of the vector space and every basis Vector E1 to e k has a negative square so E1 s = e equ the N is- one and um so therefore this element here has a negative square then there is the frequency in the K Direction and the position in the K Direction and it's just constant and you multiply these kernal factors so you have up to n of these cars here and this has been used in application now for color images there is one transform which um has been defined by uh medison here in 2011 and they apply it uh to color pictures and these two pictures relative to each other they scaled and translated and uh in their um F transform color image F transform they have a basic I Vector B and the space they work in is four dimensional and uh they project um in the result the component parallel to that y Vector parall to that plane in four dimensions and perpendicular to it um both for the signal F and for the scale and shift signal G and then they compare the parallel components and they compare the perpendicular components and compute uh the shift uh in that space here and then through face correlation of the paral part and the orthogonal part they can uh kind of score and find out uh the scaling and the shift and uh this is one way of uh kind of application for color images where you can also successfully register color images but there are other ways and depending on your application one or the other way may work better so I show you another way later if I have time then the general CA of two-sided different F transforms you see here you have the signal function and then you have a kernel on the right and the kernel on the left and f and g two elements which spr to minus one in the C algebra but there are two different elements they can be the same but they are different and then we have these pH functions u and v they have a scal result and they map a position vector and the frequency Vector to real scers here and so Bally descending from two-sided G fluid Transformations you have um if you go into the curnan algebra the two-sided cionic FL transform and uh that has been now frequently applied in the last two or three decades already the general construction is here you have the signal function then you have the position space component X1 and X2 for example the pixel coordinates of an image and here the frequency coordinates Omega one and Omega 2 in two orthogonal directions and f and g they all both Square to minus one so they are pure unit quents and there are different variations for example you can take away the factor on the left or on the right and first it was suggested um some nearly 30 years ago in spectral analysis of two dimensional NMR signals and uh then you get um two independent adjustable phas angles with respect to the to frequency variables so you can refine the image analysis and later L he formulated uh the same qf for the analysis of linear time independent in Varian systems of par different equations and then later bu of he applied it to image video and texture analysis and S to color image analysis analysis of non stationary improper complex signals and vector image processing and so on Polar signal representations and more has been app to now here you see a complex F transform in two dimensions but you may know that it's only a kind of plain wave analysis so each um element at a certain frequency in the X Direction and Y Direction Just represents an intrinsically one dimensional structure and if you instead look at the kernel of the cionic fuid transform which has here these two factors this Factor here and the factor over here you get with these two frequencies like Omega 1 here and Omega 2 here if the frequency is nonzero in both directions you get uh intrinsically two dimensional patterns and so this is ideal for analyzing texture or images which have intrinsic uh structures which are not just one dimensional and you can Define this split of a cian signal by taking the pure cian signal IJ or f and g in general from the left and the right and it gives you a really nice AAL split into two components of the cence and each of these two components is two dimensional itself and um this can be generalized replacing I and J by f and g in the transform and even FAL G makes a very nice split this split has been applied and has different name previously called Simplex perplex uh split U of pans and if you take this lit here and apply it to um this easy version of the F transform you get two parts the minus part and the plus part because it's a linear transform also the transform result can be split easily and the two parts are such that you can shift the two kernel factors on both on one side it becomes one a one-sided transform in the end which is easier to handle and here you see um discretization so that's how you would compute it with a computer and once you have this discritization you find you can also make fast transforms so computationally it's no problem to do it f as well and applications of this F transform are for images and can stretch reflect and rotate the uh image and you see what is the effect on the spectrum of the image and you can uh compute using for example aan filter minimal uncertainty and you can do texture segmentation and here I have written the transform again the kernel Factor here and here and the signal so that's under the integrant and um this is a transformation of the quic signal H and you can split H into its minus part and plus part and then these two factors they simply rotate the signal in the two opal planes in space ofans and an application here you take a color image and you encode the three colors r g and B into the components i j and k take the real part to be zero and then you take the pureing units in the gry line Direction where the three colors have equal strengths i j and k and then you do the split which I proposed previously and the minus part will give you the luminance and the Plus Part has two components and it gives you the prominance here which are traditional ways in um for engineers to um kind of analyze a color picture and you find in this formalism of pans that naturally comes up it's mathematically kind of very natural way of analyzing the picture and then we do the Quan fuid transform and um we take the result of the fuid transform and split it in a cian polar form so it has here a magnitude and here there is um exponential with uh an angle and a purean AIS which is locally dependent on the frequencies and uh this are the resulting components and now if we take here I take only to show the modulus part and you can do here a high pass filter here now this is a low pass filter a high pass filter and here this is a band pass filter and you see um the color component is actually in the low frequency part but the interesting Contours in the picture you find uh at the high frequency uh values oh yeah um recently um Dr sine gave a lecture at kak University in Tokyo and he also showed a nice video where uh he continuously changes the wids of the filter and if you go to YouTube and look for Steve S one you find the same picture and it's continuously uh change of the uh filter and you see how the granularity of the picture changes okay now um is very good for textual segmentation also disparity estimation intrinsic limitations of complex uh flu Transformations are overcome and what I don't treat today or explain there is a color sensitive Edge detection uh which is treated in this paper here and there's another way to do image registration using Quan Vena deconvolution uh and this is also very noise uh robust way in this here and now um you can use uh cionic and generaliz algebra foran types of transforms and these are very interesting so you take a kind of polar form of uh the signal in R Anda and you have r^ minus f1on unit and the frequency and e^ minus g k Theta the an and these quic in general the F Med transforms which is now here generalized to Quan is very useful for translation rotation and scale invariant um characterization of um images and objects and images and in this way it can also be generalized to color valid signals so not only black and white signals and um now I show you a comparison of the kernel patterns of these various transforms this is this standard for transforms in the complex domain and you see the col has this pattern and this are the elementary kernel patterns for different frequencies here and the X and the Y actually it's radial and um um angular Direction uh of the Fel transform and here for comparison is the cionic fu transform which has these plain wave patterns and the quion Fon transform and here um we take one pattern and we analyze its four cian directions because uh Kon are four dimensional so each uh pattern has four PLS and um now for the cic F milon transform uh you see there's a lot of intrinsic structure and the four parts have different Symmetry and the Symmetry is with respect to the rear line reflection across the rear line you have even and mod Symmetry and the unit circle you have here even Symmetry and here OD Symmetry and these symmetries can be Pure or mixed so you have another way way of analyzing the Symmetry in a signal and the transform may have high angular resolution High radial resolution or both and the CER of the transform is scale invariant so if you zoom into by factor of 10 or 100 the pattern appears the same all the time and that's this why the transation is taking okay now you can generalize it to the algebra of space and time which may be useful for satellite signals Geographic data Etc and here you have a signal which takes values in the volume time algebra of the space time algebra it's a sub algebra so e is the time Vector I3 is the Space volume and IST is a hyper volume of space and time and you have a k Factor on the left this is T is time Omega T is the frequency in the time Direction and here you have a Comm I3 is Scala of space and the three dimensional X vector and Omega Vector are the frequencies and the space for ordinates in three dimensions edian space and then um the split which I previously discussed is a split with respect to time which very important in physics or also applications and this one you can generalize say Lift to a full SpaceTime through a transformation so you Rite the same but now the signal is now full signal in the C algebra of space and time which is 16 dimensional so it has Scala and has four vectors three in space and one in the turn direction Six B vectors three of them are mixed um space time uh directions and uh four Tri vectors and one space time hyper volume component and then you take a signal in space time and you can split it with respect to the time direction and you get two signals one is propagating to the right and the other is propagating to the left and what you find it is a multiv vector wave packet analysis which naturally comes out um from um the analogy withans and this is very useful for analyzing uh for SpaceTime signal stretches Reflections rotations accelerations and boosts so changing the speed of an object in uh space and time if you have a satellite or a meteorite or something in the spectrum of based on to transform okay now the one standed um f transforms are very simple they look very similar to The Standard F transforms the only thing you change is that the F now is not the complex unit I it's a general multiv Vector with a negative square here and one application are spinorial different FID transforms and the spinner different FID transform can be used for Edge and texture detection and the square root of minus one which is used here is a local tangent by vector v to the image intensity surface so I've taken an image here and this is the intensity surface and then you take uh by Vector which is tangent to this kind of image intensity Mountain uh surface and so it changes locally depending on where you are uh in the image and this can also be used for gion um filtering now the pseudo scal for transform they have as the complex imaginary unit either the Sol Scala of a plane or of the threedimensional space and they have been studied a lot and applied a lot already so this I in is simply the product of E1 * E2 or E1 time 2 time 3 or in high dimensions and first it was used for processing electromagnetic fields and um also it can be used to define a two-dimensional analytic signal and this can be used in image structure processing or used in threedimensional Vector field processing also includ processing for sure and it has a very nice uh VOR signal convolution cem which allows you to to pointwise um multiplication of signal and filter in the F domain and then you can ful back transform with this transformation you can uh also do processing of two Dimension and uh it can also be applied for geographic information systems and climate data so now this particular case here it's a Scala which is used and you can decompose uh the components of the transform in four complex signals so the cic algebra of threedimensional space has eight components which I mentioned in the beginning Scala one Tri vectors one two scal then vectors three and B vectors three it makes all together eight and um so then based on this decomposition in four complex like uh signals with uh real and uh Scala component each you can um discretize transform and you get kind of four complex SE transforms combined and uh then you can make a fast transform as well and here's an application for Vector pattern matching so um this is um the flow field taken in a gas finess chamber and you see some flow field lines added for visualization and then you take this Vector pattern in Dimensions this um Vortex pattern here and you can um scale it from a 3 * 3 * 3 size pattern to this one here which is 5X 5 by 5 um um vectors or 8 * 8 * 8 so it's a more detailed um filter and then when you apply it uh these different colors here um indicate the dominating um results when you apply uh this filter so the um red one is a small Vortex structure and this is uh around it you have the green larger Vortex structure and in around it can have the yellow the largest no sorry the yellow is okay uh first is red then around it is yellow and B is a green one at the outside and you can use it for geographic information system data processing and so this is a kind of a big overview here and in addition you have here uhan domain for transform and this indicated in zoo are further types of hyper complex uh transforms which are kind of in a family with CPO 2 transforms and uh they have been uh studied uh recently and the Quan domain for transform has the advantage that the signal domain is not R2 so it's not just a static image with pixel coordinates in the X and Y Direction but um the signal itself um is defined over the space of perum so can be um the time and the space signal combined okay this is my uh conclusion the field has a history of some 30 years by now and I have actually only shown you part of this bind field I haven't time to treat everything and uh so there's some digits and um there's the G Net where you can get updated information on this type of research and there's a conference in Barcelona in Spain this summer and I want to invite you all uh to come to Barcelona and this conference is um from 27th to 351st of July and is a two-day School preceding the conference so if you say I'm not so confident um about using phot Etc don't worry first you attend the two day school you learn everything there are um some other teachers who can give better explanations than me and then confident so confident you can attend uh the conference as well okay thank you for my presentation and uh do you have any uh questions or comment thank you Professor I'm very interested in your presentation uh I also your information I I learn about you you uh you uh study this field for many years but uh now I'm starting to uh learn about study about the IND processing I I don't know this meod P to your presentation so I I I want some information about how to apply your for an to so can you have some suggestion because I'm the beginner yes yes so um yeah maybe I want to show you the name of the person who brought a very nice book last year yeah um sine and L these two and a third person from France his name is Nicholas Lan these three people they published the book last year I don't have it in my list of references yet and in this very nice Elementary textbook they explain the qu transform and give lots of examples for the color image processing applications I think that's a very good textbook to start with it appeared with Wy Wy yes I I oh yes please work together with Professor thank you any questions or comment I um want to um I want to um more detail and some Your Presence by V body for thank you very much