**3. Color 3D Printing**

Whether in personalized design work, advanced aerospace component manufacturing, or even in three-dimensional cells, the color and appearance issues cannot be ignored. Although in industrial design, mechanical design, or creative and cultural industries, there are not yet satisfactory color reproduction solutions or a low-cost color reproduction technology. But 3D color printing already has a huge and significant impact on life and society. Along with the development of different three-dimensional manufacturing processes, the 3D color printing will also accumulate certain color reproduction techniques.

3D color characterization is an important tool to communicate the color reproduction. Currently, the uniform color spaces are also applied in 3D color characterization, such as the CIELab system. Based on the existing color space, the color value is easily measured and computed with current devices and metrics. When color data were obtained, the color gamut of printed device, the corresponding substrate, and colorant can be described and compared. This can be easily obtained by the Profilemaker software. DE2000 and DEcmc color difference values were widely used as the contrast of color difference.

Color quality has an irreplaceable effect on the surface appearance of the color objects. Color properties and stability are the two key factors that influence the product quality. Stanic et al. had studied the effect of lightfastness experiment on color difference of powder-based 3D color printing objects. He had discussed the effect of aqueous adhesives on color of paper surface in the paper-based 3D Printing. Actually, the process parameters and basic material can achieve the color quality without a fixed trend. At the same time, the postfinished methods can affect the color quality. Carinna Parraman had presented the effect of wax and selected color patches on the color appearance of color patches even if the influence mechanism is under the water.

## **3.1. Stages of development**

and colorful eatable food, Jie Sun et al. introduced and analyzed the effect of multimaterials and multiprintheads on color food products, and provided a feasible method to optimize the

3D color printing is a promising technology that will produce all the color objects in daily life; however, there are rare relevant researches now. For the different 3D color printing processes, the colorization principles are varied from the devices and software. Like the printing graphic, the color reproduction of color 3D Printing is also a key issue for manufacturers and designers. So, 3D color measurement, 3D color characterization, and 3D color quality evaluation are the

For the 3D color measurement, Stanic et al. systematically introduced the selected measure‐ ment method based on inkjet printing parts. This method is similar to CIE guide to the color measurement of printing graphics. In color 3D Printing instances, some researches proposed their measurement devices and conditions. Stanic had adopted the GretagMacbeth (X-rite) XTH sphere spectrophotometer as the measurement for the printed powder-based color object. Xiao et al. used a Minolta CM-2600d spectrophotometer using Spectra Magic NX Color Data Software that was employed to study color measurements in CIELAB values. The illuminant was based on the CIE standard D65 to simulate skin color in daylight conditions. He et al. selected the X-rite 530 spectrophotometer for the 3D color patches printed by the Mimaki UJF-3042 UV printer. Obviously, the color measurement tool is the same as that in printing

graphics. This maybe caused the unexpected color error under the curved surface.

will also accumulate certain color reproduction techniques.

values were widely used as the contrast of color difference.

Whether in personalized design work, advanced aerospace component manufacturing, or even in three-dimensional cells, the color and appearance issues cannot be ignored. Although in industrial design, mechanical design, or creative and cultural industries, there are not yet satisfactory color reproduction solutions or a low-cost color reproduction technology. But 3D color printing already has a huge and significant impact on life and society. Along with the development of different three-dimensional manufacturing processes, the 3D color printing

3D color characterization is an important tool to communicate the color reproduction. Currently, the uniform color spaces are also applied in 3D color characterization, such as the CIELab system. Based on the existing color space, the color value is easily measured and computed with current devices and metrics. When color data were obtained, the color gamut of printed device, the corresponding substrate, and colorant can be described and compared. This can be easily obtained by the Profilemaker software. DE2000 and DEcmc color difference

Color quality has an irreplaceable effect on the surface appearance of the color objects. Color properties and stability are the two key factors that influence the product quality. Stanic et al.

three key factors to improve the color reproduction of 3D color objects.

reproduction of color Eatable food [20].

30 New Trends in 3D Printing

**3. Color 3D Printing**

**2.6. Current researches of color 3D Printing**

The evolution of 3D Printing technology and materials for inkjet printing and the development of 3D color printing can be divided into the following three stages:


#### **3.2. Gamut characteristics with 3D models**

In gamut mapping, the image gamut with a certain range of colors can be expressed. Then, it can be seen as a subset of the larger color space. The gamut mapping method based on image content is an important direction for research currently. Gamut characteristics of image determine the gamut mapping source, which is the mean issue to be considered at first as color gamut mapping is carried out based on the image content. The chromaticity of a color image is one of the input values for gamut mapping and a main parameter for the evaluation and calculation of color reproduction. Thus, a digital image can be analyzed following the steps outlined in **Figure 1**. And, then an appropriate gamut mapping algorithm can be selected for specific digital image gamut characteristics.

**Figure 1.** Digital image analyzing process.

#### **3.3. 3D Printing color management**

Consider the Z510 3D printer of ZCORP Company for example, during the 3D color inkjet printing, coloring is achieved by adding CMY or CMYK color adhesive to mixed transparent adhesives. The colorant in adhesive has its main role as coloring agent. 3D digital models can have a variety of coloring methods, such as directly on the surface, or coloration is applied as the image content, in frequently used information carrier, such as TIFF JPEG file formats files. Of course, a colorful 3D digital model need to be saved in formats that can include its color information, such as VRML, PLY, ZPR, or other proprietary formats.

Issues of coloring of 3D inkjet printing include 3D color reproduction, color consistency and control, color gamut range, effects of materials and processes, the surface characteristics of 3D Printing, and color measurement and other urgent problems to solve. Another related problem is the persistence and firmness of the finished 3D model; both are becoming increasingly important in the 3D inkjet printing and other common applications such as space planning and art and design applications.

#### **3.4. Color detection, measurement, and control**

It has been proposed for 10 years that rapid prototyping technology can produce colored objects with basic color or full spectrum of colors, such as the Zcrop Company developed the 3D inkjet printing system. While the conventional color inkjet printing is being researched and developed for a long period of time, 3D inkjet technology prints different materials with the corresponding coloring processes, which may have special problems. The overall performance is affected by the physical appearance of the object and color appearance, material's gamut, liquidity of base material before forming, the positioning accuracy of printed surface, and the accuracy of scanned digital information. The main effect on finished surface of 3D Printing objects is caused by different printing principles, printing equipment, as well as different features and rough texture of powdered print material particles.

Different positions on the surface of printed objects and different x, y, and z location coordi‐ nates on 3D model will affect the result of colored reproduction. This is due to manufacturing method and layered approach to generate model, which causes difficulties in superimposition of layers to achieve true vertical surface. The results have curved surfaces with certain angles, so it inevitably has questions of aliasing and grinning. Of course, parts of the surface effect issues can be solved during subsequent processing.

Since the finished surface has inherent optical, chemical, and physical characteristics, the finishing coloring processes provide different results. **Figure 2** shows print sample color test chart and their contrasts of saturation and lightness characteristics under three different conditions. The left sample was impregnated with cyanoacrylate, the middle sample was blank, and the right one was impregnated with epoxy resin.

**Figure 2.** 3D Printing color test charts.

gamut mapping is carried out based on the image content. The chromaticity of a color image is one of the input values for gamut mapping and a main parameter for the evaluation and calculation of color reproduction. Thus, a digital image can be analyzed following the steps outlined in **Figure 1**. And, then an appropriate gamut mapping algorithm can be selected for

Consider the Z510 3D printer of ZCORP Company for example, during the 3D color inkjet printing, coloring is achieved by adding CMY or CMYK color adhesive to mixed transparent adhesives. The colorant in adhesive has its main role as coloring agent. 3D digital models can have a variety of coloring methods, such as directly on the surface, or coloration is applied as the image content, in frequently used information carrier, such as TIFF JPEG file formats files. Of course, a colorful 3D digital model need to be saved in formats that can include its color

Issues of coloring of 3D inkjet printing include 3D color reproduction, color consistency and control, color gamut range, effects of materials and processes, the surface characteristics of 3D Printing, and color measurement and other urgent problems to solve. Another related problem is the persistence and firmness of the finished 3D model; both are becoming increasingly important in the 3D inkjet printing and other common applications such as space planning

It has been proposed for 10 years that rapid prototyping technology can produce colored objects with basic color or full spectrum of colors, such as the Zcrop Company developed the 3D inkjet printing system. While the conventional color inkjet printing is being researched and developed for a long period of time, 3D inkjet technology prints different materials with the corresponding coloring processes, which may have special problems. The overall performance is affected by the physical appearance of the object and color appearance, material's gamut,

information, such as VRML, PLY, ZPR, or other proprietary formats.

specific digital image gamut characteristics.

32 New Trends in 3D Printing

**Figure 1.** Digital image analyzing process.

and art and design applications.

**3.4. Color detection, measurement, and control**

**3.3. 3D Printing color management**

Another significant issue not yet tackled is that there is no precise equipment to measure color surface structure and characteristics of 3D Printing product. Currently, we can use a spectrophotometer with an 8-degree field of view to measure them. Spectrophotometer is widely used in textiles, dyes, paints, plastics, and automotive industry as it may require any kind of surface texture and gloss color measurement.

In order to describe 3D Printing color reproduction capability, and monitor and control color output on a 3D printer, we need professional precise testing instruments and software to obtain real sample color and test strip. The color test strip can be generic, such as Zcrop icon's color test bar or special test strips can be developed. Those special strips should be designed specifically to measure appearance and size, while a good variety of colors characteristic can be expressed. In order to facilitate the existing spectrophotometric measurement, color test strips are preferably designed by flat form. **Figure 3** shows some specific designs and printouts of color test strips.

**Figure 3.** 3D Printing color test strips.

Of course, the color test strips can also be of 3D print themselves, which is more suited to some particular measurement requirement, for example, to observe and monitor quality of color reproduction on flat structure, or curve or groove surface. Color test strips can be considered as a kind of color authenticity visual aids to help users predict the printing process. They can also be considered as a direct tool like 3D color swatch book for user to select the desired color.

#### **3.5. Color stability and durability**

3D Printing applications are being used in new areas, people discuss about a wide range of common properties, such as durability and stability of color 3D Printing objects. These issues are of more interest to art and design people, who use rapid prototyping technology to aid their work, because they publicly display their works of art, and geospatial information systems, architecture, space planning, scenic modeling, and urban design need excellent color effects. Color is a core part of printed objects, and people expect the original effects to remain for a certain time.

Duration and stability depend on several factors, such as light, humidity, temperature, and the amount of air in contact. Graphics, printing industry, and color science and other industries have been developed and used a lot of color standards and standard printing methods developed to study and predict the color durability and stability. This is more significant in inkjet printing in commercial and art fields. Whether 3D Printing is also suitable for the abovementioned research methodology, and the materials, air exposure and other factors can influence the finished colors aging mechanism are real issues focused by current researches.

Currently, the durability and stability study of 3D printed color focus on how to find the right way to simulate aging or change real 3D prints, which are exposing in different environment conditions. For example, different lighting, temperature, humidity, the finished color of 3D Printing, and the behavior of color change with different impregnating agent can be simulated and predicted. Custom color strips should be developed in test researches, as well as corre‐ sponding colors and test procedures, so that people can conduct relevant research.

### **3.6. Digital coloring method of 3D Printing**

A color model describes the displayed color effects of print. Different color models define different range of colors. The fields of their applications are different. These applications are in addition to determining the number of color inside a model, the number of color channels, and the file size of color image information. Here we present several related concepts of color images.

*Bit depth*: Bit depth, also called the pixel depth or the color depth, is used to measure color information in the image to display or print. Greater bit depth means that the digital image has more colors and more accurate color representation.

*Gamut*: This represents the range of colors capable of displaying or printed. Lab color space has the widest color gamut, which can contain all the color types from RGB and CMYK color mode. Typically, RGB color gamut can be displayed on a computer monitor or a television screen. However, some, such as pure cyan or pure yellow and other colors, cannot be accurately displayed on digital monitors. CMYK color space has narrower gamut, containing only color types used in printing, which can be printed with ink. When a color cannot be printed or displayed on the screen, it is called out of the gamut, that is, beyond the CMYK color gamut.

*Color channel*: Each color image has one or more channels, and each channel can store the color information of imagery elements. Default color channels in an image depend on their color space. The CMYK image has at least four channels, each representing the color layers cyan, magenta, yellow, and black. The RGB image has three channels, representing the color layers of red, green, and blue.

*Gray image*: This type of image can exhibit a rich range of tone. It uses up to 256 shades of gray. Each pixel has a grayscale image luminance value from 0 (black) to 255 (white). Black and white or grayscale scanning device produces images that often appear in this gray image. If a high-quality color image is converted into a black and white image, Photoshop will display all the color information in the original image. When a gray image is converted into an RGB image, the color values of pixels will be lower than the previous gray image values. A gray image can be converted to a Lab or a CMYK color image.

#### **3.7. Two-dimensional interpolation method**

Colorful images cannot scale without interpolation methods, and commonly used color interpolation methods are bilinear interpolation, cubic convolution, and B-spline interpolation.

#### *3.7.1. Bilinear interpolation*

**Figure 3.** 3D Printing color test strips.

34 New Trends in 3D Printing

**3.5. Color stability and durability**

for a certain time.

Of course, the color test strips can also be of 3D print themselves, which is more suited to some particular measurement requirement, for example, to observe and monitor quality of color reproduction on flat structure, or curve or groove surface. Color test strips can be considered as a kind of color authenticity visual aids to help users predict the printing process. They can also be considered as a direct tool like 3D color swatch book for user to select the desired color.

3D Printing applications are being used in new areas, people discuss about a wide range of common properties, such as durability and stability of color 3D Printing objects. These issues are of more interest to art and design people, who use rapid prototyping technology to aid their work, because they publicly display their works of art, and geospatial information systems, architecture, space planning, scenic modeling, and urban design need excellent color effects. Color is a core part of printed objects, and people expect the original effects to remain

Duration and stability depend on several factors, such as light, humidity, temperature, and the amount of air in contact. Graphics, printing industry, and color science and other industries have been developed and used a lot of color standards and standard printing methods developed to study and predict the color durability and stability. This is more significant in inkjet printing in commercial and art fields. Whether 3D Printing is also suitable for the abovementioned research methodology, and the materials, air exposure and other factors can influence the finished colors aging mechanism are real issues focused by current researches. Currently, the durability and stability study of 3D printed color focus on how to find the right way to simulate aging or change real 3D prints, which are exposing in different environment conditions. For example, different lighting, temperature, humidity, the finished color of 3D Printing, and the behavior of color change with different impregnating agent can be simulated and predicted. Custom color strips should be developed in test researches, as well as corre‐

sponding colors and test procedures, so that people can conduct relevant research.

As shown in **Figure 4(a)** and **(b)**, at R and B positions, the color component of G is to be interpolated; this approach uses current G values of vertically and horizontally adjacent four pixels; these four values are added and an average is taken, which is set as G interpolation value of this position. Position B uses the average of adjacent four G values. As shown in **Figure 4(c)** and **(d)**, if the G value is to be inserted inside the R and B color components, and this G position is vertically or horizontally adjacent to R and B, then the sum of vertical or horizontal two values of the same color can be used and an average is taken. This is called a color component interpolation method.

#### **Figure 4.** Bilinear interpolation.

## *3.7.2. Cubic convolution*

A cubic convolution interpolation method is one of the most commonly used functions within the grid data interpolation method. It uses the pixel to be estimated from the nearest 16-pixel value, and the result is calculated according to the formula of cubic convolution. As shown in **Figure 5**, there are many cubic convolution formulas; some produce low-pass filtering effects and some produce a high-pass filter effect. A better approach would be to achieve a balance in the tradeoffs between high and low frequency information. Meanwhile, the bandwidth of cubic convolution is wider than bilinear interpolation, and in the image of high frequency, cubic convolution shows better interpolation results than bilinear interpolation.

**Figure 5.** Cubic convolution interpolation.

The principle of cubic convolution interpolation is based on the interpolation theory, which shows that the cubic convolution of two-dimensional image with generally equally spaced sampling can be expressed as

$$s(\mathbf{x}, \mathbf{y}) = \sum\_{l} \sum\_{m} c(\mathbf{x}\_l, \mathbf{y}\_m) \times h(|\mathbf{x} - \mathbf{x}\_l|) \times h(|\mathbf{y} - \mathbf{y}\_m|) \tag{1}$$

where *h* is referred to as the interpolation weight function, *s*(*x*, *y*) is the results of unknown interpolation points, and sample *c*(*xl*, *ym*) represents the pixel (*xl*, *ym*). Obviously, the selection of interpolation function *h* is the key of any convolution interpolation method. The weight function is a cubic interval function, and the nature of its characteristics can be assessed by the range of its frequency sections. An ideal weight function has unity gain in the band-pass range, has a zero gain in the band-stop range, and thus the signal components with different fre‐ quencies can be effectively suppressed.

#### *3.7.3. B-spline interpolation*

horizontal two values of the same color can be used and an average is taken. This is called a

A cubic convolution interpolation method is one of the most commonly used functions within the grid data interpolation method. It uses the pixel to be estimated from the nearest 16-pixel value, and the result is calculated according to the formula of cubic convolution. As shown in **Figure 5**, there are many cubic convolution formulas; some produce low-pass filtering effects and some produce a high-pass filter effect. A better approach would be to achieve a balance in the tradeoffs between high and low frequency information. Meanwhile, the bandwidth of cubic convolution is wider than bilinear interpolation, and in the image of high frequency,

The principle of cubic convolution interpolation is based on the interpolation theory, which shows that the cubic convolution of two-dimensional image with generally equally spaced

cubic convolution shows better interpolation results than bilinear interpolation.

color component interpolation method.

**Figure 4.** Bilinear interpolation.

36 New Trends in 3D Printing

*3.7.2. Cubic convolution*

**Figure 5.** Cubic convolution interpolation.

sampling can be expressed as

Commonly known B-spline interpolation methods use interpolation function with discrete data to generate continuous function, and then the function can resample and thereby generate new interpolation values. A continuous function can be generated with the following formula:

$$f(\mathbf{x}) = \sum\_{k \in s} f\_k \boldsymbol{\uprho}\_{\text{int}}(\mathbf{x} - k) \qquad \forall\_x \in R \tag{2}$$

In the above formula, *fk* is the known discrete data sequence, and *ϕ* int(*x*) is the interpolation function. In order to ensure the accuracy of interpolation, usually it requires *ϕ* int(*x*) values of 1 at the origin of coordinates, and in other integer coordinate points, its value is 0, for example, one of the ideal interpolation function is sin(*x*).

#### **3.8. Three-dimensional real volume rendering**

A solid color has two types of applications: one is coloring the surface of 3D solid objects. However, sometimes in order to meet industry needs, products internal parts can have color too. For example, if each part in a car engine has its own color code, it will provide the product management, assembly workers, and maintenance workers great convenience. The other type is coloring the entire inner side of an object or add color to a three-dimensional entity internally which is a commonly used method of volume rendering.

The goal of drawing color information in two-dimensional surface graphics is to add color on the face, same as the concept of pixels in plane. The number of pixels per square inch defines the concept of resolution. Corresponding to surface graphics, there is volume graphics and its models. In the application of three-dimensional grid to directly draw a three-dimensional scene, each grid in space corresponds to a value of particular property, which can be measured.

Those three-dimensional grids are called voxel. Voxel can be seen as a two-dimensional pixel in three-dimensional space. It can be understood that an entity is composed of an amount of small cubes or other three-dimensional primaries. An entity may be a function of an image but composed by voxels. In real three-dimensional scene visualization, the information of a threedimensional coordinates is corresponding to attribute data set. In this real visualization, the three-dimensional coordinates (*x*, *y*, *z*) are expressed as dependent arguments to space entity for object geometry modeling. In that way, an established model can not only achieve true three-dimensional visualization, but it can also be used in a three-dimensional spatial analysis.

In general, three-dimensional spatial data are continuous, and numerical results or measure‐ ment data are discrete and are the results of consecutive sampling of real three-dimensional scene. A volume-rendering technique directly samples this three-dimensional space into twodimensional image on the screen as accurately as possible to represent the original 3D data. The two-dimensional image on the screen depends on the frame buffer corresponding to the brightness value of each pixel, which is a two-dimensional discrete data set.

Thus, the real volume rendering transforms the three-dimensional spatial discrete data into two-dimensional discrete lattice. If discrete three-dimensional data set can be converted into discrete two-dimensional data matrix, first it must be resampled into three-dimensional data set. Second, the contribution of each voxel in three-dimensional space for final two-dimen‐ sional image data should be considered so that the image should be composted. Therefore, real volume rendering of three-dimensional model is a process of resampling discrete data set and image composting. Resampling in theory should have the following steps:


The three-dimensional convolution calculation is very time-consuming; in the past few years, a number of scholars have proposed several rendering algorithms to be implemented, mostly discrete methods. Most noticeable is proposed by Westover, a parallel, feed-forward volumerendering algorithm (i.e., a unit projection method), and M. Levoy presented volumerendering algorithm with image space order (i.e., a ray casting method).

The conversion brings brightness values from each sampled point into an image plane, that is, to calculate the contribution of brightness values from each sampled point and then a final image can be composed. The idea is first determine the projected order of voxels and then break voxels into several smaller voxels so that these smaller voxels can only be seen from a positive side or a reverse side from the projection direction. Positive and reverse side surfaces are corresponding to projection surface with a projected polygon, so, in the full use of its twodimensional projection coherence, polygon can be interpolated with a cumulative value of light intensity and transparency. The process is shown in **Figure 6**.

**Figure 6.** Convolution calculation process.

composed by voxels. In real three-dimensional scene visualization, the information of a threedimensional coordinates is corresponding to attribute data set. In this real visualization, the three-dimensional coordinates (*x*, *y*, *z*) are expressed as dependent arguments to space entity for object geometry modeling. In that way, an established model can not only achieve true three-dimensional visualization, but it can also be used in a three-dimensional spatial analysis. In general, three-dimensional spatial data are continuous, and numerical results or measure‐ ment data are discrete and are the results of consecutive sampling of real three-dimensional scene. A volume-rendering technique directly samples this three-dimensional space into twodimensional image on the screen as accurately as possible to represent the original 3D data. The two-dimensional image on the screen depends on the frame buffer corresponding to the

Thus, the real volume rendering transforms the three-dimensional spatial discrete data into two-dimensional discrete lattice. If discrete three-dimensional data set can be converted into discrete two-dimensional data matrix, first it must be resampled into three-dimensional data set. Second, the contribution of each voxel in three-dimensional space for final two-dimen‐ sional image data should be considered so that the image should be composted. Therefore, real volume rendering of three-dimensional model is a process of resampling discrete data set

**(1)** Select the appropriate reconstruction function, three-dimensional data set for discrete three-dimensional convolution, and continuous three-dimensional reconstruction of the

**(2)** Geometric transformations of three-dimensional continuous data set should be based on

**(3)** Since sampling point with screen resolution is given, the sampled signal's Nyquist frequency limit can be calculated, and use low-pass filter function to remove frequency

The three-dimensional convolution calculation is very time-consuming; in the past few years, a number of scholars have proposed several rendering algorithms to be implemented, mostly discrete methods. Most noticeable is proposed by Westover, a parallel, feed-forward volumerendering algorithm (i.e., a unit projection method), and M. Levoy presented volume-

The conversion brings brightness values from each sampled point into an image plane, that is, to calculate the contribution of brightness values from each sampled point and then a final image can be composed. The idea is first determine the projected order of voxels and then break voxels into several smaller voxels so that these smaller voxels can only be seen from a positive side or a reverse side from the projection direction. Positive and reverse side surfaces are corresponding to projection surface with a projected polygon, so, in the full use of its twodimensional projection coherence, polygon can be interpolated with a cumulative value of

rendering algorithm with image space order (i.e., a ray casting method).

light intensity and transparency. The process is shown in **Figure 6**.

brightness value of each pixel, which is a two-dimensional discrete data set.

and image composting. Resampling in theory should have the following steps:

data field.

38 New Trends in 3D Printing

a given observation direction.

components above this limit. **(4)** Resample the filtered function. A ray casting method is also known as ray tracing method. It follows the following basic steps: start from each pixel on the screen, according to the current position of the viewpoint, launch a line through the data set, select a range of sampling points along the line, and use the interpolation method to calculate the value of the opacity points and colored sampling points. Then, from the front to the rear or from the rear to the front sequence to composite color based on their corresponding color value and opacity value at each sampling point, the screen color of each pixel is obtained. The process is shown in **Figure 7**.

**Figure 7.** Ray casting/sampling process.

#### *3.8.1. Gradient discrete solid coloring method*

There are two kinds of algorithms in 3D solid drawing; ray casting contains intersection operations of light and solid units. For a larger number of units in finite-element data set, intersection calculation is very time-consuming. Because the unit projection method requires advance knowledge of attributes to determine unit order and unit projection, this unit sorting is also very time-consuming. Because two ways of volume rendering are costly and timeconsuming, associated overhead and space operation usage are unusually large, up to the time complexity *O*(N3), so it is very complicated to achieve the right color gradient on each unit voxel. Therefore, this section proposes a new method of coloring the internal entities—gradient discrete solid coloring method. By this method, the coloring process of three-dimensional solid can be reduced to a two-dimensional coloring process; its flexibility and efficiency have been greatly improved and the most important characteristic is that it can achieve the right color gradient effect.

Internal coloring process of three-dimensional solid can be completed by adding color information to each discrete layer of the solid. Since each discrete layer shows color informa‐ tion, the solid can obtain the color features when layers superimpose together. A gradient discrete solid coloring method is shown in **Figure 8**.

Discrete three-dimensional modeling in rapid prototyping technology uses discrete method to easily and quickly finish coloring. User would like to see as much of the color information by adding color to objects, or can take the initiative to develop new color with more interactive operations, which is desired by the user. The smooth and soft effect based on the human eye make user wants to add gradient colors.

However, as the human eye can only recognize up to about 2000 types of colors, so computer color representation is not strictly continuous and generally uses only eight binary digits to represent R (8 bit), G (8 bit), B (8 bit) of colors, but for the color change, it is often represented by R, G, B three-color superposition:

$$C(R, G, B) = F\_{\mathbb{R}}(R) + F\_G(G) + F\_B(B) \tag{3}$$

where *F* function is used to obtain the channel points for one pixel. While each color is represented by only 8-bit value; however, it fully matches the amount of human eye distin‐ guishable colors.

#### *3.8.2. Color printing forming process*

*3.8.1. Gradient discrete solid coloring method*

discrete solid coloring method is shown in **Figure 8**.

**Figure 8.** Gradient discrete solid coloring process.

make user wants to add gradient colors.

by R, G, B three-color superposition:

gradient effect.

40 New Trends in 3D Printing

There are two kinds of algorithms in 3D solid drawing; ray casting contains intersection operations of light and solid units. For a larger number of units in finite-element data set, intersection calculation is very time-consuming. Because the unit projection method requires advance knowledge of attributes to determine unit order and unit projection, this unit sorting is also very time-consuming. Because two ways of volume rendering are costly and timeconsuming, associated overhead and space operation usage are unusually large, up to the time complexity *O*(N3), so it is very complicated to achieve the right color gradient on each unit voxel. Therefore, this section proposes a new method of coloring the internal entities—gradient discrete solid coloring method. By this method, the coloring process of three-dimensional solid can be reduced to a two-dimensional coloring process; its flexibility and efficiency have been greatly improved and the most important characteristic is that it can achieve the right color

Internal coloring process of three-dimensional solid can be completed by adding color information to each discrete layer of the solid. Since each discrete layer shows color informa‐ tion, the solid can obtain the color features when layers superimpose together. A gradient

Discrete three-dimensional modeling in rapid prototyping technology uses discrete method to easily and quickly finish coloring. User would like to see as much of the color information by adding color to objects, or can take the initiative to develop new color with more interactive operations, which is desired by the user. The smooth and soft effect based on the human eye

However, as the human eye can only recognize up to about 2000 types of colors, so computer color representation is not strictly continuous and generally uses only eight binary digits to represent R (8 bit), G (8 bit), B (8 bit) of colors, but for the color change, it is often represented Three-dimensional rapid prototyping and other 3D Printing all have slicing process. One appropriate direction as the slice direction must be selected, and from one end to the other end of the solid, a series of successive sections S1, S2 … Si … Sn–1, Sn can be given. Note C represents the color information of each slice so that S1(C1), S2(C2), S3(C3), … Si(Ci) … Sn(Cn) is the color information for each slice plane.

According to the differences in range of different colors recognizable by human eye, limited critical points between (*i*, *j*) are inserted so that a good transition from Ci to Cj can be achieved; in fact, in order to adapt the characteristics of 3D Printing, these critical points can be taken on the axis of the direction of slicing and the critical points are in the slice plane; and each slice corresponds to a critical point, and the color of this point is attached to the entire slice layer. For a layer-by-layer approach of 3D Printing and rapid prototyping technology, there are same thickness layer methods, adaptive hierarchical methods, but the interpolation methods are not limited to slicing methods. There are three kinds of color interpolation methods, namely, cosine color gradients method, cosine adjustable color gradient method, and power gradient method.

When a three-dimensional printer selects an appropriate image file format as its original before coloring, in the RGB mode the color image pixel is displayed; after the conversion of the image from the RGB mode to the CMYK mode, each pixel in the RGB mode will be transferred to a corresponding CMYK mode matrix for nozzle to read, where one pixel corresponds to a 3 × 3 or 4 × 4 or an *N* × *N* matrix. Print heads receive C, M, Y, K data, and when the jet is connected with the color binder, during printing, each nozzle receives "0" or "1" for control nozzle spray or without spray.

Machine uses lattice format to represent layer information. At every move, the nozzle will be given a forming region with fixed height and a variable width. The graphics are divided into several smaller areas for printing; inkjet molding operation is actually a loop operation, wherein each loop finishes a certain area of a small area and deals with them accordingly. Image information of each layer at first is loaded into the buffer buf[*x*, *y*], which is an array of small blocks of image information, known as graphics files. It should be noted that the number of points in the array buf is not the number of points required for the actual nozzle, buf only provides pixel values of the image, and it needs to go through a particular image transfer so that it can become a nozzle-forming image corresponding to a pixel matrix. If we use a 3 × 3 lattice, a pixel in buf corresponds to nine points of nozzle point data; in accordance with such corresponding relation, an image forming process is completed for each small area.

After completion of one layer of material, the printing nozzles moves back to the origin, and platform falls to a certain height in the Z axle, such as 0.5 mm. The printing nozzles restart to form the next layer. This is called one layer forming cycle, and after certain cycles, the color solid is finally completed.

#### *3.8.3. Coloring module software development*

Three-dimensional color printer with solid forming has one key technology, that is, the development of coloring software modules. The software can process STL file data, form a solid layer contour, and add an internal solid modeling tool to add gradient color information. According to the requirements of colored objects, a coloring module should have two func‐ tions:


Using UGNX software or CAD drawing software to make three-dimensional solid modeling, mapping is performed through the menu options after forming "\* .stl" file. With common 3D modeling software, in the "File" menu there are "Export" command, and it can export a "\* .stl" file, and then manually change it into ".txt" type files. To understand the file format, use notepad to open "\* .txt", and then we can see data of small triangular.

Quick slicing algorithms create a data structure of triangular with vertices and edges. After the data structure is created, software will reorder all distinct vertexes, and then topology data are completely setup. Finally, the above-mentioned interpolation functions can be used to show the results.
