CN102360399B - Generation method of printed fabric patterns based on generalized Mandelbrot set - Google Patents

Abstract

The invention relates to a generation method of patterns being used for printed fabric and having fractal textures, and in particular relates to a generation method of printed fabric patterns based on generalized Mandelbrot set. According to a complex variable function theory, a spinning pattern is designed by adopting methods of copying, rotating, splicing and the like through constructing a generalized Mandelbrot set-based fractal graph and local detail graphs thereof as basic elements by using computer programming, and then is directly transferred to the fabric by an electronic inkjet printing machine or subjected to color separation by adopting CAD (computer aided design) software for carrying out plate-making printing to obtain the printed fabric with the fractal textures. The invention provides an iteration function and a real part and imaginary part separation method. Through changing arrangement of parameters and color values in an iteration formula, an image amplification region, amplification factors and the like, images with different styles are obtained, and innumerable and changeful image series can be conveniently and rapidly obtained, thus rapid design of the printed pattern is realized.

Description

Generation method based on the PRINTED FABRIC pattern of broad sense Mandelbrot collection

Technical field

The present invention relates to a kind of design method of pattern that can be used for PRINTED FABRIC with fractal texture, particularly a kind of design method of pattern of the PRINTED FABRIC based on broad sense Mandelbrot collection.

Background technology

Textile melts artistry and technical being integrated, and has good decorative effect and high added value.Textile can incorporate daily life with art in prevailing mode, namely uses textile design expression art figure.In the PRINTED FABRIC design, the design of grain pattern is very important, and it is the first operation of producing, and is basis attractive in appearance, also is an important step that improves the PRINTED FABRIC quality, and the design of grain pattern is the key of textile designs.

Textile art creation is nothing more than being that the designer seeks the creation subject matter in the Nature and human society, by the fine things of fabric decorative pattern pattern reflection nature and human society life, embody appreciation temperament and interest and psychology pursuit that people and the tendency of the day, social background and local conditions and customs adapt.Its mode of production then is subject to the restriction of production technology development level, because from finishing design to manufacturing a finished product, fully take manually-operated as essential characteristic, so emerging of a width of cloth works often needs the cycle of growing.

Along with the development of computer technology, the weaving electric mechanical equipment such as electronic jacquard machine, electronics ink-jet decorating machine, electronic embroidery machine and various CAD software appear on the scene one after another.The appearance of these electronic products has solved the problem of complex manufacturing, has begun to take shape the advantage of automated production, has greatly reduced labour intensity and the cycle of whole process.But textile art design itself or a chronic process, designer's design still depends on flashing of creation inspiration, and fully self-dependent " subjectivity " designs.Therefore, the design of Textile Pattern often becomes a bottleneck in the product design, the advantage of automatic electronic can't be fully played, obviously, this traditional design mode with distinct disadvantage such as mentality of designing are narrow, the flower type is few, the cycle is long has been difficult to adapt to modern life rhythm and the market demand.

Before the present invention makes, document " research of Mandelbrot collection visual information characterizing method in the fabric pattern design " ([J] University Of Suzhou journal industry science version, 2002, No.2, a kind of computer generating method of Mandelbrot collection graftal is disclosed p.24-27)), step comprises: adopt iterative zn+1=zn2+v, with real part and imaginary component from, selected parameter v=p+iq, figure is shown in specified scope, the result draws each point with different colours according to iteration, obtains Mandelbrot collection graftal; Document " based on the decalcomania design pre-test of trigonometric function Mandelbrot collection " ([J] silk 2005, No.1, p.13,51) in, the generation method that has related to a kind of trigonometric function Mandelbrot collection pattern, step comprises: adopt iterative zn+1=sinzn+ μ, utilize Euler's formula with real part and imaginary component from, draw trigonometric function Mandelbrot collection graftal.Its style of the image that prior art provides changes very limited, can't satisfying the market to the designing requirement of textile patter.

Based on broad sense Mandelbrot collection, adopt different iteration functions, and different real parts and the separation method of imaginary part, to obtain the diverse image of style, therefore, how research obtains the abundant image of changing style, quantity, making the design of decalcomania more versatile and flexible, is very necessary.

Summary of the invention

The object of the invention is to overcome the deficiency that prior art exists, provide a kind of changing style, amount of images to enrich, make the more flexile method for generating pattern of design of printing in textiles pattern.

Realize that technical scheme that the object of the invention adopts provides a kind of generation method of the PRINTED FABRIC pattern based on broad sense Mandelbrot collection, comprises the steps:

(1) structure broad sense Mandelbrot collection graftal

1) take a, b as pixel, obtain the visual in-scope of a * b, to three color components of red, green, blue respectively with the numeral 0,1,2 ..., L represents, amounts to the L+1 kind, with 0 expression black;

2) in complex plane, set up iterative

Wherein, z is iteration variable, and c is iteration constant, and n is iterations, and w is iterative exponential term, and w=α+i β, (α, β ∈ R), R is real number, and i is imaginary unit, and z and c are plural number, and they are respectively:

Z=x+iy, (x, y ∈ R), x is the real part of z, y is the imaginary part of z,

C=p+iq, (p, q ∈ R), p is the real part of c, q is the imaginary part of c;

Get iterative initial value z ₀=x ₀+ iy ₀=0, wherein, 0＜| x ₀|≤0.1, y ₀=0;

3) get

Wherein, pmin≤p≤pmax, qmin≤q≤qmax;

With n _xAnd n _yBe horizontal ordinate and the ordinate of drawing point, n _x=0,1 ... a-1 and n _y=0,1 ..., b-1 is to all drawing point (n _x, n _y) carry out following circulation:

①p ₀＝p _min+n _x·Δp，q ₀＝q _min+n _y·Δq ，n＝0，x ₀＝0.01，y ₀＝0；

2. separate

Real part and imaginary part, get iterative A:

Real part: $x_{n+1}=\exp [\mathrm{\α}\ln \sqrt{{x_n}^2+{y_n}^2}-\mathrm{\β}({\mathrm{\θ}}_z+2\mathrm{k\π})]\·\cos [\mathrm{\α}({\mathrm{\θ}}_z+2\mathrm{k\π})+\mathrm{\β}\ln \sqrt{{x_n}^2+{y_n}^2}]+p,$

Imaginary part: $y_{n+1}=\exp [\mathrm{\α}\ln \sqrt{{x_n}^2+{y_n}^2}-\mathrm{\β}({\mathrm{\θ}}_z+2\mathrm{k\π})]\·\sin [\mathrm{\α}({\mathrm{\θ}}_z+2\mathrm{k\π})+\mathrm{\β}\ln \sqrt{{x_n}^2+{y_n}^2}]+q,$

θ _z∈ (π, π] be the main argument of plural z, | z| is the mould of z, and π is circular constant, and k is integer, main argument θ _zBe one of following three kinds of situations:

To iterative A from (x _n, y _n) calculate (x _N+1, y _N+1), counting n=n+1;

3. get

If the result who obtains is judged, s＞M, or 4. n=L goes to step; If 2. s≤M, and n＜L go to step; M 〉=1000;

4. to point (n _x, n _y) press the n value and select color dyes, go to next point, repeating step 1., until all drawing point processing finish;

(2) in the p that has image to show, q interval, the interval of p, q is dwindled, obtain the meticulous topography of Mandelbrot collection; Change interval, obtain different topographies;

(3) fractal pattern that integrates with the broad sense Mandelbrot that obtains or topography be as fundamental element, adopts to copy, rotate or the method for piecing together obtains Textile Pattern, saves as image file with BMP or JPG or TIF form; Image file is input to the electronics ink-jet decorating machine, the grain pattern pattern is directly printed on the fabric, or with CAD software image file is carried out color separation processing, make a plate again, stamp, obtain PRINTED FABRIC.

Compared with prior art, technical scheme provided by the invention has following characteristics:

1, computer visualization technique utilizes computer program to generate broad sense Mandelbrot collection graftal, the pattern that can obtain quickly and easily having fine structure.

2, iteration function z ← z ^wW is plural number among the+c, behind real part and imaginary part in the theory of complex functions separation iteration function, can obtain the Mandelbrot collection graftal of arbitrary number of times power, makes the textile patter design obtain the abundant fountain-head of literary and artistic.

3, Mandelbrot collection graftal has characteristics local and whole self similarity, and therefore the meticulous topography of different parts is different in a certain general image, but becomes the similar series of style, and this is very convenient in the PRINTED FABRIC design of seriation.

Description of drawings

Fig. 1 is the broad sense Mandelbrot collection graftal that the embodiment of the invention provides;

Fig. 2 is the graftal that the embodiment of the invention adopts different amplification to obtain to zones of different;

Fig. 3 is the Textile Pattern figure that partial, detailed view that the embodiment of the invention provides is combined into.

Embodiment

The invention will be further described below in conjunction with drawings and Examples.

Embodiment 1

One, according to the composite principle of Mandelbrot collection, construction analysis figure

Mandelbrot collection (being called for short the M collection) fractal pattern is the foremost fractal pattern in fractal field.In complex plane, it is by iterative Produce, wherein, z is iteration variable, and c is iteration constant, and n is iterations; Z and c are plural numbers, are designated as:

z＝x+iy

c＝p+iq；

From z _nTo z _N+1Iterative process be exactly

$x_{n+1}=x_n^2-y_n^2+p,$ Be real part,

y _N+1=2x _ny _n+ q is imaginary part;

In M collection iterative process, make primary iteration value z ₀=0, z namely ₀=0+i0; C is changed in certain zone of complex plane regularly, and to different c values (c ≠ 0), i.e. different p and q are by the above-mentioned iterative iteration (wherein p and q go on foot in iteration and remain constant at each) of repeatedly carrying out.

Repeatedly carry out iteration, consequent z with different c values _nSequence has two kinds of situations:

z _nSequence freely towards infinitely-great direction diffusion, is namely dispersed;

z _nSequence is limited in a certain zone of complex plane, i.e. convergence.

Set up and differentiate convergence and the judgment criterion of dispersing, for the z of those convergences _nThe point of sequence arranges the tone of certain color, just can show the computer picture of M collection; For those z that disperses _nThe point of sequence according to the difference of divergence speed, different colors according to given rule, just can show M collection image on every side.

In a certain regional extent of complex plane, make c do regular variation, different c values is carried out iteration, infinitely great if result of calculation reaches, then c is become white, otherwise, into black, so just can demonstrate the contour shape of M collection image.If the point that will disperse according to the speed of its speed of dispersing, represents with different colors, just form the extremely attracting color image of a width of cloth.

Two, according to the theory of complex functions structure broad sense Mandelbrot collection graftal

In fact, shape such as z ← z ²Rerunning a movie of+c penetrated, and just shines upon z ← z on the parameter plane ^wThe special case of+c when w=2.When w gets other positive integers, negative integer, plural number, or when getting the mapping such as trigonometric function, logarithmic function, can obtain the M collection image of broad sense, these images will become the abundant fountain-head of literary and artistic of textile patter.

Present embodiment is got w=α+i β (α, β ∈ R).

According to the theory of complex functions, make θ _z∈ (π, π] expression plural z main argument, | z| is the mould of z, then by z ^a=exp{alnz}=exp{a[ln|z|+i (θ _z+ 2k π)] }

(a is any given plural number, z ≠ 0, and k=0, ± 1, ± 2 ...) obtain:

z ^w＝z ^α+iβ＝exp{(α+iβ)[ln|z|+i(θ _z+2kπ)]}

＝exp{αln|z|-β(θ _z+2kπ)+i[α(θ _z+2kπ)+βln|z|]}

＝exp[αln|z|-β(θ _z+2kπ)]·exp{i[α(θ _z+2kπ)+βln|z|]}

Again according to Euler (Euler) formula

Obtain

z ^a+iβ＝exp[αln|z|-β(θ _z+2kπ)]·{cos[α(θ _z+2kπ)+βln|z|]+isin[α(θ _z+2kπ)+βln|z|]}＝exp[αln|z|-β(θ _z+2kπ)]·cos[α(θ _z+2kπ)+βln|z|]

+iexp[αln|z|-β(θ _z+2kπ)]·sin[α(θ _z+2kπ)+βln|z|]

Work as z=x+iy, during c=p+iq,

$z^{\mathrm{\α}+\mathrm{i\β}}+c$

$=\exp [\mathrm{\α}\ln \sqrt{x^2+y^2}-\mathrm{\β}({\mathrm{\θ}}_z+2\mathrm{k\π})]\·\cos [\mathrm{\α}({\mathrm{\θ}}_z+2\mathrm{k\π})+\mathrm{\β}\ln \sqrt{x^2+y^2}]+p$

$+i\left\{\exp \right[\mathrm{\α}\ln \sqrt{x^2+y^2}-\mathrm{\β}({\mathrm{\θ}}_z+2\mathrm{k\π})]\·\sin [\mathrm{\α}({\mathrm{\θ}}_z+2\mathrm{k\π})+\mathrm{\β}\ln \sqrt{x^2+y^2}]+q\}$

Separate z ← z ^{α+i β}The real part of+c and imaginary part get iterative (1):

$x\←\exp [\mathrm{\α}\ln \sqrt{x^2+y^2}-\mathrm{\β}({\mathrm{\θ}}_z+2\mathrm{k\π})]\·\cos [\mathrm{\α}({\mathrm{\θ}}_z+2\mathrm{k\π})+\mathrm{\β}\ln \sqrt{x^2+y^2}]+p$

$y\←\exp [\mathrm{\α}\ln \sqrt{x^2+y^2}-\mathrm{\β}({\mathrm{\θ}}_z+2\mathrm{k\π})]\·\sin [\mathrm{\α}({\mathrm{\θ}}_z+2\mathrm{k\π})+\mathrm{\β}\ln \sqrt{x^2+y^2}]+q;$

Main argument θ _zThree kinds of situations are arranged:

α and β can get any real number, and when α and β variation, image also changes thereupon.Mapping z ← z ^{A+i β}+ c has almost comprised all scenario except trigonometric function.

The core process of broad sense Mandelbrot collection image generating algorithm is as follows:

Take a, b as pixel, suppose that image size is a * b point, the color component of RGB has respectively the L+1 kind, respectively with numeral 0,1,2 ..., L represents, and with 0 expression black, n is iterations.

According to the iteration theorem of classical Mandelbrot collection, get iterative initial value z ₀=0, i.e. x ₀=y ₀=0; But the present invention the situation that x is denominator occurs after according to the theory of complex functions conversion, for avoid occurring in the program operation process denominator be 0 can't computing the result, the present invention is with iterative initial value x ₀Be modified to one and be slightly larger than 0 value, such as x ₀=0.01.Specify p, q within the specific limits, namely calculate the iteration result of a certain c point in the specified scope (value by p, q is determined); Get again a constant M, just be considered to reach " infinity " when the mould of iteration variable surpasses M, corresponding c point is the point of dispersing.

Get

To all drawing point (n _x, n _y), n wherein _xAnd n _yBe respectively horizontal ordinate and the ordinate of drawing point, n _x=0,1 ... a-1 and n _y=0,1 ..., b-1, finish following circulation:

(1) gets p ₀=p _Min+ n _xΔ p, q ₀=q _Min+ n _yΔ q; N=0, x ₀=0.01, y ₀=0;

(2) by iterative (1) from (x _n, y _n) calculate (x _N+1, y _N+1), counting n=n+1;

Get

If s＞M, or n=L goes to step (4), if s≤M, and n＜L, go to step (2), M 〉=1000;

(4) to point (n _x, n _y) select color dyes according to the n value, go to next point, repeating step (1) is until calculated all drawing points.

Present embodiment adopts Visual Basic Programming with Pascal Language to draw Mandelbrot collection graftal.For obtaining the image of different color effect, by Pset (x, y), RGB (r, g, b) function is controlled the drawing color in program, x wherein, and y is the coordinate figure of drawing point, r, g, b are respectively three kinds of color components of red, green, blue.Get r, g, b is the function of n.As r, g, when b gets same functions, what obtain is gray scale image; As r, g, when b gets different function, what obtain is color image.In order to make the map architecture levels are rich distinct, present embodiment is multiplied by large (more than a 30) coefficient to n, to enlarge the color distortion between the different divergence speed points according to the color of n value selected element the time; In order to guarantee color value within displayable scope, logarithm value is carried out modulo operation simultaneously.As get the RGB color value and be respectively (60*n) Mod 255, (60*n) Mod 255,0.Different α and the corresponding Mandelbrot collection of β value graftal as shown in Figure 1, among Fig. 1, a figure: α=6, β=0; B figure: α=-3, β=0; C figure: α=-4, β=0; D figure: α=-5, β=0.5.

Three, in the zone that has image to show, Mandelbrot collection graftal is amplified, to obtain new image.

Mandelbrot collection graftal has infinite meticulous characteristics, and the amplification of each part all is the image of a width of cloth fineness in the zone that has image to show.Interval with p, q in the p that has image to show, q interval dwindles, and obtains the meticulous topography of Mandelbrot collection; Change interval, obtain different topographies, therefore, the different parts of choosing different images amplifies, can obtain innumerable, protean image series, referring to accompanying drawing 2, it is the broad sense Mandelbrot collection graftal that adopts different amplification to obtain by the zones of different among Fig. 1 b figure.

Four, utilize broad sense Mandelbrot collection graftal or its enlarged drawing design grain pattern and be applied to textile

Integrate graftal or its enlarged drawing as fundamental element with Mandelbrot, by the design such as the method that copies, rotates, pieces together grain pattern, referring to accompanying drawing 3, it is that the Textile Pattern figure that is combined into is amplified in the part of the b figure among the Fig. 1 that is provided by present embodiment.Grain pattern has multiple use, as carrying out textile printing according to the design grain pattern.

According to the design grain pattern can carry out textile printing by two kinds of approach, the one, utilize the electronics ink-jet decorating machine directly with the grain pattern design transfer to fabric; The 2nd, the method that adopts tradition to make a plate utilizes first CAD software that grain pattern is carried out color separation processing, makes a plate, then stamp again.Fractal Textile Pattern is optimum to be stamp be used to the silk kind fabric with fine structure, can demonstrate fully so the beautiful meticulous characteristics of fractal pattern.

Key of the present invention is, by change α and β parameter in iterative, change the RGB color value configuration, change zooming zone and enlargement factor etc., innumerable, protean image series can be obtained quickly and easily, thereby the rapid Design of stamp grain pattern can be realized.

Claims (1)

1. the generation method based on the PRINTED FABRIC pattern of broad sense Mandelbrot collection is characterized in that comprising the steps:

(1) structure broad sense Mandelbrot collection graftal:

1) take a, b as pixel, obtain the visual in-scope of a * b, to three color components of red, green, blue respectively with the numeral 0,1,2 ..., L represents, amounts to the L+1 kind, with 0 expression black;

2) in complex plane, set up iterative

Wherein, z is iteration variable, and c is iteration constant, and n is iterations, and w is iterative exponential term, and w=α+i β, (α, β ∈ R), R is real number, and i is imaginary unit, and z and c are plural number, and they are respectively:

Z=x+iy, (x, y ∈ R), x is the real part of z, y is the imaginary part of z,

C=p+iq, (p, q ∈ R), p is the real part of c, q is the imaginary part of c;

Get iterative initial value z ₀=x ₀+ iy ₀=0, wherein, 0＜| x ₀|≤0.1, y ₀=0;

3) get Wherein, p _Min≤ p≤p _Max, q _Min≤ q≤q _Max

With n _xAnd n _yBe horizontal ordinate and the ordinate of drawing point, n _x=0,1 ... a-1 and n _y=0,1 ..., b-1 is to all drawing point (n _x, n _y) carry out following circulation:

①p ₀＝p _min+n _x·Δp，q ₀＝q _min+n _y·Δq ，n＝0，x ₀＝0.01，y ₀＝0；

2. separate

Real part and imaginary part, get iterative A:

Real part: $x_{n+1}=\exp [\mathrm{\α}\ln \sqrt{{x_n}^2+{y_n}^2}-\mathrm{\β}({\mathrm{\θ}}_z+2\mathrm{k\π})]\·\cos [\mathrm{\α}({\mathrm{\θ}}_z+2\mathrm{k\π})+\mathrm{\β}\ln \sqrt{{x_n}^2+{y_n}^2}]+p,$

Imaginary part: $y_{n+1}=\exp [\mathrm{\α}\ln \sqrt{{x_n}^2+{y_n}^2}-\mathrm{\β}({\mathrm{\θ}}_z+2\mathrm{k\π})]\·\sin [\mathrm{\α}({\mathrm{\θ}}_z+2\mathrm{k\π})+\mathrm{\β}\ln \sqrt{{x_n}^2+{y_n}^2}]+q,$

θ _z∈ (π, π] be the main argument of plural z, | z| is the mould of z, and π is circular constant, and k is integer, main argument θ _zBe one of following three kinds of situations:

To iterative A from (x _n, y _n) calculate (x _N+1, y _N+1), counting n=n+1;

3. get

The result who obtains is judged, if, s＞M, or 4. n=L goes to step; If 2. s≤M, and n＜L go to step; M 〉=1000;

4. to point (n _x, n _y) press the n value and select color dyes, go to next point, repeating step 1., until all drawing point processing finish;

(2) in the p that has image to show, q interval, the interval of p, q is dwindled, obtain the meticulous topography of Mandelbrot collection; Change interval, obtain different topographies;

(3) fractal pattern that integrates with the broad sense Mandelbrot that obtains or topography be as fundamental element, adopts to copy, rotate or the method for piecing together obtains Textile Pattern, saves as image file with BMP or JPG or TIF form; Image file is input to the electronics ink-jet decorating machine, the grain pattern pattern is directly printed on the fabric, or with CAD software image file is carried out color separation processing, make a plate again, stamp, obtain PRINTED FABRIC.

Images