MATLAB codes and specification of cods for Example 1*.
\r\n\tThis project will provide an opportunity for all faculty members from all over the world to present their learning outcomes.
\r\n\tAfter all, modern society needs information in the field of psychological and pedagogical training of future students, teachers, scholars. The presented developments will add to the pedagogical case of knowledge, additional information that will be useful not only to teachers, teachers, and students but also parents.
I would like to begin the chapter with two quotations, I came across during my study, I like very much and I am certain, they literally express holography, its properties, its beauty and its role today, at the beginning either of the 21st century or the third millenium, we have an opportunity to be eye witnesses.
“Holography is the only visual recording and playback process that can record our 3D world on a two-dimensional recording medium and playback the original object or scene to the unaided eyes, as a 3D image. The image demonstrates complete parallax and depth-of-field. The image floats in space either behind, in front, or straddling the recording medium.” – is the first one. It is hard to find its origin, since it can be found at the beginnings of many papers dealing with holography.
“Since its first commercial usage in the 70’s the demand has only increased with each passing year. Holography has found its applications in almost all industrial sectors including commercial and residential applications. Next years might bring us to see a new world of holograms in every aspect. The use of holograms is the representation of a new visual language in communication and we are moving into the age of light as the media of the future. Holography will soon be an integral part of the light age of information and communications.“ [1]
Well, people today come across the term hologram and can meet holograms on banknotes, various cards and products. A holographic technology comes along with advertisements, promotions,... People specialized in photonics [2], an interdisciplinary field dealing with utilizing photons, may know that holography is closely related to a special kind of light – laser light, which possesses a special property – coherence, i.e. all the light waves coming from a laser have the same properties. It is very important, because there are two basic physical light wave phenomena, enabling holography.
To record, i.e. to create a hologram, the phenomenon of two-wave interference of light is applied. The term hologram tells us that all the information transported by light wave is recorded. What does it mean – all the information? Let us repeat some basic terms [3]. Modelling light as a light wave A(r, t) depending on space (r) and time (t) means, that there are two main parameters – the amplitude\n\t\t\t\tA0 of a wave and its phase\n\t\t\t\tΦ
where r(x, y, z) is a displacement vector determining a point in a space and k(kx, ky, kz) denotes for a wave vector, determining direction of wave propagation. Its absolute value k = 2π/λ defines the wave number. Φ0 is the initial phase of the wave. The distance between the two neighbouring amplitude peaks of the same kind is wavelength λ [m]. ω [rad.s-1], is angular frequency, which is related to the linear frequency, ν [s-1], by the formula ω = 2πν. It lasts T = λ/c seconds to pass the path λ at the speed c. Such a time interval is called period and T = 1/ν.
A wave front is another useful term for us, else. It is the surface upon which the wave has equal phase. It is perpendicular to the direction of propagation, i.e. to the wave vector k.
For example, considering a wave propagating in the +z direction, the wave vector k is parallel to the z–axis everywhere. Because of that, wave fronts are parallel planes, perpendicular to the z–axis. Such a wave is known as a plane wave. When the k(kx, ky, kz) direction is general, the phase in a point determined by a displacement vector r(x, y, z) includes the scalar product of k.r.
Let us mention also a spherical wave
which is irradiated from a point light source in homogeneous medium. In such a case wave fronts are centrally symmetrical spheres, so it is enough to consider only radial coordinate r of spherical ones. Moreover, k and r are parallel, so k.r = kr. Increasing distance from the source the surface of the sphere increases and amplitude A0 decreases proportionally to 1/r.
It is more convenient to represent the light wave expression in a complex form
As for light wave recording, one has to realize a special property of light waves. The instantaneous amplitude A, which varies with both, time and space, cannot be measured experimentally in a direct way. The frequency of the light wave is too high for any known physical mechanism to reply to the changes of the instantaneous amplitude A. Any known detector or recording medium replies only to the incident energy. When denoting energy transferred by a wave as E, it can be got as square of the amplitude, i.e. E = (A0)2 = A.A*, when using the complex representation, *-symbol represents the complex conjugate quantity.
The value known as intensity\n\t\t\t\tI of light, is proportional to the energy per unit of surface and unit of time. It is very important to realise that the time averaged light intensity is a measurable value, only. Because of that it is said that both, light detection and light recording are quadratic.
It is just holography, which provides us with a method allowing to record both, the amplitude and phase information despite quadratic recording. Naturally, it is impossible when there is only one light wave. Dennis Gabor realized, that there is included the phase information in intensity interference pattern of two waves and it can be utilized in a new – holographic method of recording [4].
When two light waves meet and are able to keep a constant phase difference at any point for a proper time interval (long enough to make a record), the interference pattern, i.e. a space redistribution of the resulted energy, can be recorded. Such a situation can come truth only for two coherent waves. Just because of that boom of practical holography started after laser had been invented [5].
To reconstruct, i.e. to see what information is hidden in a hologram, another basic physical phenomenon is applied – diffraction of light. There are various types of holograms. Some of them can be reconstructed only by laser light. However, there is also a group of holograms, which can be reconstructed also using common sunlight or another source of white light, in which, all the visible spectrum is included. Naturally, that group takes the greatest interest among public and this chapter is going to deal just with such holograms, named here white light reconstructed holograms (WLRH).
In the beginning, the basic properties of diffraction of light by a periodic structure are going to be noticed and correlation between diffraction and holographic reconstruction shown. Attention is concentrated especially to the white light diffraction. Understanding of the principles will help the reader to understand problems arising during white light reconstruction of a hologram and also give a hint how to proceed when recording a hologram to avoid such problems. Attention is going to be given to Denisyuk’s holograms, image holograms and Benton’s rainbow holograms.
The last part of the chapter is focused on some applications of WLRH, especially public ones. To mention also a scientific application, one of our former works, dealing with determination of index of refraction radial profile of a fibre is described briefly.
The simple fact that there are two groups of holograms, one of them reconstructed only by coherent light, another one by common white light, tells us that the secret must be hidden in the hologram recording process. To reveal the secret of recording, it would be usable to understand phenomenon of light diffraction, which is the physical principle of hologram reconstruction.
Firstly - it is said that any deviation from rectilinear propagation of light that cannot be explained because of reflection or refraction is included into diffraction. Such a deviation can be met when light either passes through or reflects from a structure, i.e. a space distribution either of the transmittance, or the refractive index.
Telling more precisely - diffraction is the spreading of waves from a wave-front limited in extent, occurring either when a part of the wave-front is removed by an obstacle, or when all, but a part of the wave-front is removed by an aperture or a stop. The Fraunhofer theory of diffraction, which is interesting for us from the point of view of hologram reconstruction, is concerned with the angular spread of light leaving an aperture of arbitrary shape and size [3].
Secondly – remember that a hologram is, in fact, a two-wave interference pattern recorded, i.e. a kind of structure of intensity distribution. It depends on the type of recording medium and related light-matter interaction mechanism, which of its optical properties distribution follows the interference pattern intensity distribution.
And thirdly – when reconstructing a hologram, it has to be illuminated with light that either passes through or reflects from the hologram. Now, it seems to be obvious to put the equal sign between reconstruction and diffraction.
An introduction to phenomenon of light diffraction and related basic relations can be found in any basic book of optics/photonics, e.g. [3] and even in [6]. Let us show briefly some basic results, related to white light reconstructed holograms.
For the readers, taking a deeper interest - exact solutions of diffraction problems are given by solving Maxwell’s equations. However, well-known Kirchhoff’s scalar theory gives very good results if period of diffraction structure does not approach a wavelengths size and amplitude vector does not leave a plane.
Let the plane (x0, y0 ) is the plane of a structure and diffraction is observed in the plane (x, y). To find resulting amplitude AP in P(x, y), amplitudes of spherical waves from all the point sources in the plane of the structure have to be summed (Fig. 1). The idea is expressed by Kirchhoff’s diffraction integral
where the distance r2 = [(x - x0)2 + (y - y0)2+ z2]1/2 and S is size of the illuminated structure. The experimentally observable interference pattern is given by
To the principle to solve scalar diffraction problems
To calculate the integral (4), two approximations for r2 are used – Fresnel’s approximation and Fraunhofer’s ones. For the purposes of this chapter, let us mention Fraunhofer’s aproximation: x0, y0, « z, i.e. next mathematical approximation is used to express the phase
and integral (4) turns into
A holographic record can be regarded as a record of a general structure. The light diffraction theory (5) gives us a detail description of light diffraction at a regular plane grating. It is widely used, especially in spectroscopy (looking for various wavelengths).
Diffraction by a plane grating (d – grating interval, b – slit width)
The grating is an ensemble of single equal slits, parallel to each other and having the same distance between each other.
There are two parameters, which define the grating – the slit width (b) and the grating interval (d) – distance between centres of any two adjacent slits (Fig. 2.). Symbols θi and θ denote angles of incident and diffracted waves respectively.
Diffraction at such a grating is, in fact, the interference of many “diffractions” by single slits. The number of interfering “diffractions” depends on the number N of illuminated slits. A detailed calculating of (5) results in the angular intensity distribution
and it’s normalized value can be get in the form
The angle θ appeared because of the Fraunhofer’s approximation, where x/z ~ sinθ. In fact, relation (6) expresses N-wave interference (IR2) modulated by diffraction at a single slit (IR1). Practically usable is especially the condition for interference maxima
where k = 2πn/λ0, n – refractive index of the space of diffraction, λ0 – wavelength in vacuum and m – order of diffraction. It is obvious that real values of m can be achieved only when grating constant d > λ0/n = λ. For simplicity, only one-dimensional (x) structure distribution was considered.
Study in the region of X-rays contributed to the phenomenon of diffraction. The short wavelengths of X-rays are not well suited for diffraction by optical gratings. They are, however, conveniently close to the spacing of atoms in crystal lattices, which therefore provide excellent three-dimensional diffraction gratings for X-rays. It was shown, that the diffraction pattern is intimately connected with the arrangement and spacing of atoms within a crystal, so that X-rays can be used for determining the lattice structure [3].
It was Max von Laue who first suggested that a crystal might behave towards a beam of X-rays rather as does a ruled diffraction grating to ordinary light. It is interesting, that at the time it was not certain either crystals really were such regular arrangements, or X-rays were short-wavelength electromagnetic radiation [3]. W. L. Bragg proved the idea in 1912.
Diffraction by a volume grating (h – thickness, n – index of refraction)
It can be derived in a simply way taking into account the result for plane grating and the influence of a thickness h of the grating (Fig. 3). Let us consider angles of incidence θi and angle of refraction θ. To get the intensity maximum in passing light, the condition (7) has to be fulfilled, i.e.
Moreover, because of the thickness of the grating, all the waves irradiated along the z-axis should be in phase in direction of the angle θ, i.e.
The condition has to be satisfied for all the z coordinates. It is possible only for θ = ±θi that leads to the well known Bragg’s law for X-ray reflection from a crystal
used also to describe diffraction at optical volume gratings.
Concluding this paragraph, let us try to describe the diffraction phenomenon using holographic terms. Two coherent plane waves
interfere (Fig.4a). The interference pattern that will work as a grating can be expressed in the form, which corresponds with the amplitude transmission t of produced grating
Let us suppose the grating to be very thin and illuminated by a plane wave w0 in the z axis direction (Fig. 4b). The phases of all the waves can be expressed as
The light wave w passing the grating consists of three parts
The phases of three waves in (14) are
Diffraction as holography. a - 2-wave interference produces a grating; b - diffraction at the grating
Supposing φ0 = 0, φ1 = α/2 and φ2 = -α/2 one can obtain phases
Let us compare (16) to the light diffraction at a plane grating with the grating interval d. The grating interval in our case is
Taking into account θi = 0, the condition (7) for intensity maximum turns into
The relation (18) gives the same angles of propagation sinθA = 0, sinθB = 2sin(α/2), and sinθC = 2sin(-α/2), as relations (16).
Now we posses all the knowledge to realize what kind of problems can be met when reconstructing a hologram using white light and even to find a way to avoid them. There are two important results that had been derived above – relation (7) expressing the conditions for existence of a diffracted wave of the order m at a plane grating that can be rewritten into:
The second one is the same kind of relation, valid for a volume grating - relation (10)
Let us consider white light reconstruction of a plane hologram, which deals with the relation (19). It is obvious that when using waves with various wavelengths λj, they will be diffracted at various angles θj. When considering reconstruction as a diffraction, m =1. The angle θ of diffraction depends on the wavelength λ0j and various angles θj(λ0j) will cause various positions of the reconstructed object (Fig. 5.) Naturally, it results in “a blurred” reconstructed object both, horizontally and vertically. However, looking at Fig. 5, it could have occurred to us that if the distance between the object and hologram would have been smaller a less blurred reconstruction could be obtained (image holograms).
White light reconstruction – the scheme and white light & HeNe laser reconstruction
Moreover, not only a colour blur appears. It is also a space blur present. Both, lens and hologram create an image, so hologram is often compared to a lens. So-called hologram formula\n\t\t\t\t
similar to the lens formula can be derived, too. A hologram also can be characterised by its focal length fH. It depends on the wavelength of light when either recording (λ1), or reconstructing (λ2), object distance from the hologram (R1) and distance of the source of both, reference (L1) and reconstructing (L2) beam from the hologram [7]. Paraxial approximation is supposed, all the distances are measured perpendicularly to the plane of hologram and indices (signs) r (±→-), v (±→+) in (21) are related to real and virtual reconstruction.
In fact, the relation (21) defines a space blur (various R2j) because of various wavelengths λ0j. Remembering of various horizontal and vertical resolution of a human eye, it gives us another hint – to try to limit a possibility of the reconstruction in whole (Benton’s holograms).
Now, let us take an interest in volume holograms, where results obtained for diffraction at a volume grating were obtained. Ideally there are only two diffraction grating order numbers relevant – m = 0, 1. Intensity of higher diffraction grating orders diminish to zero because of mutual interference and the condition (20) turns into
It is a very interesting and for WLRH important result. According to the relation (22) there is an unambiguous relation among the grating period d, wavelength λ0j and angle θj = θij. That means – when reconstructing by white light there is the only special angle for every wavelength. This way, the volume hologram picks from the white-light spectrum different reconstruction angle for every wavelength, i.e. prepares time coherent light for reconstruction (Denisyuk’s hologram). It depends on the method of hologram recording if the reconstructed object is observed either in single colours or colourfully.
Let us proceed following the history of holography and start with Denisyuk’s holograms. In 1962, Yuri Denisyuk combined holography with 1908 Nobel Laureate Gabriel Lippmann\'s work in natural color photography [8]. Denisyuk\'s approach produced a white-light reflection hologram which, for the first time, could be viewed in light from an ordinary incandescent light bulb [9].
To explain the principle briefly, at least: Isaac Newton found out colours produced by a very thin layer due to interference of light. Colours in butterflies are the result of interference phenomena, too. And in 1886, when the photography was still struggling to transfer the colours of nature to adequate tonal values in black and white, the French physicist Gabriel Lippmann conceived a method to record and reproduce colour images directly through the wavelengths from the lighted object. He introduced a photographic colour process that demanded no colorants, dyes, or pigments, based on light waves interference principles, too.
Principle of Lippmann’s colour photography. a - colour photography (1-object, 2-lens, 3-image); b - two-wave interference (w1 and w2-interfering waves, 1-photographic glass plate, 2-photographic emulsion, 3-mercury mirror, n(h)-refractive index (thickness) of the emulsion.
He used a photographic emulsion between a photographic glass plate and a mercury mirror (Fig. 6a). The glass plate and emulsion are nearly transparent. Light waves coming from the object reflect from the HG-mirror (w1) and interfere with coming light waves (w2) – Fig. 6b. Since the Hg index of refraction prevails the emulsion index of refraction n, standing waves are created [3]. The interference pattern consists of nodes-surfaces and loops-surfaces, i.e. dark and bright interference surfaces that are recorded in the fine grain emulsion. Fig. 6b demonstrates a simple section of the interference surfaces to evaluate their spacing λ0/2n, which depends on the wavelength λ0.
It can be derived in a very simply way, when realizing the phase difference ΔΦ(z) between two plane waves meeting at the coordinate z
If ΔΦ(zm) = 2mπ, where
This way the processed photographic emulsion became a layer where beside an image also volume gratings with various grating intervals, related to the local wavelength, were recorded.
Taking into account the former paragraph (2.2), it is obvious that after the photograph is illuminated by white light, the image in its original colours can be observed because of diffraction of white light at such a structure.
It is hardly imaginable that in the late 1800s such an advanced photographic technique was already conceived and realized. The resemblance with volume holography, published in 1962 by Yuri Denisyuk, is striking. Lippmann photography actually is the ancestor of volume reflection holography or Denisyuk holography, which is sometimes also referred to as Denisyuk-Lippmann-Bragg holography.
Yu. Denisyuk, inspired by the ingenious Lippmann’s colour photography technique did extensive theoretical analysis and experimented with mercury lamp sources since 1958 and even enlisted colleagues to develop a special thick, high-resolution, and relatively sensitive photographic emulsion to record the wave pattern in depth [10].
Later, realizing the potential of wave-front reconstruction, devised a different approach to record a hologram. A laser beam illuminated an object through a photographic plate (reference beam) interfered with light reflected from the object (object beam) to produce a hologram in the layer of the photographic emulsion (Fig. 7a).
Reconstruction can be performed by sunlight or another white-light source lighting the hologram. If the direction is the same as when recording, the Bragg’s diffractive reflection provides us with a monochromatic reconstructed image of the object in the wavelength used when recording the hologram (e.g. Fig. 7b). Fig. 7c demonstrates a possible two-beam experimental set-up.
Denisyuk’s 1-beam holographical set-up. a – recording (1-object, 2-recording medium, 3-reference beam, 4-object beam), b – reconstruction (1-hologram, 2-reconstructing beam, 3-reconstructed wave, 4-holographic image), c – 2-beam set-up (1-object, 2-recording medium, 3-reference beam, 4-object beam, 5-beam splitter, 6-mirror).
However, as mentioned above, such a reconstruction is a monochromatic one, only (Fig. 8). What about colour holograms?
To produce colourful hologram, one needs three lasers generating on three basic wavelengths (red, green, blue) to record such a hologram. Each of the waves creates own interference structure (Fig. 9). After illuminating such a hologram with white light, each structure helps to reconstruct the object wave in related wavelength. The reconstruction can be seen in three colours and thanks to sophisticated activity of our brain as colourful.
Yu. Denisyuk reconstructed from a hologram and alive [11]
A colourful hologram recording (1-object, 2-recording medium, 3-optics to spread light, 4-mirror, 5-semitransparent mirrors)
Yu. N. Denisyuk presented his technique as a generalized form of Lippmann photography, or as a colour-dependent optical element. This technique, using reflection holography and the white-light reconstruction technique, seems to be the most promising one as regards the actual recording of colour holograms [12].
Writing on white-light reconstructed holograms it is necessary to take into account that depending on the relation between the thickness h of the recording medium and the average spacing
introduced in [13]. Q < 1 corresponds to thin gratings, while Q > 1 corresponds to volume gratings. Moreover the objects around us can be defined as either 2D or 3D objects.
It is obvious that Denisyuk’s method of holography is applicable to any object and the recording media suitable to create a volume hologram. Moreover, it represents a reflection holography, i.e. the reconstructed wave seems like reflection of the reconstructing wave from the hologram. When recording, the reference beam and the object beam come from opposite sides of the recording medium. Because of that, when reconstructing, observer and the source of the reconstructing wave are on the same side of the hologram. The next two paragraphs are dealing with so-called transmission thin holograms.
Let us remember of the physical principle of reconstruction of a hologram. It is diffraction of either transmitted or reflected incident, i.e. reconstructing light. Due to that, the white-light reconstruction of transmission thin holograms brings a possible colour and space blur of the reconstructed image (Fig. 5). The illustrative Fig.5 gives us a hint to decrease the object – hologram distance when recording the hologram as much as possible to avoid the colour blur. It would be the best to decrease the distance to zero. However, the reference beam must not be blocked.
Image holography. a – experimental set-up (1-laser & collimator, 2-beam splitter, 3-mirror, 4-hologram, 5-ground glass, 6-object, 7-projecting lens; b - objects, c - reconstruction
There is a simple possibility to put the recorded object (Fig. 10b) even “into” the recording medium and not to restrict the reference wave while recording the hologram – to project it there by a projecting lens (Fig. 10a). When reconstructing in white light the reconstruction is observed in various (rainbow) colours from various directions at the same place (Fig. 10c.).
When the image coincides with the hologram plane, it is perfectly sharp. Naturally, projection of a 3D object has 3D properties, too. In practice, display holograms up to around 2 cm in depth can be acceptable. This type of hologram is called an open-aperture hologram [14]. However, such a simple method is used mostly for approximately “2D” objects like medals, coins, photos, and so on.
A more detailed estimation of the image blur can be found in [15]. It can be shown that if the source used to reconstruct the hologram is located at the same position as the reference source used to record it and has very nearly the same wavelength, the blur ΔxR2 of the reconstructed image along the x axis for a source size ΔxL2 is
Similarly, if the source used to reconstruct the hologram has a mean wavelength λ2 very nearly equal to λ1, the wavelength used to record the hologram, and a spectral bandwidth Δλ2, the transverse image blur ΔxR2 due to the finite spectral bandwidth of the source of reconstructing wave, located at xL2 can be shown to be
The image blur increases with the depth of the image and the interbeam angle.
It follows from (25) and (26) that if the interbeam angle and the depth of the object are small, it is possible to use an extended white-light source to view the image.
When a real 3D object is recorded the well-known Benton’s method [16], which allows us to get rid of vertical blur of the reconstructed image, has to be used. The Benton (rainbow) hologram is a transfer transmission hologram, which reconstructs a bright, sharp, monochromatic image when illuminated with white light [17]. Benton holograms are produced by means of an optical technique that sacrifices the vertical parallax of the holographic image in favour of a sharp monochromatic reconstruction by a white light point source. In other words - the physical basis of this method is to reduce the amount of information on the hologram. The vertical parallax is eliminated. The method relays on the fact that human beings have two eyes in horizontal position, i.e. people are less sensitive to vertical parallax.
In fact, it is a hologram of a hologram. The first (master) hologram H1 of the object O is produced (Fig. 11a). The object and reference waves are directed around the horizontal plane.
Then, H1 is masked with a narrow horizontal slit S and wave 1, conjugated to the reference wave 2 from Fig. 11a is used to reconstruct the free part of H1. The conjugate wave [6] of the original object wave is reconstructed and the real holographic image HI serves as an object to record the second hologram H2 (the part a) in Fig. 11b). The part b) of Fig. 11b demonstrates the view to the process of recording H2 from the side. The 3D object, represented by reconstructed HI straddles the plane of medium to record H2 and a converging reference wave 3 is now inclined in the vertical plane.
When viewing such a hologram in the same monochromatic light as the recording one, the holographic image from H2 that straddles the plane of H2 is reconstructed. However, while recording H2 the width of the slit S limited the vertical parallax of the object and it will be limited when observing the reconstruction, too. That means, there is a small interval of angles in vertical direction that allows observing the reconstruction of H2, i.e. the perspective information in vertical axis is lost. The horizontal parallax is much more wider, given by the width of H1 determined by the length of the slit S.
Principle of Benton’s rainbow hologram. a – object O, hologram H1, object wave 1, reference wave 2; b – hologram H1, slit S, reconstructing wave 1, reconstructed wave 2 as a new object wave, holographic image HI, reference wave 3, hologram H2; c – hologram H2, reconstructing white light 1, colour reconstructed waves 2 related to various angles α.
It seems like a kind of limitation, but remember – when recording H2, the reference wave 3 was inclined in the vertical plane. This way, the hologram H2, which is a set of arbitrary gratings with horizontal fringes in principle, became a dispersion element for vertical angle variations. That provides the hologram H2 with colour dispersion. With a white-light source, located approximately in the convergence point of 3 (Fig. 11c), the reconstruction is dispersed in the vertical plane to form a continuous spectrum (hence the term "rainbow"). An observer whose eyes are positioned at any part of this spectrum then sees a sharp, three-dimensional image of the object in the corresponding colour. In [18] is presented an optical system that permits both steps of the recording process to be carried out with a minimum of adjustment.
The image reconstructed from a rainbow hologram is free from speckle, because it is illuminated with incoherent light. However, it is not free from blur. A more detailed analyse of rainbow holograms can be found in [19].
One cause of image blur is the finite wavelength spread in the image. The maximum wavelength spread observed when the rainbow hologram is illuminated with white light can be estimated as
where θ is the angle between object and reference waves when recording H2, L is the distance between H1 and H2, D is eye pupil diameter and b is the slite S width. The eyes of the person viewing the image are supposed to be placed in the position, where the image of the slit is formed. The relation (27) leads to the image blur due to wavelength spread of about
where z0 reprezents the H2-to-image distance generally, when the image does not straddle the H2 plane. With z0 → 0 the image blur (28) also goes to zero.
Another cause of image blur is the finite size of the source used to illuminate the hologram. If the source has the angular subtense ΩS, as viewed from the hologram, the resultant image blur is approximatelly
A final cause of image blur is diffraction at the slit
White light reconstruction of Benton’s hologram “Holography Blocks” (made of tens of 1 inch transparent acrylic cubes) photographed from two directions to demonstrate its 3D property [20].
However, this can be neglected unless the width of the slit is very small.
Concluding this part, I would like to memory the inventor of this kind of holography, Benton S. A., and include the reconstruction of his Holography Blocks, the transmission rainbow hologram that he made of tens of 1 inch transparent acrylic cubes (1975). The pictures (Fig. 12) were photographed from his hologram from the left and from the right to demonstrate its 3D property by R. L. van Renesse and published in his monograph written in remembrance of Steve Benton after he passed away, in 2003 [20].
This paragraph consists of two parts. The first one mentions some situations briefly, at least, when anyone can meet white light reconstructed holograms. For those, taking an interest, there is great amount of Internet information accessible now, even pointing to details related to methods and technologies to produce WLRH.
Naturally, there are many scientific applications of WLRH, too. To solve the problem of commercial application can, also, be considered as a scientific application. I decided to use the second part to present short information, at least, on one of our former works, which was published in Slovak, only. It is focused on using the phase-shifting image holographic interferometry to study an optical fibre refractive index radial distribution. The image holography was used for us to be able to profit from possible magnification of the studied region.
When considering current applications of holographic technology enabling white light hologram reconstruction, i.e. displaying and viewing holograms using a common white light source, consumer products and advertising materials have to be mentioned firstly [1].
Security and product authentication seem to be the most popular growing areas for the use of holograms, especially of white-light reconstructed holograms. Why is it so?
Generally speaking, holograms can reconstruct one of two waves used to record them, when illuminated by the second one. That means optical reconstruction of 3D space from a 2D record. After four milestones in history of holography, represented by D. Gabor (father of holography), Yu. N. Denisyuk (invented volume reflection holography), E. N. Leith and J. Upatnieks (invented off-axis holography), and S. Benton (invented rainbow holography), the subsequent development of the micro-embossing technique allowed their mass replication [21].
For example a very popular variation became the embossed hologram. Such holograms are easily produced in mass quantity and at a very low cost. The holographic structure is recorded on a light sensitive medium (a photo-resistor), which can be processed by etching and a microscopic relief is created. By electro-deposition of a metal (nickel) on the relief a stamper is made and its surface relief is copied by impressing it onto another material (e.g. polyester base film, thermoplastic film).
It was soon realized that holograms might be used as security features on valuable documents and products and this way classical hologram became the first of a manifold of diffractive structures developed to thwart counterfeiting. Many of these diffractive structures can no longer be called holograms in the strict sense and some of them are no longer made by laser interference techniques, but created by advanced electron beam lithographic techniques [21].
These holograms, i.e. diffractive elements provide a powerful obstacle to counterfeiting. One can meet holograms either on various goods itself or in the packaging of products, on banknotes, various types of cards, and so on. For example - almost all credit cards carry a hologram, which is a good sign that security holography has proven to be very effective. There are various kinds of holographic labels and stickers [1].
Holographic labels are firmly affixed on the desired place of a product to verify its genuineness. Such labels cannot be copied, altered, either adapted or manufactured in an easy way. Moreover, hidden information can be embedded, visible at special circumstances, only. To increase the uniqueness of holographic labels, there are used another specific techniques, e.g. custom etching and overwriting of labels. To give some examples, holographic labels are used in various kinds of cards, artwork, banknotes, bank checks, product packaging for brand protection, alcohol, cosmetics, and so on.
Holographic stickers (HS) – most of them are self adhesive, providing authentication, security and protection against counterfeit, too. To increase their security level, various techniques are used. Let us mention some of them, like laser beam engraved dots (dot-matrix HS), double-exposed hologram of two objects from two directions (flip-flop HS) that displays two images from two different viewing angles), combinations of holograms, micro-information included, visible only by magnifier (micro text/image HS). Locally different thickness of the hologram layer gives reconstruction in various colours. A hidden text or image invisible to the naked eye but visible by means of a laser reader, serial numbers, overprinting left after the sticker is peeled off, are next examples of holographic stickers security increasing.
To minimise counterfeiting in holograms various methods during recording are used. It is possible to include hidden information or make the image so complicated that it is not worth to duplicate it, considering the time and money involved [1]. However, hidden information is of great value only if the cheater cannot find it or duplicate it. So, effective use of hidden information or any kind of complex images requires some sort of relatively simple and inexpensive reading device or decoding device.
Another possibility profits from using variable processing parameters, like randomly changed exposure, development time or other processing parameters to produce variable shrinkage all over the hologram. This results into a hologram whose colour varies from point to point and when using a monochromatic source - laser the brightness changes dramatically with the shrinkage of the film.
Holograms are not easy to counterfeit either if variable information like serial numbers, encoded personal information or dates are included or they are made of some special materials. Combined countermeasures can be another effective approach, too.
To show another ways of WLRH using, let us remember that in the beginning, holograms, a kind of "windows with a memory" with unfamiliar properties, were regarded as similar to daguerreotypes, photography predecessors [22]. Like early photography, holography was expected to develop technically and become cheaper, more capable and widespread. Where possible, people worked to realize these expectations of progress, especially for trade show displays. A pulse ruby laser had been developed to enable recording a human portrait in a darkened room to create static 3D human scenes for advertising. While such holograms attracted interest their display requirements (laser) made them too unwieldy to be sold or even displayed outside the laboratories.
Some artists, supported by scientists at first, began to take up holography. Inspired by the art and technology movement that was then exploring videotape, architecture, and other influences, they sought to make fine-art holograms. A second group - artisans intended to make holography an expressive medium for anyone.
However, the cost of the laser, needed for both, recording and viewing holograms, was a crucial constraint for artists and advertisers. Moreover, the monochromacity of laser light provided portraits inferior to the panchromatic black-and-white films. For photographers then, holographic portraiture represented problems not progress. These limitations restricted holography to a narrow class of subjects and applications [22].
Denisyuk\'s holograms offered a solution of the most pressing problem for aesthetic and commercial users - the need for a laser to display the hologram. Then Benton’s rainbow hologram also became widespread, which could be viewed in white light in a spectrum of colours.
As mentioned above, embossed holograms provided new audiences, manufactured by the millions on metal foil, they became ubiquitous in packaging, graphic arts, and security applications. Embossed holograms were inexpensive, reducing the cost of copies by a hundredfold. They could be mass-produced reliably by using a number of proprietary techniques. And they were chemically and mechanically stable, unlike most previous hologram materials that were susceptible to breakage, humidity, or aging [22]. Together, these technical advantages promoted the widespread application of embossed holograms.
On the other hand, their flexibility, particularly on magazine covers, caused colour shifts and image distortion. Moreover, the holograms were usually viewed in uncontrolled lighting, images could appear fuzzy or dim. In response to these limitations, their producers progressively simplified the imagery. This way embossed holograms promoted low-cost mass production but had relatively poor image quality. However, these characteristics were deemed to be a serious defect for imaging purposes. Fine-art holograms declined in popularity, with artists complaining that embossed holograms irreparably devalued the aesthetic attraction of the medium [22].
Despite all of that, holograms are used in advertisement to attract potential buyers. They can be met on magazine and book covers. Display holograms are widely used wherever an audience needs to be reached (e.g. at trade fairs, presentations). Holograms found their place in the entertainment industry (movies), became popular in the area of packaging and for promotional purpose. This type of holography can be found as a part of some either pure or combined holographic artist works (in special galleries), too [22, 23].
There is a special interest focused to optical fibres all over the world. They became of great importance as parts of many photonic devices and various methods are used to study their properties. Radial profile of fibre optical thickness, i.e. of its refractive index is one of the most important fibre characteristics. Many papers arise contemporary. Mostly they are based on classic interferometric microscopy of a fibre located in a wedge shaped layer of an immerse liquid. When applying optical imaging, methods are limited by the diffractive resolution λ/2n.sinθ, where n is the refractive index of the object space, θ is the angle related entrance aperture of the objective. The limit follows from the Rayleigh criterion, applicable in the Fraunhofer approximation. Taking into account a conception of near-field optics it would be a better solution [12].
The experimental method elaborated in our paper is based on double-exposure phase-shifting interferometry. Moreover, the image holographic interferometry was used to profit from both the advantages - lower demands on optical quality of all the set-up elements and also the possibility of white-light reconstruction and magnifying the object size. Moreover, phase-shifting interferometry allows determining both, the size and direction of optical path changes. Fig. 13 demonstrates the experimental set-up. The laser beam is divided into collimated waves, object (o) and reference (p) ones. The object V is projected on the recording medium with a proper magnification and hologram H is recorded. To apply the phase-shifting interferometry the wedge K inclines the reference beam between two expositions. V represents two states of the object – either a glass cell filled with glycerine (nK = 1,4534 ± 0.0005) and the fibre embedded into it, or the glass cell filled with glycerine without the fibre.
The glycerine helped to reduce somehow the influence of the basic index of refraction n0 of the fibre on the interference pattern. Accurately, the cell exit was projected on the plane of hologram. The object wave passes through the fibre perpendicularly to its axis.
The glass wedge K, passed through by the reference wave enabled phase-shifting interferometry. It was a very simple method to change the object-reference-beam angle in the order of the hundredths of a degree [24].
Two fibre specimen of the same kind from Slovak Academy of Sciences were used with radii R1 = 0.825 mm and R2 = 0.55 mm. The numerical aperture of them was 0.22 and relation between radii RJ = 0.9R (Fig. 13).
The basic fibre index of refraction was n0 = 1,45718 (at λ = 632.8 nm).
Interferograms (e.g. Fig. 14) were analysed supposing radial refractive index increment distribution Δn(x, y) within the core radius RJ and the thickness of the glycerine layer d. The change of optical thickness Δl at coordinate y (Fig. 13) can be expressed in the form
Experimental set-up. L–He-Ne laser, D-beam splitter, U-switch, Ki-collimators, Zi-mirrors, K-wedge, V-the object, OS-objective, H-hologram, o (p)-object (reference) beam. In left bottom corner - the scheme of the fibre cross-section in a cell.
Demonstration of interferograms.
Supposing axially symmetrical refractive index distribution, it is convenient to introduce the substitution
Interference maxims appear when Δl is equal to an integral number of wavelengths - mλ. Due to that the interference order m becomes a function of the coordinate y
The shift Δm of the interference order was estimated from the interference pattern obtained (Fig. 14), which were digitalized and numerically analysed using the relation (33). Fig. 15 is to demonstrate analysis of one set of experimental data. It is necessary to realize that also the change ΔmC(y) caused by the cylindrical shape of the fibre, denoted as y0 in Fig. 15, is included in the experimentally obtained distribution of m(y) expressed by ym in Fig. 15. The distribution ΔmC(y) = y0 was determined theoretically using the known fibre parameters.
Analysis of experimental data
This way the producer’s data, related to the original preform were approved (a stepped-index fibre with core radius RJ = 0.9R and Δn = 0.016). The results obtained confirmed possibilities provided by holographic interferometry to analyse refractive index profile of an optical fibre in the region of sizes not demanding near-field optics concepts. The numerical analysis had been improved applying method of genetic algorithm and extended to a gradient-index fibre [25].
Today, holography seems to be a common method of optical information recording and especially - information advertising. Not only holograms can be met everywhere, moreover, terms, like “hologram” and “holographic” can be found in many fields of human life.
I suppose white-light reconstructed holograms to be of great interest especially for common public, without a special optical education. However, it is not possible to restrict their usage for public reasons, only. They are applicable as both, scientific and measuring tools, too.
Great amount of information is accessible by the Internet. However, it is important to read patiently, to find the truth. Due to that I tried to explain especially the basic principles to understand the matter.
To conclude, I would like to use another quotation from a very interesting paper, devoted to a historian’s view of holography [22]: „From a historian\'s point of view, then, holography represents a fascinating case of modern science and technology. It is a complex example of a surprisingly common but little noticed situation in modern science, in which a technical subject has created new communities and grown with them. Its evolution has been distinctly different from what most historians of science-and even holographers-might have expected, which can help us to better understand how modern sciences emerge, and how to more realistically chart their future trajectories. And because of the rich variety of communities that the subject has embraced, ranging from artists to defense contractors, its history is likely to be of enduring interest to broad audiences.“
I would like to express this way my sincere thanks to Faculty of Mathematics and Physics, Comenius University at Bratislava, which provided me with a possibility to get familiar with beauty of new, contemporary optics. I, also, would like to express my thanks to all of my close colleagues and students, for transforming the atmosphere of my working place into a home atmosphere, indeed.
Artificial neural networks (ANNs) are a type of the computing system that mimics and simulates the function of the human brain to analyze and process the complex data in an adaptive approach. They are capable of implementing massively parallel computations for mapping, function approximation, classification, and pattern recognition processing that require less formal statistical training. Moreover, ANNs have the ability to identify very high complex nonlinear relationships between outcome (dependent) and predictor (independent) variables using multiple training algorithms [1, 2]. Generally speaking, in terms of network architectures, ANNs tend be classified into two different classes, feedforward and recurrent ANNs, each may have several subclasses.
Feedforward is a widely used ANN paradigm for, classification, function approximation, mapping and pattern recognition. Each layer is fully connected to neurons in another layer with no connection between neurons in the same layer. As the name suggests, information is fed in a forward direction from the input to the output layer through one or more hidden layers. MLP feed forward with one hidden layer can virtually predict any linear or non-linear model to any degree of accuracy, assuming that you have a appropriate number of neurons in hidden layer and an appropriate amount of data. Adding more neurons in the hidden layers to an ANN architecture gives the model the flexibility of fitting extremely complex nonlinear functions. This also holds true in classification modeling in approximating any nonlinear decision boundary with great accuracy [2].
Recurrent neural networks (RNNs) emerged as an operative and scalable ANN model for several learning problems associated with sequential data. Information in these networks incorporate loops into the hidden layer. These loops allow information to flow multi-directionally so that the hidden state signifies past information held at a given time step. Thus, these network types have an infinite dynamic response to sequential data. Many applications, such as Apple’s Siri and Google’s voice search, use RNN.
The most popular way to train a neural network is by backpropagation (BP). This method can be used with either feedforward or recurrent networks. BP involves working backward through each timestep to calculate prediction errors and estimate a gradient, which in turn is used to update the weights in the network. For example, to enable the long sequences found in RNNs, multiple timesteps are conducted that unrolls the network, adds new layers, and recalculates the prediction error, resulting in a very deep network.
However, standard and deep RNN neural networks may suffer from vanishing or exploding gradients problems. As more layers in RNN containing activation functions are added, the gradient of the loss function approaches zero. As more layers of activating functions are added, the gradient loss function may approach zero (vanish), leaving the functions unchanged. This stops further training and ends the procedure prematurely. As such, parameters capture only short-term dependencies, while information from earlier time steps may be forgotten. Thus, the model converges on a poor solution. Because error gradients can be unstable, the reverse issue, exploding gradients may occur. This causes errors to grow drastically within each time step (MATLAB, 2020b). Therefore, backpropagation may be limited by the number of timesteps.
Long Short-Term Memory (LSTM) networks address the issue of vanishing/exploding gradients and was first introduced by [3]. In addition to the hidden state in RNN, an LSTM block includes memory cells (that store previous information) and introduces a series of gates, called input, output, and forget gates. These gates allow for additional adjustments (to account for nonlinearity) and prevents errors from vanishing or exploding. The result is a more accurate predicted outcome, the solution does not stop prematurely, nor is previous information lost [4].
Practical applications of LSTM have been published in medical journals. For example, studies [5, 6, 7, 8, 9, 10, 11, 12, 13, 14] used different variants of RNN for classification and prediction purposes from medical records. Important, LSTM does not have any assumption about elapsed time measures can be utilized to subtype patients or diseases. In one such study, LSTM was used to make risk predictions of disease progression for patients with Parkinson’s by leveraging longitudinal medical records with irregular time intervals [15].
The purpose of this chapter is to introduce Long Short-Term Memory (LSTM) networks. This chapter begins with introduction of multilayer feedforward architectures. The core characteristic of an RNN and vanishing of the gradient will be explained briefly. Next, the LSTM neural network, optimization of network parameters, and the methodology for avoiding the vanishing gradient problem will be covered, respectively. The chapter ends with a MATLAB example for LSTM.
Artificial Neural Networks (ANNs) are powerful computing techniques that mimic functions of the human brain to solve complex problems arising from big and messy data. As a machine learning method, ANNs can act as universal approximators of complex functions capable of capturing nonlinear relationships between inputs and outcomes. They adaptively learn functional structures by simultaneously utilizing a series of nonlinear and linear activation functions. ANNs offer several advantages. They require less formal statistical training, have the ability to detect all possible interactions between input variables, and include training algorithms adapted from backpropagation algorithms to improve the predictive ability of the model [2].
Feed forward multilayer perceptron (Figure 1) is the most used in ANN architectures. Uses include function approximation, classification, and pattern recognition. Similar to information processing within the human brain, connections are formed from successive layers. They are fully connected because neurons in each layer are connected to the neurons from the previous and the subsequent layer through adaptable synaptic network parameters. The first layer of multi-layer ANN is called the input layer (or left-most layer) that accepts the training data from sources external to the network. The last layer (or rightmost layer) is called the output layer that contains output units of the network. Depending on prediction or classification, the number of neurons in the output layer may consist of one or more neurons. The layer(s) between input and output layers are called hidden layer(s). Depending on the architecture multiple hidden layers can be placed between input and output layers.
(adapted from Okut, 2016). Artificial neural network design with 4 inputs (pi). Each input is connected to up to 3 neurons via coefficients wkjl (l denotes layer; j denotes neuron; k denotes input variable). Each hidden and output neuron has a bias parameter bjl . Here P = inputs, IW = weights from input to hidden layer (12 weights), LW = weights from hidden to output layer (3 weights), b1 = Hidden layer biases (3 biases), b2 = Output layer biases (1 bias), n1 = IWP + b1 is the weighted summation of the first layer, a1 = f(n1) is output of hidden layer, n2 = LWa1 + b 2 is weighted summation of the second layer and t̂ = a2 = f(n2) is the predicted value of the network. The total number of parameters for this ANN is 12 + 3 + 3 + 1 = 19.
Training occurs at the neuronal level in the hidden and output layers by updating synaptic strengths, eliminating some, and building new synapses. The central idea is to distribute the error function across the hidden layers, corresponding to their effect on the output. Figure 1 demonstrates the architecture of a simple feedforward MLP where P identifies the input layer, next one or more hidden layers, which is followed by the output layer containing the fitted values. The feed forward MLP networks is evaluated in two stages. First, in the feedforward stage information comes from the left and each unit evaluates its activation function f. The results (output) are transmitted to the units connected to the right. The second stage involves the backpropagation (BP) step and is used for training the neural network using gradient descent algorithm in which the network parameters are moved along the negative of the gradient of the performance function The process consists of running the whole network backward and adjusting the weights (and error) in the hidden layer. The feedforward and backward steps are repeated several times, called epochs. The algorithm stops when the value of the loss (error) function has become sufficiently small.
Because the two stages, outlined in Figure 1 above, are used in all neural networks, we can extend MLP feedforward neural networks for use in sequential data or time series data. To do so, we must address time sequences. Recurrent neural networks (RNNs) address this issue by conducting multiple time steps that unrolls the network, adds new layers, and recalculates the prediction error, resulting in a very deep network. First, the connections between nodes in the hidden layer(s) form a directed graph along a temporal sequence allowing information to persist. Through such a mechanism, the concept of time creates the RNNs memory. Here, the architecture receives information from multiple previous layers of the network.
Figure 2 outlines the hidden layer for RNN and demonstrates the nonlinear function of the previous layers and the current input (p). Here, the hyperbolic tangent activation function is used to generate the hidden state. The model has memory since the bias term is based on the “past”. As a consequence, the outputs from the previous step are fed as input to the current step. Another way to think about RNNs is that a recurrent neural network has multiple copies of the same network, each passing a message to a successor (Figure 3). Thus, the output value of the last time point is transmitted back to the neural network, so that the parameter estimation (weight calculation) of each time point is related to the content of the previous time point.
A typical RNN that has a hyperbolic tangent activation function ex−e−xex+e−x to generate the hidden state. Because of the hidden state RNNs have a “memory” that information has been calculated so far is captured. The information in hidden state passed further to a second activation function 11+e−x to generate the predicted (output) values. In RNNs, the weight (W) calculation of each time point of the network model is related to the content of the previous time point. We can process a sequence of vectors of inputs (p) by applying a recurrence formula at every time step.
An unrolled RNN with a hidden state carries pertinent information from one input item in the series to others. The blue and red arrows in the figure are indicating the forward and the backward pass of the network, respectively. With backward pass, we sum up the contributions of each time step to the gradient. In other words, because W is used in every step up to the output, we need to backpropagate gradients from t = 4 through the network all the way to t = 0.
Similar to feedforward MLP networks, RNNs have two stages, a forward and a backward stage. Each works together during the training of the network. However, structures and calculation patterns differ. Let us first consider the forward pass.
Stage 1: Forward pass.
The forward pass will be summarized into 5 steps:
Summation step. In this step two different source of information are combined before nonlinear activation function will be take place. The sources are the values of weighted input
The weight of the previous hidden state and current input are placed in a trainable weight matrix. Element-wise multiplication of the previous hidden state vector with the hidden state weights (
Hyperbolic tangent activation function is applied to the summed of the two parameterized vectors
The network input to the output unit at time t with element-wise multiplication of output weights and with updated (current) hidden state
The output of the network at time t is calculated (the activation function applied to the output layer depends on the type of target (dependent) variable and the values coming from the hidden units. Again, a second activation function (mostly sigmoid) is applied to the value generated by the hidden node. The predicted value of a RNN block with sigmoid:
During the training of the forward pass of the RNN, the network outputs predicted value (
Then the error (loss function, cost function) at each time step is calculated to start the “backward pass”:
Here y and
Stage 2: Backward Pass.
After the forward pass of RNN, the calculated error (loss function, cost function) at each time step is injected backwards into the network to update the network weights at each iteration. The idea of RNN unfolding in Figure 3 takes place the bigger part in the way RNNs are implemented for the backward pass. Like standard backpropagation in feed forward MLP, the backward pass consists of a repeated application of the chain rule. For this reason, the type of backpropagation algorithm used for an RNN to update the network parameters is called backpropagation through time (BPTT). In BPTT, the RNN network is unfolded in time to construct a feed forward MLP neural network. Then, the generalized delta rule is applied to update the weights W(p), W(h) and W(y) and biases b(h) and b(y). Remember, the goal with backpropagation is minimizing the gradients of error with respect to the network parameter space (W(p), W(h) and W(y) and biases b(h) and b(y)) and then updates the parameters using Stochastic Gradient Descent. The following equation is used to update the parameters for minimizing the error function:
Here, a is learning rate and
The total error is calculated by the summation of the error from all time steps as:
The value of gradients
To calculate the error gradient given in Eq. (6):
To calculate the overall error gradient, the chain rule of differentiation given in (7) is used.
Then the network weights can be updated as follow:
Note that, as given in (2), the current state (ht) = tanh
The Jacobians of any time
while the Jacobian matrix for hidden state is given by:
Putting the Eqs. (7) and (8) together, we have the following relationship:
In other words, because the network parameters are used in every step up to the output, we need to backpropagate gradients from last time step (t = t) through the network all the way to t = 0. The Jacobians in (10),
As mentioned before, the output from RNNs is dependent on its previous state or previous N time steps circumstances. Conventional RNN face difficulty in learning and maintaining long-range dependencies. Imagine the unfolding RNN given in Figure 3. Each time step requires a new copy of the network. With large RNNs, thousands, even millions of weights are needed to be updated. In other word,
The long short-term memory networks (LSTMs) are a special type of RNN that can overcome the vanishing gradient problem and can learn long-term dependencies. LSTM introduces a memory unit and a gate mechanism to enable capture of the long dependencies in a sequence. The term “long short-term memory” originates from the following intuition. Simple RNN networks have long-term memory in the form of weights. The weights change gradually during the training of the network, encoding general knowledge about the training data. They also have short-term memory in the form of ephemeral activations, which flows from each node to successive nodes [17, 18].
The neural network architecture for an LSTM block given in Figure 4 demonstrates that the LSTM network extends RNN’s memory and can selectively remember or forget information by structures called cell states and three gates. Thus, in addition to a hidden state in RNN, an LSTM block typically has four more layers. These layers are called the cell state (Ct), an input gate (it), an output gate (Ot), and a forget gate (ft). Each layer interacts with each other in a very special way to generate information from the training data.
Illustration of long short-term memory block structure. The operator “⨀” denotes the element-wise multiplication. The Ct-1, Ct, ht and ht are previous cell state, current cell state, current hidden state and previous hidden state, respectively. The ft; it; ot are the values of the forget, input and output gates, respectively. The C∼t is the candidate value for the cell state, W(f), W(i), W(c), W(o) are weight matrices consist of forget gate, input gate, cell state and output gate weights, and b(f), b(i), b(c), and b(o) are bias vectors associated with them.
A block diagram of LSTM at any timestamp is depicted in Figure 4. This block is a recurrently connected subnet that contains the same cell state and three gates structure. The pt, ht−1, and Ct−1 correspond to the input of the current time step, the hidden output from the previous LSTM unit, and the cell state (memory) of the previous unit, respectively. The information from the previous LSTM unit is combined with current input to generate a newly predicted value. The LSTM block is mainly divided into three gates: forget (blue), input-update (green), and output (red). Each of these gates is connected to the cell state to provide the necessary information that flows from the current time step to the next.
A sigmoid activation function
As shown in the upper part of Figures 4 and 5, the cell state is the key to LSTMs and represents the memory of LSTM networks. The process for the cell state is very much like to a conveyor belt or production chain. The information about the parameters runs straight forward the entire chain, with only some linear interactions, such as multiplication and addition. The state of information depends on these interactions. If there are no interactions, the information will run along without changes. The LSTM block removes or adds information to the cell state through the gates, which allow optional information to cross [19].
The cell state, the horizontal line running through the top of the diagram of an LSTM.
The Forget Gate (ft) decides the type of information that should be thrown away or kept from the cell state. This process is implemented by a sigmoid activation function. The sigmoid activation function outputs values between 0 and 1 coming from the weighted input (Wfpt), previous hidden state (ht-1), and a bias (bf). The forget gates (Figure 6) can be described by the equation given in (11). Here, σ is the sigmoid activation function, W(f) and b(f) are the weight matrix and bias vector, which will be learned from the input training data.
The forget gate controls what information to throw away from the memory.
The function takes the old output (ht−1) at time t − 1 and the current input (pt) at time t for calculating the components that control the cell state and hidden state of the layer. The results are [0,1], where 1 represents “completely hold this” and 0 represents “completely throw this away” (Figure 6).
The Input Gate (it) controls what new information will be added to the cell state from the current input. This gate also plays the role to protect the memory contents from perturbation by irrelevant input (Figures 7 and 8). A sigmoid activation function is used to generate the input values and converts information between 0 and 1. So, mathematically the input gate is:
The input-update gate decides what new information should be stored in the cell state, which has two parts: A sigmoid layer and a hyperbolic tangent (tanh) layer. The sigmoid layer is called the “input gate layer” because it decides which values should be updated. The tanh layer is a vector of new candidate values C∼t that could be added to the cell state.
Memory update is done using old memory via the forget gate and new memory via the input gate.
where W(i) and b(i) are the weight matrix and bias vector, pt is the current input timestep index with the previous time step ht-1. Similar to the forget gate, the parameters in the input gate will be learned from the input training data. At each time step, with the new information pt, we can compute a candidate cell state.
Next, a vector of new candidate values,
In the next step, the values of the input state and cell candidate are combined to create and update the cell state as given in (14). The linear combination of the input gate and forget gate are used for updating the previous cell state (Ct-1) into current cell state (Ct). Once again, the input gate (it) governs how much new data should be taken into account via the candidate (
The Output Gate (ot) controls which information to reveal from the updated cell state (Ct) to the output in a single time step. In other words, the output gate determines what the value of the next hidden state should be in each time step. As depicted in Figure 9, the hidden state comprises information on previous inputs. Moreover, the calculated value of the hidden state for the given time step is used for the prediction (
The output state decides what information will be output using a sigmoid σ and tanh (to push the values to be between −1 and 1) layers.
First, the previous hidden state (ht-1) is passed to the current input into a sigmoid function. Next newly updated cell state is generated with the tanh function [15, 18]. Finally, the tanh output is multiplied with the sigmoid output to determine what information the hidden state should carry (16). The final product of the output gate is an updated of the hidden state, and this is used for the prediction at time step t. Therefore, the aim of this gate is to separate the updated cell state (updated memory) from the hidden state. The updated cell state (Ct) contains a lot of information that is not necessarily required to be saved in the updated hidden state. However, this information is critical as the updated hidden state at each time is used in all gates of an LSTM block. Thus, the output gate does the assessment regarding what parts of the cell state (Ct) is presented in the hidden state (ht). The new cell and new hidden states are then passed to the next time step (Figure 9).
Summary of forward pass
Forget gate: Controls what information to throw away and decides how much from the part should be remember.
Input-Update Gate: Controls information to add cell state from current input and decides how much should be added to the cell state
Output gate: Determines the part of the current cell state makes it to the output
Current cell state:
Current hidden state:
LSTM block prediction:
Calculate the LSTM block error for the time step:
Like the RNN networks, an LSTM network generates an output
Illustration of the (A) an LSTM unit from 3-time steps with input data (demographic and clinical data). LSTM network takes inputs from to the current time step to update the hidden state and (LSTM(ptht−1) with relevant information. The “x” in the circles denote point-wise operators, σ and tanh are sigmoid (11+e−x, generates between 0 and 1) and hyperbolic tangent (ex−e−xex+e−x, generates between −1 and 1) activation functions. (B) an RNN with 3-time steps. It has only a tangent, ex−e−xex+e−x, activation function in the block.
Similarly, the value of gradients
and for the overall error gradient using the chain rule of differentiation is:
As Eq. (19) illustrates, the gradient involves the chain rule of
The problem of gradient vanishing
Recall the Eq. (14) for cell state is
Note the term
Several modifications to original LSTM architecture have been recommended over the years. Surprisingly, the original continues to outperform, and has similar predictive ability compared with variants of LSTM over 20 years.
This is a type of LSTM by adding “peephole connections” to the standard LSTM network. The theme stands for peephole connections needed to capture information inherent to time lags. In other words, with peephole connections the information conveyed by time intervals between sub-patterns of sequences is included to the network recurrent. Thus, peephole connections concatenate the previous cell state (Ct-1) information to the forget, input and output gates. That is, the expression of these gates with peephole connection would be:
This configuration was offered to improve the predictive ability of LSTMs to count and time distances between rare events [21].
Gated recurrent units (GRU) is a simplified version of the standard LSTM designed in a manner to have more persistent memory in order to make it easier for RNNs. A GRU is called gated RNN and introduced by [22]. In a GRU, forget and input gates are merged into a single gate named “update gate”. Moreover, the cell state and hidden state are also merged. Therefore, the GRU has fewer parameters and has been shown to outperform LSTM on some tasks to capture long-term dependencies.
This configuration of LSTM was introduced by Krause et al., [23]. The architecture is for sequence modeling combines LSTM and multiplicative RNN architectures. mLSTM is characterized by its ability to have different recurrent transition functions for each possible input, which makes it more expressive for autoregressive density estimation. Krause et al. concluded that mLSTM outperforms standard LSTM and its deep variants for a range of character-level language modeling tasks.
The core idea behind the LSTM with Attention frees the encoder-decoder architecture from the fixed-length internal illustration. This is one of the most transformative innovations in sequence to uncover the mobility regularities in the hidden node of LSTM. The LSTM with attention was introduced by Wu et al., [24] for Bridging the Gap between Human and Machine Translation in Google’s Neural Machine Translation System. The LSTM with attraction consists of a deep LSTM network with 8 encoder and 8 decoder layers using attention and residual connections. Most likely, this type of LSTM continues to power Google Translate to this day.
Two examples using MATLAB for LSTM will be given for this particular chapter.
Example 1 This example shows how to forecast time series data for COVID19 in the USA using a long short-term memory (LSTM) network. The variable used in training data is the rate for the number of positive/number of tests for each day between 01/22/2020–2112/22/2020. Data set was taken from publicly available
Total daily number of positively tested COVID19 and the rate (positively tested/number of test) conducted in the USA. (a) Plot of the training time series of the number of positively tested COVID19 with the forecasted values, (b) compare the forecasted values of the number of positively tested with the test data set. This graph shows the total daily number of virus tests conducted in each state and of those tests, how many were positive each day. (c) Plot of the training time series of the rate of positively tested COVID19 (d) compare the forecasted values of the rate of positively tested with the rates in the test data set. The trend line in blue shows the actual number of positive cases and the trend line in red shows the number predicted for the last 38 days.
Example 1. MATLAB Codes and descriptions of the codes | |
---|---|
a. Data prepretion | Descriptions |
numTimeStepsTrain = floor(0.95*numel(Tpositive)); dataTrain = Tpositive(1:numTimeStepsTrain+1); dataTest = Tpositive(numTimeStepsTrain+1:end); | Partition the training and test data. Train on the first 95% of the sequence and test on the last 5% |
b. Define LSTM | Descriptions |
numHiddenUnits = 300; layers = […sequenceInputLayer(numFeatures) lstmLayer(numHiddenUnits) fullyConnectedLayer(numResponses) regressionLayer]; | Define LSTM Network Architecture and create an LSTM regression network. Specify the LSTM layer to have 300 hidden units |
c. Specify the training options | Descriptions |
options = trainingOptions(‘adam’, … ‘MaxEpochs’,1000, … ‘GradientThreshold’,1, … ‘InitialLearnRate’,0.005, … ‘LearnRateSchedule’,\'piecewise’, … ‘LearnRateDropPeriod’,125, … ‘LearnRateDropFactor’,0.2, … ‘Verbose’,0, … ‘Plots’,\'training-progress’); | Training with Adam (adaptive moment estimation) To prevent the gradients from exploding, set the gradient threshold to 1. The software updates the learning rate every certain number of epochs by multiplying with a certain factor. Number of epochs for dropping the learning rate Factor for dropping the learning rate when ‘LearnRateSchedule’,\'piecewise’ |
d. Train the LSTM network | Descriptions |
net = trainNetwork(XTrain,YTrain,layers,options); | Train the LSTM network with the specified training options |
e. Initialize -training and loop over | |
net = predictAndUpdateState(net,XTrain); [net,YPred] = predictAndUpdateState(net,YTrain(end)); numTimeStepsTest = numel(XTest); for i = 2:numTimeStepsTest [net,YPred(:,i)] = predictAndUpdateState(net,YPred(:,i-1),\'ExecutionEnvironment’,\'cpu’); end; | The training and update data Next, make the first prediction using the last time step of the training response YTrain(end). Loop over the remaining predictions and input the previous prediction to predictAndUpdateState. |
f. Update Network State with observed values | Descriptions |
net = resetState(net); net = predictAndUpdateState(net,XTrain); | Update Network State with Observed Values |
g. Predict on each time step | |
YPred = []; numTimeStepsTest = numel(XTest); for i = 1:numTimeStepsTest [net,YPred(:,i)] = predictAndUpdateState(net,XTest(:,i),\'ExecutionEnvironment’,\'cpu’); end | Predict on each time step. For each prediction, predict the next time step using the observed value of the previous time step. |
MATLAB codes and specification of cods for Example 1*.
All descriptions are based on MATLAB 2020b and related examples from the MATLAB.
This example trains an LSTM network to forecast the number of positively tested given the number of cases in previous days. The training data contains a single time series, with time steps corresponding to days and values corresponding to the number of cases. To make predictions on a new sequence, reset the network state using the “resetState” command in MATLAB. Resetting the network state prevents previous predictions from affecting the predictions on the new data. Reset the network state, and then initialize the network state by predicting on the training data (MATLAB, 2020b). The solid line with red color in Figure 11a and c indicates the number of cases predicted for the last 30 days.
Example 2: Data from example 2 is from SEER 2017 for different age groups. SEER collects cancer incidence data from population-based cancer registries of the U.S. population. The SEER registries collect data on patient demographics, primary tumor site, tumor morphology, stage at diagnosis, and the first course of treatment, and they follow up with patients for vital status. The example given in this chapter is the cancer type and non-cancer causes of death identified from the survey.
Example 2. MATLAB Codes and descriptions of the codes | |
a. Data prepretion | Descriptions |
filename = “filename” data = readtable(filename,\'TextType’,\'string’); | To import the text data as strings, specify the text type to be \'string’. |
b. Partition data | Descriptions |
crossval = cvpartition(‘DataName.ClassVaribleName’,0.30); dataTrain = DataName(training(crossval),:); dataValidation = DataName (test(crossval),:); | Partition data into sets for training and validation. Partition the data into a training partition and a held-out partition for validation and testing. Specify the holdout percentage to be 30%. |
c. Extract the text data | Descriptions |
textDataTrain = dataTrain.TextVariableName_; textDataValidation = dataValidation. TextVariableName _; YTrain = dataTrain. TextVariableName; YValidation = dataValidation. TextVariableName; | Extract the text data and labels from the partitioned tables. Here TextVariableName is column name for the group variable (for example age group) and TextVariableName is the column name for the text variable (in this example the name of cancer types) |
d. Create a cloud for the text | Descriptions |
figure wordcloud(textDataTrain); | To check that you have imported the data correctly, visualize the training text data using a word cloud. |
e. Preprocess Text Data | |
documentsTrain = preprocessText(textDataTrain); documentsValidation = preprocessText(textDataValidation); | Preprocess the training data and the validation data using the preprocessText function. |
f. Convert document to sequences | Descriptions |
enc = wordEncoding(documentsTrain); sequenceLength = 10; XTrain = doc2sequence(enc,documentsTrain,\'Length’,sequenceLength); XTrain(1:5) XValidation = doc2sequence(enc,documentsValidation,\'Length’,sequenceLength); | Encoding to convert the documents into sequences of numeric indices. |
g. Create and Train LSTM | Descriptions |
inputSize = 1; embeddingDimension = 30; numHiddenUnits = 200; numWords = enc.NumWords; numClasses = numel(categories(YTrain)); layers = [… sequenceInputLayer(inputSize) wordEmbeddingLayer(embeddingDimension,numWords) lstmLayer(numHiddenUnits,\'OutputMode’,\'last’) fullyConnectedLayer(numClasses) softmaxLayer classificationLayer] | Initialize the embedding weights |
h. Specify training options | Descriptions |
options = trainingOptions(‘adam’, … ‘MiniBatchSize’,30, … ‘GradientThreshold’,8, … ‘Shuffle’,\'every-epoch’, … ‘ValidationData’,{XValidation,YValidation}, … ‘Plots’,\'training-progress’, … ‘Verbose’,false); | MiniBatchSize: Classifies data using mini batches of size ‘GradientThreshold: Clip gradient values for the threshold |
i. Train the LSTM network | Descriptions |
net = trainNetwork(XTrain,YTrain,layers,options); | Train the LSTM network using the trainNetwork function. |
j. Predict using new data | Descriptions |
reportsNew = [… “The text definition here.”]; | Classify the event type of three new reports. Create a string array containing the new reports. |
k. Preprocess Convert and Classify | Descriptions |
documentsNew = preprocessText(reportsNew); XNew = doc2sequence(enc,documentsNew,\'Length’,sequenceLength); labelsNew = classify(net,XNew) |
MATLAB codes and specification of cods for Example 2*.
All descriptions are based on MATLAB 2020b and related examples from the MATLAB.
Visualizing the training data text file for SEER-2017 cancer types by age groups using a word cloud of LSTM. The MATLAB codes to create this figure are given in Table 2. The bigger the word, the more often diagnosed cancer type in 2017.
LSTM is a very powerful ANN architecture for disease subtypes, time series analyses, for the text generation, handwriting recognition, music generation, language translation, image captioning process. The LSTM approach is effective to make predictions as equal attention is provided for all input sequences by the information flows through the cell state. Because of the mechanism adopted, the small change in the input sequence does not harm the prediction accuracy done by LSTM. Future work on LSTM has several directions. Most LSTM architectures are designed to handle data evenly distributed between elapsed times (days, months, years, etc.) for the consecutive elements of a sequence. More studies are needed to improve the predictive ability of LSTM for nonconstant consecutive observations elapsed times. Moreover, further studies are needed for possible overfitting problems for training with smaller data sets. Rather than using early stopping to avoid the overfitting, Bayesian regularized approach would be more effective to ensure that the neural network halts training at the point where further training would result in overfitting. As the Bayesian regularized approach uses a different loss function with different hyperparameters, this approach demands costly computation resources.
The author would like to sincerely thank to Rosey Zackula for reading and revising the manuscript carefully.
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