JPH06102952A - Optical fractal figure generation method and generation apparatus - Google Patents

Abstract

(57) [Abstract] [Purpose] High-speed image reproduction of an image encoded by a reduction conversion group. A method for generating a fractal figure coded by a reduced affine transformation, wherein a plurality of images are created from an optically presented image, and magnification conversion, image movement, and image rotation are performed by parameters of the reduced affine transformation. Magnification conversion of each image by multiple optical systems with each coefficient set,
It is characterized by performing an optical process of moving and rotating an image and synthesizing, reading a synthetic image to feed back, presenting again, and repeating the optical process to optically generate a fractal figure.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an optical fractal figure generation method and an apparatus for reproducing fractal figure information represented by a reduction conversion group by an optical method.

[0002]

2. Description of the Related Art Humans visually obtain most of the information from the outside world, and from that point of view, various applications of the technology for handling images are considered. Due to the rapid progress of electronic computers, the technology for handling images has made great progress in recent years. However, the amount of data of image information is enormous, and it is difficult for some electronic computers to handle it as it is.

Therefore, it is considered important to develop a technique for compressing image information. Images contain a lot of information, but also a lot of redundant information. By removing the redundant part, the image information can be compressed. However, an effective compression technique has not been established for a complicated figure in the natural world, and an effective method is desired.

[0004]

BB Mandelbrot uses fractal geometry (Nikkei Science (1985), translated by Heisuke Hironaka, "Fractal geometry").
It was shown that complex figures in the natural world can be expressed. This is a method that pays attention to the fact that natural patterns have self-similarity. M. Barnsley Iterat fractal figures
We proposed a method of effectively compressing an image by expressing it by a reduction transformation group called ed Function System (IFS) (Fractals Everywhere, Academic Press, Inc (1988)).
With this technique, fractal figures can be expressed only with IFS parameters, and as a result, image compression as high as 1:10 ⁴ can be achieved (MFBarnsley, ADSloan, "A better way to co
mpressimages, "Byte 13 215-223 (1988)) However, the realization of IFS by a computer requires a huge amount of calculation, and the compression and reproduction of images imposes a heavy burden on the computer.

The present invention is intended to solve the above problems, and an optical fractal figure that enables high-speed image reproduction by parallelly converting an image encoded by a reduction conversion group by an optical method. The purpose is to provide a generation method.

[0006]

SUMMARY OF THE INVENTION The present invention is a method for generating a fractal figure encoded by a reduced affine transformation, wherein a plurality of images are created from an optically presented image and the parameters of the reduced affine transformation are used. Magnification conversion,
Optical processing that combines the images by magnification conversion, image movement, and image rotation is performed by a plurality of optical systems in which the coefficients of image movement and image rotation are set, and the combined image is read, fed back, and re-presented. The fractal figure is optically generated by repeating the above optical processing. Further, the present invention is a fractal figure generation device coded by reduction affine transformation, comprising image presenting means for presenting an optical image, and a multiple imaging optical system for producing a plurality of images from the presented image, Coefficients for magnification conversion, image movement, and image rotation are set by the parameters of the reduction affine transformation, and a plurality of optical systems for magnification conversion, image movement, and image rotation are combined with the output images from each optical system. Synthesizing means,
An image reading unit for reading a composite image is provided, and the read image is fed back to the image presenting unit to optically generate a fractal figure.

[0007]

The present invention prepares a plurality of sets of optical systems each of which is composed of optical elements whose coefficients for magnification conversion, image rotation, and image movement are set on the basis of the image described by the parameters of the reduced affine transformation. , By performing image magnification conversion, image rotation, and image movement on any initial image in parallel to combine the output images of each optical system, and repeating similar optical processing on the combined image, self-similarity Generates a fractal figure that has the property of having a similar relationship with a small portion.
In the generation of the fractal figure of the present invention, due to the parallelism and high speed of the optical processing, it is possible to remarkably speed up the image reproduction compared with the processing by the conventional electronic computer, and the continuity of the data is achieved. The calculation accuracy in the image rotation and reduction processing can be improved.

[0008]

First, before describing the embodiments of the present invention, the handling of images by fractal geometry will be described.
In current image processing technology, image information is typically sampled and digitized for recording on a computer. This has been done because it was believed that spatial frequency information below a certain frequency of the image could be supplemented by interpolation, ie linear approximation. However, the fractal geometry proposed by BB Mandelbrot shows that many natural objects have fine structures that cannot be linearly approximated.

(A) When expressing the shape of a fractal geometrical object, it is common to take out the lines and surfaces that compose it and approximate it. This can be represented by a straight line if you look at any complicated figure in detail, that is,
It is based on the assumption that "continuous curves are piecewise differentiable". However, many of the objects in the natural world have a small deviation that cannot be ignored when such an approximation is used.

Consider, for example, approximating a cloud with a sphere.
Here, the sphere is considered to be one of the "piecewise differentiable figures". First, the cloud is approximated by a sphere of a certain size. At this time, naturally there is a deviation in details. If the deviation of this detail is approximated by a finer sphere, a deviation of the same degree as that of the first approximation will occur. This happens no matter how many approximations are made. In other words, the assumption that "a continuous curve is piecewise differentiable" does not hold for natural objects such as clouds. This means that Euclidean geometry is not very effective when expressing things in nature. BBMa
Based on the above, ndelbrot proposed fractal geometry as a method of describing natural figures. Fractal geometry targets "continuous and non-differentiable figures at all points".

(A-1) Definition of fractal figure In (a), it was stated that the figure in the natural world is a fractal figure, but it is not correct in a strict sense. A figure in the natural world can be regarded as a fractal figure within a certain scale range, and is not a fractal figure in the strict sense. One of the characteristics of fractal figures is that they do not have a characteristic length (Hideki Takayasu "Fractal" Asakura Shoten (1987)). The characteristic length is the size that represents the figure, such as the diameter and radius of a circle or sphere, the vertical and horizontal lengths of a rectangle, the vehicle height, vehicle width, and vehicle length.

Here, consider a Koch curve which is a typical fractal figure. To make a Koch curve,
The triangle shown in (a) is prepared. 1/3 of this triangle
Same as the one reduced to ^1/2 and rotated 150 degrees
Fig. 1 (b) shows that it is reduced to 3 ^1/2 and rotated by -150 degrees.
Put like. The same transformation is again performed on the two resulting triangles by the above procedure. (FIG. 1 (c)) A figure obtained by repeating this infinitely is a Koch curve. It can be seen that the figures created in this way are figures approximated by triangles of infinitely different sizes from 1 to infinitesimal size. In other words, this curve has no characteristic length. One of the characteristics of fractal figures is that they are expressed by elements of infinitely different sizes and do not have characteristic sizes.

Next, what kind of figure can be treated as a fractal figure will be described. BB Mandelbrot, who proposed fractal geometry, defines a set as a fractal figure if the fractal dimension (Hausdorff-Beskovich dimension) of the set is larger than the topological dimension (Nikkei Science, Inc. (1985), Hironaka Hironaka). Supervision, "Fractal Geometry"). The definition of the Hausdorf-Beskovich dimension is as follows.

Let D> 0. Cover set E with a countable number of spheres with diameter ε> 0. At this time, if the diameter of each sphere is d ₁ , d ₂ ..., d _K , the D-dimensional Hausdorff measure is M _D (E) = lim inf Σ d _k ^D …… (1) ε → 0 d _k <Defined by ε. Here, when the Hausdorff measure transits from 0 to infinity, D is called the Hausdorff-Besskovich dimension of the set E and is represented as D _H. The topological dimension is an integer dimension, which is consistent with everyday feeling. The Hausdorff-Beskovich dimension can take a non-integer value, and a figure that is larger than the topological dimension is called a fractal figure.

(A-2) Fractor for figures in the natural world
As mentioned in Le Geometry Application (a-1), many of the natural figures are fractal within a certain scale. Here, we describe the application of fractal geometry to natural figures. Any object has a finite size, and is made up of very small but finite size components. Therefore, the applicable range of fractal geometry is a scale from the size of the object to the smallest constituent unit. (1)
When the Hausdorff measure was defined in the equation, it was set to ε → 0, but in the case of general figures, ε → ε ₀ must be considered. Here, ε ₀ is the size of the minimum structural unit. In such a case, the Hausdorff measure is not strictly defined, but the value D _{H at} which this amount transits from 0 to infinity can be determined (Nikkei Science Co. (1985), translated by Heisuke Hironaka, “Fractal”. Geometry").

Now, consider a cloud as an example of an actual natural figure. I said that clouds can be approximated by spheres of innumerable sizes, but this is only true within a certain size range. Because when you observe the structure of the cloud very closely, it becomes water vapor and eventually reaches the water molecule. However, if you take a closer look, the cloud shape becomes meaningless. When considering the shape of clouds, the approximation by the infinite number of spheres agrees very well, and it is safe to say that the clouds have a fractal structure between some maximum and minimum scales. If such an assumption is made, the figure in the natural world can be regarded as fractal.

Usually, the maximum scale and the minimum scale cannot be easily determined. Further, there are various ways to obtain the fractal dimension D _{H, and} there is no completely fixed method. However, many of the figures in the natural world have a fractal structure (Nikkei Science (1985), translated by Heisuke Hironaka, “Fractal Geometry”), and various studies (MFBarnsley, RCVevaney, BBM)
andelbrot et al, The Science of FRACTAL IMAGES, Spri
nger-Verlag, (1988), Alex.P.Pentland, "Fractal-Based
Description of Natural Scenes. "IEEE PAMI-6661 (19
84), Masaya Yamaguchi "Chaos and Fractals" Kodansha (1986)
).

(A-3) Self-similar figure An important property of a fractal figure is self-similarity. Self-similarity is the property that when a part of a figure is enlarged, it matches the whole figure or a larger part. Here, this property will be described using the Shelbinski triangle, which is one of the fractal figures.
The figure in FIG. 2 is the Shelbinski triangle. Multiplying the lower right triangle of this figure by 3 will match the original figure. In other words, this figure has a similar nested structure. A figure having self-similarity is called a self-similar figure. Fractal figures always have self-similarity, and self-similarity is one of the important properties of fractal figures.
Such properties are also found in general natural figures. For example, taking a cloud as an example, the outer shape of the cloud does not change when viewed with the naked eye or through a telescope. This is not a homology in a strict sense, but it can be said that it has self-similarity when viewed statistically. This property is also found in many other things, such as mountain ridges, river tributaries, coastlines, blood vessels of certain plants and animals, and branching structures such as lungs.

(B) Reduction Transformation Group and Self-Similar Figure Next, a method of expressing a self-similar figure by a set of two or more reduction transformations (referred to as a reduction transformation group) will be described. 2 points P ₁ ,
When P ₂ is present, the reduction conversion f satisfies the following condition.

| P ₁ −P ₂ |> | f (P ₁ ) −f (P ₂ ) | (2) That is, the contraction conversion is a conversion in which the distance between two points is shortened by the conversion. There are various kinds of such transformations, but here, the transformation of R ² ⇒R ² , that is, the transformation from the two-dimensional plane to the two-dimensional plane, and only the linear one is taken up. This is considered to be conversion to a reduced similar figure based on the linear conversion. A conversion in which two or more of these reduction conversions are combined is defined as a reduction conversion group.

(B-1) Mathematical Expression of Reduced Linear Transform The linear transform of R ² ⇒R ² is expressed in matrix form as follows. When the R ² plane is regarded as a complex plane, the complex number z = x + i
It is expressed as follows using y (however, x and y represent coordinates on a plane). f (z) = _αz + βz _conj + γ (4) where z _conj represents a complex conjugate and αβγ represents a complex number.

This form is well studied mathematically,
Various figures are expressed by the reduction conversion expressed in this form. For example, the Koch curve shown in FIG. 1 can be expressed as follows using two transformations.

F ₀ (z) = _{αz conj} (5) F ₁ (z) = (1-α) z _conj + α (6) However, α = ^1/2 + i3 _1/2/6 (b- 2) Representation of self-similar figure by reduction transformation group Next, representation of self-similar figure by reduction transformation group (MFBarn
sley, ADSloan, "A better way to compress images,"
Byte 13 215-223 (1988), Masayoshi Hata, "Fractal Patterns: The Theory of Pattern Growth", Separate Volume "Mathematical Science", Fractal, 11
8 Science Company (1986)). First, consider the behavior of the mapping of the point p _n by the reduction transformation group. As a specific example, consider the Koch curve described in FIG. The position at the initial time t = 0 is p ₀ , and the time t = n
The position at time is p _n . The rules of the movement from the p _n to p _{n + 1} by using a uniform random variable P (5) below, determined as follows to select the F _0, F ₁ as defined in equation (6).

P _{n + 1} = F ₀ (p _n ) P ≦ 1/2 p _{n + 1} = F ₁ (p _n ) P> 1/2 (7) By repeating this trial, p _{0 is} sequentially obtained. , P ₁ , p ₂
The dot moves. At this time, the behavior of the point sequence { _pn } is observed. Considering time t = n, it means that trials of probability P are repeated n times. At this time, there are 2 ^{n possible} positions of p _n . For simplicity, the initial position p ₀ = (1 / 2,3 ^1/2
As / 6), at most 2 ⁿ places (this is designated as Q _n ) where p _n is located when t = n is plotted. Figure 3 shows a diagram depicting this plot, Fig. 3 (a) Q _0, FIG. 3 (b) Q _1, FIG. 3 (c) is a diagram depicting the Q _5, increasing the n Then, it can be seen that Q _n approaches the Koch curve K of FIG. In this, p ₀ is the Koch curve K
This is because it is located above.

Next, consider the case where the point p ₀ ′ not on the Koch curve K is the initial value. | Α | = | 1-α | = 1/3 ^1/2 ...... (8) Due to | Fi (z) -Fi '(z) | = 1/3 ^1/2 | z-z' | (9) holds for arbitrary complex numbers z and z '. So P
Regardless of the trial _{results, | p n -p n '|} = (1/3 1/2) n | p 0 -p 0' | ...... (10) is established. When n → ∞, the right side of equation (8) becomes sufficiently small, so that when n → ∞, { _pn } and { _pn ′} draw substantially the same locus. Therefore, it can be seen that the point sequence {p _n '} approaches the Koch curve K as much as possible. That curve K is established in an integrated point of 'Q _n regardless of the "initial position p _0. Therefore, the reduced maps F ₀ and F ₁ given by the equation (4)
It can be seen that one fractal figure corresponds to. This also applies to general reduction conversion groups.

As described above, a self-similar figure can be defined using the reduction transformation group. Suppose that there are two or more reduction transformations f ₁ , f ₂ ... F _n that do not have the same fixed point, for this reduction transformation group {f _n }, set X that satisfies the following equation is {f _n } Invariant set.

X = f ₁ (X) ∪f ₂ (X) ∪ ... ∪f _n (X) (11) where ∪ represents a union. Such a set X is uniquely determined for the reduced transformation group {f _n }. Then, X is a self-similar figure and becomes a fractal figure. Therefore, as a method of expressing a fractal figure, it is possible to use {f _n }. Focusing on this, Michael Barnsley proposed a method of expressing a general image by {f _n } Ite-rated Function S
We proposed ystem (IFS) and applied it to image compression.

(C) In Ite-rated Function System (IFS)
Image Coding and Compression by IFS The image compression by M. Barnsley's IFS suggests that general images containing natural figures that can be approximated as fractal figures can also be represented by the reduction transformation group due to the fact that fractal figures can be represented by the reduction transformation group. Born from consideration. This is to encode an image by extracting a reduction conversion group from the image information and perform information compression. That is, if it is possible to find a unique {f _n } for general image information, it is expected that the image information can be efficiently compressed.

(C-1) Image reproduction by IFS For image reproduction by IFS, see MF Barnsley, ADSlo.
an, "A better way tocompress images," Byte 13 215-2
23 (1988), Steve Kocsis, "Digital compression and i
terated function systems. "SPIE 1153 Applications
of Digital Image Processing XII 19 (1989), Takeo Miyazawa, Fractal-Self-similarity and non-integer dimension (second part).

There are various types of reduction conversion, but IFS is limited to only reduction affine transformation. The affine transformation is a general linear transformation including the movement of the position, and can be expressed as the same as the equation (3) as follows.

[0031] The reproduction of the image by IFS is the same as the method of obtaining { _pn } described in (b-2). The occurrence probability P _n is determined for each of {f _n } by the following equation.

P _n = | a _n d _n −b _n c _n | / Σ _{i = 0} ^N | a _i d _i −b _i c _i | (13) A transformation is selected according to this occurrence probability, and the point sequence {p _n } is obtained.
As described in the item (b), the position of p ₀ may be arbitrary. Therefore, if the 6N parameters (a _i to f _i : i = 1,2,3, ... You can do it.

(C-2) IFS coding IFS coding is performed by M. Barnsley, Fractals Everywhere, Acade.
mic Press, Inc (1988), Steve Kocsis, "Digital compre
ssion iterared function systems. "SPIE 1153 Applica
tions of Digital Image ProcessingXII 19 (1989),
Miyazawa Takeo, Fractal-Self-similarity and non-integer dimension (Part 2), etc.

The encoding and compression of an image by IFS is basically to find a similar figure. If a similar figure is found, the parameter of {f _n } can be determined from the corresponding relationship. This principle has been shown by M. Barnsley (M. Barnsley, Fractals Everywhere, Academic Press, I.
nc (1988)). The specific procedure is as follows.

1) The target image is decomposed into parts based on a normal image processing method. For example, an image such as a human face photograph is divided into parts such as a background, face, hair, eyes, nose, and mouth. 2) Find reduced similar figures for each part. The method of finding reduced similar figures can be done by reducing the whole and pasting it like a collage. 3) Obtain each reduction conversion. At this time, it is proved that a reduced affine transformation can be found if there is information of three points or more (M.Barnsley, Fractals Everywhere, Academic
Press, Inc (1988)).

(C-3) Problems of Current IFS Calculation Method The IFS calculation method described in (b-2) is point sequence calculation, which is very suitable for current computers that perform time series processing. Is. Moreover, since an image can be expressed by a plurality of reduction conversions, a considerable effect can be expected in image encoding and compression. Compression ratios of 10 ⁴ : 1 have been reported for normal images. However, tens of thousands to millions of points are necessary for the reproduced image to be regarded as an image. Further, according to the effect of IFS compression for general photographs, the required number of reduction conversions is about 100 to 200. Dealing with such a large number is a heavy burden even if the current computer advances.

However, when considering a method of expressing a fractal figure by a reduction conversion group, it is not necessary to sequentially reproduce an image. Barnsley's method of finding { _pn } is based on current computers that perform time series processing. Here, considering {Q _n }, that is, hardware that performs conversion from 100 to 200 at the same time, it is considered that an image can be reproduced at extremely high speed.

The present invention is realized by an optical method as a parallel conversion method, and a specific method will be described below with reference to the drawings. The elements of the transformations that constitute the reduction affine transformation are reduction transformation, image rotation, image movement, and image inversion. These transformations are expressed in matrix form as follows.

The image magnification conversion is Can be expressed as Here, r _x and r _y respectively represent magnifications in the X-axis and Y-axis directions.

Image rotation is a well-known equation. It is represented by.

The movement of the image is Is. Here, (Sx, Sy) represents the center coordinates of the conversion.

The inversion of the image is It is represented by.

Reduction affine transformation can be performed by combining the transformations described so far.

[0044] (E) Optical realization of the reduced affine transformation group As described in the previous section, the reduced affine transformation can be realized if the image can be reduced, rotated, moved, and inverted. Also, since IFS is expressed as the logical sum of multiple reduced affine transformations, multiple transformations must be realized in parallel. The optical realization method will be described below.

(E-1) Multiplexing Generation In an optical system, image formation is considered as one conversion. Also in each of the methods described below, image conversion is realized based on the image forming optical system. In order to optically realize the reduction conversion group, a multiple imaging optical system that creates a plurality of images from one image is required. Although various proposals have already been made, this can be classified into three types.

Method by wavefront division (Karl-Heinz Brenn
er, Alan Huang, and Norbert Streibl, "Digital optical
computing with symbolic substitution "Appl.Opt.25
3054 (1986)) Split the wavefront using a beam splitter or half mirror to create multiple images. Hologram method (ASKumar and RMVasu, "Mu
ltiple Imaging withan Aberration Optimized Hololes
Array, "Opt.Eng.28,903 (1989)) A multifocal lens is made by a hologram and an image is multiplexed.

Microlens array method (Kenj
iro Hamanaka, Hiroyuki Nemoto, Masahiro Oikawa, and T
akashi Kisimoto, "Multiple Imaging and multiple Fou
riertransformation using planer microlens arrays. "
Appl, Opt 29 4064 (1990)) Construct a proof system and an imaging system by a lens array and make multiple images.

Each of these methods has advantages and disadvantages.
The easiest method is the one that is easy to create. However, this method is not practical because it is difficult to obtain many images. Although the method of (1) can obtain many images, it is disadvantageous in terms of aberration. Although the method of (1) can obtain many images without aberration, the optical system tends to be complicated.

In the present invention, it is easy to prepare for the purpose of confirming the principle of optical fractal figure processing.
Method was adopted. Of course, the present invention may use the method of.

(E-2) Magnification Conversion Magnification conversion is one of the easiest processes to realize in an optical system. Even with a simple imaging system, magnification conversion can be easily performed. In FIG. 4, if the ratio of the distance from the principal point of the lens 1 to the object 2 and the image 3 is b / a, the magnification by imaging is b / a.

If the anamorphic optical system shown in FIG. 5 is used, an optical system having a variable vertical / horizontal ratio can be produced. In FIG. 5, after collimating the light with the lens 10,
The aspect ratio can be changed by changing the vertical or horizontal (parallel or vertical to the paper surface) luminous flux width by the anamorphic optical system configured by combining the prisms 11 and 12. In the figure, the lenses 10 and 13 form an image forming optical system, and the prisms 11 and 12 arranged between the lenses 10 and 13 respectively change the light flux width to change the light flux width in the direction parallel to the paper surface.

According to the matrix display described in (d), these can be written as follows.

[0053] In the reduced affine transformation, the scaling factors r _x and r _y must be smaller than 1.

(E-3) Movement of image Movement of the image can be easily realized by using a deflection mirror. That is, as shown in FIG. 6A, in the optical system in which the image of the object 23 is formed by the lenses 20 and 22,
By arranging the deflection mirror 21 in the middle and changing the angle of the mirror 21 as shown by the arrow to change the deflection angle, the image forming position can be moved.

However, the method using the deflecting mirror has a drawback that the optical system becomes large. In order to make the optical system more compact, the optical system using the hologram shown in the conceptual diagram of FIG. 6B is desirable. That is, the hologram 31 on which the image of the object 32 is recorded is irradiated with the reproduction light, and the ± 1st-order diffraction image or the high-order diffraction image is used as the reproduction image 13 to move the image. In addition, it is necessary to consider that the problem of aberration occurs when using a hologram.

(E-4) Inversion of the image The inversion of the image can be easily realized by the odd number of reflections by the mirror surface. That is, as shown in FIG.
In the optical system for forming 3 as the image 44, the image is inverted by disposing the reflection mirror 41 in the middle. In this method, the optical system is likely to be large as in the case of moving the image, and therefore it may be combined with an element such as a reflection hologram. In both methods, the influence of aberration becomes large unless reflection by parallel light is used.

(E-5) Image Rotation There is a method of rotating the image by a plurality of reflections, but the system becomes too large and not practical. In the present invention, an image rotating prism is used. That is, as shown in FIG.
In the optical system that forms the object 53 as an image 54 at 0 and 52, the image rotation prism 51 is arranged in the middle and the angle is rotated about the optical axis by the angle θ. Since the image rotation prism 51 has one reflecting surface, the incident image is rotated by 2θ and further inverted.

(E-6) Conversion by MSLM Conversion by MSLM is disclosed in Junpei Tsujiuchi, Yoshiki Ichioka and Komine Minemoto, Optical Information Processing, Ohmsha (1988). MSLM (Multi-ch) is used as a spatial light modulator composed of a combination of a photocathode, an electron lens, a microchannel plate (MCP), and a ferroelectric crystal plate that exhibits the electro-optical effect.
annel Spatial Light Modulator) has been developed.
This MSLM is capable of inverting, reducing, and rotating an image by using an electronic lens for optical input information, and it is possible to configure a compact optical system of a reduction conversion group.

(E-7) Summary By combining the optical systems (e-1) to (e-5) described above, it is possible to create an arbitrary reduced affine transformation group. In order to generate a figure from this reduced affine transformation group, it is necessary to feed back an image. Next, the problems of the feedback system, especially the problem of the temporary image storage element will be described.

(F) Problems of Feedback System It is necessary to configure a feedback system when an image is generated by the reduction conversion group. However, in the system envisioned in the present invention, there is a loss of light due to the optical system, and it is necessary to amplify the image to compensate for this. If the number of feedbacks is not limited, the image may disappear depending on the state of the optical system. There is a problem such as. Therefore, it is conceivable to use a temporary image storage element in order to facilitate the control of the feedback process. A frame memory is a widely used image storage element at present. The frame memory is often used because it is easy to use with an electronic computer, and image data can be written directly from an image pickup device such as a TV camera. However, there are the following drawbacks. -The image information, which is originally a continuous quantity, can be handled only as a discretized image information.・ Because it has a lattice structure, it causes stripes and moire other than the lattice direction. -Generally available frame memories are at most 1000 * 1000, and the resolution of the optical system cannot be fully utilized.

As an element for solving these problems, an optical writing type spatial light modulator (SLM [Spatial Light Modula
tor]). In the past, SLM had a slower operating speed than CCDs and other image sensors, but ferroelectric liquid crystal SLMs with operating speeds of several ms and semiconductor SLMs of several ns have also been developed. These elements do not have a lattice structure, and the number of resolution points has been developed to be more than that of a frame memory. Therefore optically IFS
When calculating, it is desirable to use the optical writing type SLM.

By combining the above optical systems,
A reduced affine transformation group can be constructed. Therefore, we conducted an experiment to confirm the principle of image generation, and describe the configuration of the system and the result of the experiment. (G) System Configuration (g-1) As a system for confirming the element principle of the optical system, the following optical system was adopted for multiplexing conversion, magnification conversion, image rotation, and movement. -Multiplexing of conversion At least 2 reduction conversions are required. In a microlens array or an optical system using a hologram, it is difficult to make adjustments, so wavefront division by a beam splitter is used.・ Magnification conversion Anamorphic optical system is prone to astigmatism,
Since it is difficult to adjust the angle of the prism, a simple imaging system that uses a zoom lens was used. Image rotation An image rotation prism was used for this.・ Movement of image This was realized by the deflection mirror. An experimental system was constructed with a combination of these four optical elements and a TV feedback system via a frame memory.

(G-2) Device Configuration FIG. 9 is a diagram showing a device configuration in the present invention. A feedback system using a CRT monitor and a CCD camera is used for feedback, and a frame memory IMG PC9801VM (ImagePC (256 * 240 * 8bit)) is used as a temporary storage device for images.
Was used. Table 1 shows the optical components used. (H) Characteristics of optical system (h-1) Expression of optical system Here, the form of reduction conversion that can be expressed by the experimental system will be described. The image rotating prism used in the experiment is called a Dove prism and has one reflecting surface. Therefore, the rotation and the reversal of the image occur at the same time. Therefore, if the rotation of the prism is φ, the conversion by the Dove prism is expressed as follows. The optical system in Fig. 9 is made up of two conversion systems, which are arm1 and arm2, respectively. arm1 is converted into image → reflection → Dove prism → reflection → image formation → image formation, so the formula expressing this is Similarly, arm2 is transformed into image formation → Dove prism → reflection → reflection → image formation → image formation. Is. Hereinafter, the conversion by arm1 is represented by f1 and the conversion by arm2 is represented by f2.

(H-2) In the light emission characteristic experiment system of the TV monitor, in order to compensate for the loss of light due to the optical system, it was decided to perform binarization in the temporary storage element of the image. For this purpose, the characteristics of the monitor through the optical system are CC
It was measured with a D camera. In FIG. 10, the horizontal axis represents the pixel value in the frame memory given to the monitor, and the vertical axis represents the pixel value captured in the frame memory after passing through the CCD, arm1,
This is for arm2. It can be seen from FIG. 10 that the threshold value for binarization is preferably about 120 to 150.

(I) Image Generation Experiment An experiment was conducted to confirm that an image can be generated with the reduced affine transformation optical system.

Experiments are as follows: 1) Binary processing is performed using a temporary storage element 2) Binary processing is performed using a temporary storage element 3) CRT monitor CCD camera without a temporary storage element It was done in three modes, though it was directly connected. In each experiment, the initial image was bright on the entire surface. Further, in order to determine the parameters of f1 (arm1) and f2 (arm2), the rotation angle, the magnification of the image, and the center of the converted image were measured on the frame memory.

(I-1) Step operation (with binarization)
An image was generated using the parameters shown in Table 2. From each parameter, f ₁ and f ₂ are given as follows.

[0068] 20 iterations were performed using these parameters. FIG. 11 shows an image for each step.

Next, as another example of parameters, an image was generated with the parameters shown in Table 3. From this parameter, f ₁ and f ₂ are respectively given as follows. In this case as well, the repeated treatment was performed 20 times. FIG. 12 shows an image for each step.

A simulation experiment was conducted to confirm that the image synthesized by the optical system was equivalent to that obtained by a digital computer. Images obtained by the simulation are shown in FIGS. 13 and 14. It should be noted that the upper and lower sides are inverted in the figure because the coordinate system is inverted.

(I-2) Step operation (without binarization)
An experiment without binarization was performed using the same parameters as the code (i-1). 15 and 16 are generated using the parameters of Table 2 and Table 3, respectively. FIG. 15 shows a generated image in the step operation (without binarization) mode, and FIG. 16 shows a generated image in the step operation (without binarization) mode.

(I-3) Continuous operation mode Even if an image is not binarized, the image converges after 10 to 15 times of feedback. Further, even after the feedback, the image does not disappear due to the condition of the optical system, and a stable image can be obtained. Therefore, the CCD camera and the CRT monitor were directly connected to each other without using the temporary storage element, and the repetitive processing was performed. A stable image can be obtained by opening the CCD aperture brightly. This is because the light of the monitor is strong CC
It is believed that the image bleeds on D and is stable.
However, this method reduces the resolution of the image. The result is shown in FIG.

(J) Convergence Speed The processing speed was as shown in Table 4. (K) Image synthesis To show the possibility of application to image compression technology by IFS,
An experiment was conducted to synthesize an image from IFS with different parameters. FIG. 18 shows the result. The parameters of images 1 to 4 are shown in Tables 5-1 to 5-4, respectively, and the composite image was generated from only these parameters. It should be noted that the pixel-wise logical sum by a digital computer was used for image synthesis. If such a method is used, for example, a complicated figure can be divided into a plurality of subfigures, coded by IFS, reproduced for each subfigure, and synthesized, and it is generated only from different parameters. It can be seen that the image compression rate is realized.

[0074]

[0075]

The present invention aims to confirm the principle of optical realization of the reduction conversion group and to compare the speeds of optical reproduction and reproduction by a computer. Regarding the results obtained by the optical system, Comparing the results of the simulations using the same parameters and comparing the two, the images are almost the same, but the image obtained by the simulation has a stronger contour. It is considered that this is due to the aberration of the optical system or the influence of the quantization error during the simulation.
With respect to, for a figure in which the number of transformations is several, it can be obtained at almost the same speed as the optical system by using a workstation class computer. However, in order to reproduce a normal image, a conversion of 100 to 200 is required, and in that case, even if a workstation is used, it takes 30 minutes or more for calculation time. On the other hand, the optical method of the present invention can be processed very efficiently by multiplexing the optical system. From the above, it can be seen that the method of the present invention utilizing an optical system is very effective as a reproduction method of an image coded by IFS.

[Brief description of drawings]

FIG. 1 is a diagram illustrating a method of creating a Koch curve.

FIG. 2 shows a Shelbinski triangle.

FIG. 3 is a diagram showing formation of a figure by IFS.

FIG. 4 is a diagram illustrating magnification conversion by image formation.

FIG. 5 is a diagram showing an optical system with an aspect ratio variable by an anamorphic optical system.

FIG. 6 is a diagram illustrating an optical system that moves an image.

FIG. 7 is a diagram illustrating inversion of an image due to reflection.

FIG. 8 is a diagram illustrating rotation of an image by an image rotation prism.

FIG. 9 is a diagram showing a configuration of an embodiment of an optical fractal figure generation device of the present invention.

FIG. 10 is a diagram showing characteristics of the optical system of the present invention.

FIG. 11 is a diagram showing a generated image in a step operation (with binarization) mode.

FIG. 12 is a diagram showing a generated image in a step operation (with binarization) mode.

FIG. 13 is a diagram showing an image obtained by simulation.

FIG. 14 is a diagram showing an image obtained by simulation.

FIG. 15 is a diagram showing a generated image in a step operation (no binarization) mode.

FIG. 16 is a diagram showing a generated image in a step operation (no binarization) mode.

FIG. 17 is a diagram showing images generated in a continuous operation mode.

FIG. 18 is a diagram illustrating image composition from IFS.

[Explanation of symbols]

1 ... Lens, 2 ... Object, 3 ... Image, 11, 12 ... Prism, 21 ... Deflection mirror, 31 ... Hologram, 41 ... Mirror, 51 ... Image rotation prism, LI ... Camera lens, L2
~ L4 ... Lens, BS ... Beam splitter, DP ... Image rotation prism, ZL ... Zoom lens, Mir ... Mirror.

Claims (10)

[Claims] 1. A method of generating a fractal figure coded by a reduced affine transformation, wherein a plurality of images are created from an optically presented image, and magnification conversion, image movement, and image are performed by parameters of the reduced affine transformation. Optical processing of combining the images by magnification conversion, image movement, and image rotation is performed by a plurality of optical systems in which the respective coefficients of rotation are set, the combined image is read and fed back, and re-presented to perform the optical processing. An optical fractal figure generation method characterized in that a fractal figure is optically generated by repeating. 2. The optical fractal figure generation method according to claim 1, wherein the read image is stored in a temporary storage means and fed back. 3. A fractal figure generation device coded by a reduced affine transformation, comprising image presentation means for presenting an optical image, and a multiple imaging optical system for producing a plurality of images from the presented images. Coefficients for magnification conversion, image movement, and image rotation are set by the parameters of the reduction affine transformation, and a plurality of optical systems for magnification conversion, image movement, and image rotation are combined with the output images from each optical system. An optical fractal figure generation device characterized in that the fractal figure is optically generated by feeding the read image back to the image presenting means. 4. The optical fractal figure generation device according to claim 3, wherein the multiplex imaging optical system comprises a beam splitter. 5. The optical fractal figure generation device according to claim 3, wherein the optical system for moving the image comprises a deflection mirror. 6. The optical fractal figure generation device according to claim 3, wherein the image rotating optical system comprises an image rotating prism. 7. The optical fractal figure generation device according to claim 3, wherein the image reading means is a CCD camera. 8. The optical fractal figure generation device according to claim 3, wherein the image reading means, the temporary storage means, and the image presentation means are spatial light modulators. 9. The optical fractal figure generation device according to claim 3, further comprising a temporary storage means, wherein the read image is stored in the temporary storage means and fed back to the image presenting means. apparatus. 10. The optical device according to claim 9, wherein the temporary storage means comprises a frame memory, and the image read by the TV feedback system via the frame memory is fed back to the image presentation means. Fractal figure generator.