WO2017188999A2  Density rank matrix normalization for threedimensional printing  Google Patents
Density rank matrix normalization for threedimensional printing Download PDFInfo
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 WO2017188999A2 WO2017188999A2 PCT/US2016/030166 US2016030166W WO2017188999A2 WO 2017188999 A2 WO2017188999 A2 WO 2017188999A2 US 2016030166 W US2016030166 W US 2016030166W WO 2017188999 A2 WO2017188999 A2 WO 2017188999A2
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 rank
 density
 threshold value
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 239000011159 matrix materials Substances 0 title claims abstract description 132
 238000010146 3D printing Methods 0 title claims abstract description 19
 230000000875 corresponding Effects 0 claims abstract description 56
 238000000034 methods Methods 0 description 12
 239000000463 materials Substances 0 description 10
 238000004519 manufacturing process Methods 0 description 6
 230000001276 controlling effects Effects 0 description 1
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Classifications

 B—PERFORMING OPERATIONS; TRANSPORTING
 B29—WORKING OF PLASTICS; WORKING OF SUBSTANCES IN A PLASTIC STATE IN GENERAL
 B29C—SHAPING OR JOINING OF PLASTICS; SHAPING OF MATERIAL IN A PLASTIC STATE, NOT OTHERWISE PROVIDED FOR; AFTERTREATMENT OF THE SHAPED PRODUCTS, e.g. REPAIRING
 B29C64/00—Additive manufacturing, i.e. manufacturing of threedimensional [3D] objects by additive deposition, additive agglomeration or additive layering, e.g. by 3D printing, stereolithography or selective laser sintering
 B29C64/30—Auxiliary operations or equipment
 B29C64/386—Data acquisition or data processing for additive manufacturing
 B29C64/393—Data acquisition or data processing for additive manufacturing for controlling or regulating additive manufacturing processes

 B—PERFORMING OPERATIONS; TRANSPORTING
 B33—ADDITIVE MANUFACTURING TECHNOLOGY
 B33Y—ADDITIVE MANUFACTURING, i.e. MANUFACTURING OF THREEDIMENSIONAL [3D] OBJECTS BY ADDITIVE DEPOSITION, ADDITIVE AGGLOMERATION OR ADDITIVE LAYERING, e.g. BY 3D PRINTING, STEREOLITHOGRAPHY OR SELECTIVE LASER SINTERING
 B33Y10/00—Processes of additive manufacturing

 B—PERFORMING OPERATIONS; TRANSPORTING
 B33—ADDITIVE MANUFACTURING TECHNOLOGY
 B33Y—ADDITIVE MANUFACTURING, i.e. MANUFACTURING OF THREEDIMENSIONAL [3D] OBJECTS BY ADDITIVE DEPOSITION, ADDITIVE AGGLOMERATION OR ADDITIVE LAYERING, e.g. BY 3D PRINTING, STEREOLITHOGRAPHY OR SELECTIVE LASER SINTERING
 B33Y50/00—Data acquisition or data processing for additive manufacturing
 B33Y50/02—Data acquisition or data processing for additive manufacturing for controlling or regulating additive manufacturing processes
Abstract
Description
DENSITY RANK MATRIX NORMALIZATION FOR THREEDIMENSIONAL
PRINTING
BACKGROUND
[0001] In threedimensional (3D) printing, one technique to create an object is with successive layers of material. The successive layers of material may be applied under computer control. The object may be formed from a variety of sources. For example, the object may be formed from a 3D model. The object may be formed of virtually any geometry. Further, the object may be formed of any material, including ceramics, metals, polymers, and composites.
BRIEF DESCRIPTION OF DRAWINGS
[0002] Features of the present disclosure are illustrated by way of example and not limited in the following figure(s), in which like numerals indicate like elements, in which:
[0003] Figure 1 illustrates an architecture of an apparatus for density rank matrix normalization for threedimensional printing, according to an example of the present disclosure;
[0004] Figure 2 illustrates an environment including the apparatus 100, according to an example of the present disclosure.
[0005] Figure 3 illustrates a high level overview of density rank matrix generation, normalization, and rendering for the apparatus of Figure 1 , according to an example of the present disclosure;
[0006] Figure 4 illustrates details of a density threshold matrix generator of the apparatus of Figure 1 , according to an example of the present disclosure;
[0007] Figure 5 illustrates mapping of rank to threshold for an identity transformation, according to an example of the present disclosure;
[0008] Figure 6 illustrates mapping of rank to threshold for a nonuniform transformation, according to an example of the present disclosure;
[0009] Figure 7 illustrates manytoone mapping of rank to threshold for an example transformation, according to an example of the present disclosure;
[0010] Figure 8A illustrates a tetrahedraloctahedral lattice, and Figures 8B8D illustrate 2x2x2 period output for different low density values, according to an example of the present disclosure;
[0011] Figures 9A9B illustrate 2x2x2 period output for different mid to high density values, according to an example of the present disclosure;
[0012] Figure 10 illustrates density regions that are to be addressed, according to an example of the present disclosure; [0013] Figure 11 illustrates clipping a rankthreshold transformation, according to an example of the present disclosure;
[0014] Figure 12 illustrates stretching the rankthreshold transformation, according to an example of the present disclosure;
[0015] Figure 13 illustrates a general rankthreshold transformation, according to an example of the present disclosure;
[0016] Figure 14 illustrates an example rankthreshold transformation, according to an example of the present disclosure;
[0017] Figure 15 illustrates a radial cross section of a density map of a sphere, according to an example of the present disclosure;
[0018] Figure 16 illustrates one octant of an output object of a variable density sphere input, according to an example of the present disclosure;
[0019] Figure 17 illustrates a flowchart of a method for density rank matrix normalization for threedimensional printing, according to an example of the present disclosure;
[0020] Figure 18 illustrates another flowchart of a method for density rank matrix normalization for threedimensional printing, according to an example of the present disclosure;
[0021] Figure 19 illustrates a further flowchart of a method for density rank matrix normalization for threedimensional printing, according to an example of the present disclosure; and
[0022] Figure 20 illustrates a computer system, according to an example of the present disclosure. DETAILED DESCRIPTION
[0023] For simplicity and illustrative purposes, the present disclosure is described by referring mainly to examples. In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present disclosure. It will be readily apparent however, that the present disclosure may be practiced without limitation to these specific details. In other instances, some methods and structures have not been described in detail so as not to unnecessarily obscure the present disclosure.
[0024] Throughout the present disclosure, the terms "a" and "an" are intended to denote at least one of a particular element. As used herein, the term "includes" means includes but not limited to, and the term "including" means including but not limited to. The term "based on" means based at least in part on.
[0025] With respect to the manufacture of 3D objects, reducing an overall amount of material used for the manufacture of a 3D object may be desirable from a weight reduction perspective. Reducing fused material in a core of an otherwise solid object may also be desirable from a thermal management perspective.
However, reducing the amount of overall material used and/or reducing fused material in a core of an otherwise solid object may reduce the object's structural strength and/or the object's ability to accommodate stresses.
[0026] In order to address the aforementioned technical challenges with respect to the manufacture of 3D objects to maintain structural strength and stress accommodation properties, an apparatus for density rank matrix normalization for 3D printing and a method for density rank matrix normalization for 3D printing are disclosed herein.
[0027] For the apparatus and method disclosed herein, a density rank matrix may be received and normalized to generate a density threshold matrix. The density threshold matrix may be analyzed with respect to details of a continuous density 3D input object to generate a bistate output object. Accordingly, the density threshold matrix may be used in a 3D object manufacturing process, where the 3D object that is manufactured includes variable density, and includes a specified structural strength, specified stress accommodation properties, and other such properties.
[0028] The attribute of density may be described as solidness or its inverse sparsity. A density of a 3D object may range from 0%, which may be referred to as empty space, to 100%, which may be referred to as solid.
[0029] The density rank matrix may be described as a periodic 3D matrix of size X by Y by Z (i.e., (X, Y, Z)), where each element is an integer indicating the order that the element will turn on a voxel with increasing density. The density rank matrix may include rank values that range from 1 to n, where n represents a total number of elements of the density rank matrix. For example, for a X by Y by Z density rank matrix, the total number of elements n may include X times Y times Z elements. The density rank matrix may be referred to herein as a rectangular periodic matrix, where the density rank matrix provides a rectangular period that tiles all of threedimensional space. The density rank matrix may provide for the rendering of a variable density 3D object. The density rank matrix may be applied to objects for which the interior needs less material or weight, while still maintaining the object strength.
[0030] The density rank matrix may be generated from a variety of matrix ordering techniques. Examples of techniques that are used to generate the density rank matrix include blue noise structures, contiguous lattices, etc. For blue noise structures, voxels may be uniformly and homogeneously distributed in space. For contiguous lattices, elements of the density rank matrix may be generated form a linebased skeleton lattice using a line dilation technique. For the line dilation technique, a minimum distance to a skeleton line to each point may be assigned. Each of the points may be rank ordered in the density rank matrix.
[0031] The density threshold matrix may be described as a matrix for which the rank values in the density rank matrix are normalized for conversion to threshold values. The density threshold matrix may be the same size as the density rank matrix (i.e., (X, Y, Z)).
[0032] The 3D input object may be described as any object that is to be printed and includes a threedimensional shape. The 3D input object may include discrete space values that describe the density at each point of the object, from empty space, represented by zero, to solid, represented by a maximum value.
[0033] The bistate output object may be described as an object that includes discretespace where each point is represented by one of two states: zero for empty space, or one for printavoxel.
[0034] In order to generate the bistate output object, a rendering system may be used. An example of a rendering system may include a 3D printer, and other such devices. For the rendering system, each element of the 3D input object, input(x, y, z), may be compared against a corresponding threshold value in the density threshold matrix, threshold (x, y, z), to print the bistate output object.
[0035] In order to generate the density threshold matrix, order, or rank of matrix positions may be determined such that the resulting 3D printed object includes a printed voxel as opposed to empty space as the input density increases. In this regard, a density rank matrix that is used to generate the density threshold matrix may be relatively large. For example, a cubic inch period structure may include R = 1200x1200x254 = 365,760,000 rank elements. As the values of such a density rank matrix are the ranks 0, 1 , ... , (R1 ), four bytes are needed to store the elements, which needs approximately 1 .46 terabytes of storage. If the input continuous density object has an 8bit amplitude resolution, a density threshold matrix would need to be scaled to match the input. If the density rank matrix is merely scaled to the input resolution, matrix storage is reduced but subsequent nonlinear transformations would result in the loss of output amplitude resolution due to coarse quantization.
[0036] In order to address the aforementioned technical challenges with respect to the manufacture of 3D objects to maintain structural strength and stress accommodation properties, and with respect to quantization loss, given a density rank matrix, the apparatus and method disclosed herein may generate a density threshold matrix that is customized for a particular structure and target printer. The density threshold matrix may provide a variable density structure definition. The density threshold matrix may also provide for implementation of policies for controlling minimum and maximum structure densities, and thus sizes, in a single matrix.
[0037] Figure 1 illustrates an architecture of an apparatus for density rank matrix normalization for threedimensional printing (hereinafter also referred to as "apparatus 100"), according to an example of the present disclosure. Figure 2 illustrates an environment 200 for the apparatus 100, according to an example of the present disclosure.
[0038] Referring to Figure 1 , the apparatus 100 is depicted as including a density rank matrix normalization module 102 to receive a density rank matrix 104, normalization specification 106, and information with respect to a 3D input object 108 to generate a density threshold matrix 110. The operations disclosed herein with respect to the density rank matrix normalization module 102 may be performed by a processor (e.g., the processor 2002 of Figure 20). The density rank matrix 104, the normalization specification 106, and the information with respect to the 3D input object 108 may be stored in and retrieved from a storage 120.
[0039] A rendering module 112 may compare each element of the 3D input object 108, input(x, y, z), against a corresponding threshold value in the density threshold matrix 110, threshold(x, y, z), at each location as follows: if lnput(x, y, z) > Threshold (χ', y', z') then Output(x, y, z) = 1 (i.e., printer voxel) else Output(x, y, z) = 0 (i.e., empty space) where x' = x mod X; y' = y mod Y; and z' = z mod Z. The "mod" may represent the modulo operation. The operations disclosed herein with respect to the rendering module 112 may be performed by a processor (e.g., the processor 2002 of Figure 20).
[0040] In some examples, the modules and other elements of the apparatus 100 may be machine readable instructions stored on a nontransitory computer readable medium. In this regard, the apparatus 100 may include or be a non transitory computer readable medium. In some examples, the modules and other elements of the apparatus 100 may be hardware or a combination of machine readable instructions and hardware.
[0041] Referring to Figure 2, the output of the rendering module 116 (i.e., Output(x, y, z) = 1 (i.e., printer voxel), or Output(x, y, z) = 0 (i.e., empty space)) may be used by a rendering system 202 to generate a bistate output object 204.
[0042] Determination of the density threshold matrix 110 is described in further detail with reference to Figures 1 , and 316.
[0043] In order to determine the density threshold matrix 110, referring to Figure 4, the map generator 400 may output a function H(r) that converts, as shown at block 402, a rank value of the density rank matrix 104 to a threshold value. In this regard, a normalization specification 106 may include an indication to convert, as shown at block 402, a rank value of the density rank matrix 104 to a threshold value according to the function H(r). The normalization specification 106 may further include minimum and maximum structure sizes as specified by minimum and maximum nonzero thresholds as disclosed herein. For conversion of a rank value of the density rank matrix 104 to a threshold value according to the function H(r), for H(r), r may represent a rank value of the density rank matrix 104. The function H(r) may be used to map all of the rank values in the density rank matrix 104 to values in the density threshold matrix 110 for all (x, y, z) addresses from (0, 0, 0) to (X1 , Y1 , Z1 ). [0044] Quantization may be used to convert a relatively large number of rank values to the smaller number of thresholds. For example, a cubic inch density rank matrix 104 may include 1200 x 1200 x 254 = 365,760,000 ranks, but the
continuous density 3D input object 108 may include T = 256 levels, for a more than a million to one mapping. For example, consider pretruncating the ranks to 3 bits, or T = 8 levels. In this case, all 1200 x 1200 x 254 matrix elements of the density rank matrix 104 would each be one of 8 values.
[0045] Referring to Figure 5 which illustrates mapping of rank to threshold for an identity transformation, such a pretruncating would be acceptable if the mapping includes the identity function as shown in Figure 5. For the identity function, the rank and threshold may identically correspond to each other.
[0046] However, referring to Figure 6 which illustrates mapping of rank to threshold for a nonuniform transformation, a nonuniform gain mapping results in a loss of amplitude resolution, where the curve at 600 represents the desired mapping. The curve at 600 may be used to account for nonlinear gain in structure size. In this case, approximately 25% of the possible output levels may be lost because of the prequantization.
[0047] Another option for mapping of rank to threshold includes retaining the full range of rank values, and segmenting ranges of ranks for conversion to one of the eight output threshold values according to the desired mapping function. As shown in Figure 7 which illustrates manytoone mapping of rank to threshold for an example transformation, the points at 700 may represent the values to which ranges of ranks are mapped. The use of the manytoone mapping may preserve the intended function to a greater degree, without loss of output levels.
[0048] In addition to the accommodating of a gainrelated curve, the map generator 400 may also address low and high ends of the density range. By way of example, if a density rank matrix that defines a tetrahedraloctahedral lattice as shown in Figure 8A is used with a linear normalization, low density samples are displayed in Figures 8B8D for 0.5%, 5%, and 10% density, respectively. As illustrated in Figure 8A, the tetrahedraloctahedral lattice may be described as a lattice that includes tetrahedral shapes in the interior thereof. Depending on the material used and the size of the lattice period, there may be some density below which a printed object will be too fragile. In this case, it may be desirable to force input values below some minimum density to be mapped to empty space (i.e., no voxels), or some predetermined higher density.
[0049] At the high end of the density range, certain adjustments may also be needed. In Figures 9A and 9B, the tetrahedraloctahedral lattice is shown rendered at 50% and 99%, respectively. In the case of the 99% rendering for Figure 9B, the structure predominately fills all space with the exception of some residual gaps that are illustrated in Figure 9B. Somewhere between 50% and 99%, such gaps become closed, thus prohibiting the removal of unused material used in the manufacturing process. In this regard, it may be desirable to set input values above a maximum density to be mapped to 100% solid, or to a predetermined lower density.
[0050] Figure 10 illustrates a graph of the rank to threshold mapping with these low and high end of the ranges circled indicating the potential density areas that are to be addressed. The low end of the range may be designated as a region of fragile structure at 1000, and the high end of the range may be designated as a region of enclosed holes at 1102. The minimum nonzero threshold allowed may be designated as ti, and the maximum nonsolid threshold allowed may be designated as t∑.
[0051] In order to enforce these minimum and maximum nonsolid threshold limits in the mapping, a first technique may be described as a clipping technique. Referring to Figure 11 which illustrates clipping the rankthreshold transformation, all ranks, r, which would have mapped to values below ti are set be 0, or empty space as illustrated at 1100. Further, all ranks which would have mapped to values above f_{2} are set to the maximum threshold, T1, as illustrated at 1102, which corresponds to a solid output. [0052] In order to enforce these minimum and maximum nonsolid threshold limits in the mapping, a second technique may be described as a stretching technique. Referring to Figure 12 which illustrates stretching the rankthreshold transformation, the original curve is stretched as indicated. The "empty space" rank r = 0 is mapped to t = 0 as illustrated at 1200, and the solid rank r = (R1) is mapped to the maximum threshold T1 as illustrated at 1202. All other rank values are mapped between and f_{2}
[0053] Figure 13 illustrates a general rankthreshold transformation, according to an example of the present disclosure. Referring to Figure 13, along with the threshold limits and f_{2}, the length of rank clipping regions may be specified as r_{?} and r_{2} as illustrated, respectively at 1300 and 1302. The rankthreshold mapping may be specified as follows:
If a function g(r) is defined for the entire range from (0,0) to ((R1 ), (T1 )), then the compressed version may be specified as follows: f(r) = (t_{2}  ) g( (r _{ri}) / (r_{2}  n) ) +
[0054] Figure 14 illustrates an example rankthreshold transformation, according to an example of the present disclosure. Referring to Figure 14, for the mapping, g(r) = (r/(R1))^{r}, where γ is an exponent that is used for describing tone scale transformation, for example, γ 1.7. This mapping is used to normalize the density rank matrix 104 for a tetrahedraloctahedral lattice to generate the density threshold matrix 110. As illustrated in Figure 15, the 3D input object 108 may include a sphere that uniformly increases in density from 0 at the center at location 1500, to solid at the surface with a density value of 1 at location 1502. A radial cross section of part of the input object is depicted in Figure 15, with bands representing changes in density. The mapping forces the minimum density to be 10%, and avoids structures with closed holes. The resulting rendering is illustrated for one octant of the sphere in Figure 16.
[0055] The nature of the rankthreshold mapping may depend on the nature of the density rank matrix structure. For example, the density rank matrix structure may be based on the shape that is to be produced by the density rank matrix. For example, for a tetrahedraloctahedral lattice shape that is to be produced, as disclosed herein, a specified density may yield an output structure that is too thin. Alternatively, for avoiding unfused powder, the density may need to be specified to avoid enclosed voids. In this regard, the apparatus 100 may be tailored to any periodic structure, and to the characteristics of any 3D printer.
[0056] Figures 1719 respectively illustrate flowcharts of methods 1700, 1800, and 1900 for density rank matrix normalization for threedimensional printing, corresponding to the example of the apparatus 100 whose construction is described in detail above. The methods 1700, 1800, and 1900 may be
implemented on the apparatus 100 with reference to Figures 1 16 by way of example and not limitation. The methods 1700, 1800, and 1900 may be practiced in other apparatus. The steps disclosed herein with respect to the methods 1700, 1800, and 1900 may be performed by a processor (e.g., the processor 2002 of Figure 20).
[0057] Referring to Figures 1 17, and particularly Figure 17, for the method 1700, at block 1702, the method may include receiving the density rank matrix 104. The density rank matrix 104 may include elements that include rank values. The rank values may specify an order by which a voxel is to be turned on to generate the output object 204.
[0058] At block 1704, the method 1700 may include receiving the normalization specification 106 of maximum and minimum structure sizes for structures that are to form the output object 204. For example, the normalization specification 106 may further include minimum and maximum structure sizes as specified by minimum and maximum nonzero thresholds ti and t_{2}, respectively, as disclosed herein. [0059] At block 1706, the method 1700 may include converting, according to the normalization specification 106, each of the rank values to a corresponding threshold value to generate the density threshold matrix 110. For example, as disclosed herein with reference to Figure 13, the function H(r) may be used to convert, according to the normalization specification 106, each of the rank values to a corresponding threshold value to generate the density threshold matrix 110.
[0060] For the method 1700, converting, according to the normalization specification 106, each of the rank values to the corresponding threshold value to generate the density threshold matrix 110 may further include specifying higher and lower rank clipping values (e.g., r_{2} and r_{?}, respectively, as disclosed herein with respect to Figure 13). In response to a determination that a rank value of the rank values is less than or equal to the lower rank clipping value, the method 1700 may include specifying the corresponding threshold value for the density threshold matrix 110 as zero (e.g., H(r) for r < ri). In response to a determination that the rank value is greater than or equal to the higher rank clipping value, the method 1700 may include specifying the corresponding threshold value for the density threshold matrix 110 as a maximum threshold value (e.g., H(r) for r > r_{2}). Further, in response to a determination that the rank value is greater than the lower rank clipping value and less than the higher clipping value, the method 1700 may include specifying the corresponding threshold value according to a function (e.g., H(r) = f(r) for r? < r < r_{2}).
[0061] For the method 1700, specifying the corresponding threshold value according to the function may further include specifying the corresponding threshold value according to the function of the higher and lower rank clipping values, and higher and lower threshold value limits that correspond to the maximum and minimum structure sizes, respectively.
[0062] For the method 1700, the function may include a nonlinear function (e.g., H(r) = f(r) for n < r < r_{2}).
[0063] For the method 1700, the density rank matrix 104 may define a tetrahedraloctahedral lattice (e.g., Figures 8A9B).
[0064] The method 1700 may further include receiving specifications of the threedimensional input object 108, and comparing each of the specifications of the threedimensional input object 108 to the corresponding threshold value in the density threshold matrix 110 to determine whether to turn on the voxel. For example, the rendering module 112 may compare each element of the 3D input object 108, input(x, y, z), against a corresponding threshold value in the density threshold matrix 110, threshold(x, y, z), at each location, to determine whether to turn on the voxel.
[0065] For the method 1700, comparing each of the specifications of the three dimensional input object 108 to the corresponding threshold value in the density threshold matrix 110 to determine whether to turn on the voxel may further include applying a modulo operation to each threshold value in the density threshold matrix 110. Further, a determination may be made as to whether a specification of the threedimensional input object 108 is greater than a corresponding modulo based threshold value. The corresponding modulo based threshold value may represent a threshold value to which the modulo operation is applied. In response to a determination that the specification of the threedimensional input object 108 is greater than the corresponding modulo based threshold value, the method 1700 may include indicating that the voxel is to be turned on. In response to a
determination that the specification of the threedimensional input object 108 is less than or equal to the corresponding modulo based threshold value, the method 1700 may include indicating that the voxel is not to be turned on. For example, if lnput(x, y, z) > Threshold (χ', y', z'), then Output(x, y, z) = 1 (i.e., printer voxel), else Output(x, y, z) = 0 (i.e., empty space), where x' = x mod X; y' = y mod Y; and z' = z mod Z.
[0066] Referring to Figures 1 16 and 18, for the method 1800, at block 1802, the method may include receiving the density rank matrix 104. The density rank matrix 104 may include elements that include rank values. The rank values may specify an order by which a voxel is to be turned on to generate an output object.
[0067] At block 1804, the method 1800 may include receiving the normalization specification 106 of maximum and minimum structure sizes for structures that are to form the output object. For example, the normalization specification 106 may further include minimum and maximum structure sizes as specified by minimum and maximum nonzero thresholds ti and t_{2}, respectively, as disclosed herein.
[0068] At block 1806, the method 1800 may include converting, according to the normalization specification, each of the rank values to a corresponding threshold value to generate the density threshold matrix 110. For example, as disclosed herein with reference to Figure 13, the function H(r) may be used to convert, according to the normalization specification 106, each of the rank values to a corresponding threshold value to generate the density threshold matrix 110.
[0069] At block 1808, the method 1800 may include receiving specifications of a threedimensional input object 108.
[0070] At block 1810, the method 1800 may include comparing each of the specifications of the threedimensional input object 108 to the corresponding threshold value in the density threshold matrix 110 to determine whether to turn on the voxel. For example, the rendering module 112 may compare each element of the 3D input object 108, input(x, y, z), against a corresponding threshold value in the density threshold matrix 110, threshold(x, y, z), at each location, to determine whether to turn on the voxel.
[0071] Referring to Figures 1 16 and 19, for the method 1900, at block 1902, the method may include receiving the density rank matrix 104. The density rank matrix 104 may include elements that include rank values. The rank values may specify an order by which a voxel is to be turned on to generate the output object 204.
[0072] At block 1904, the method 1900 may include receiving the normalization specification 106 of maximum and minimum structure sizes for structures that are to form the output object 204. For example, the normalization specification 106 may further include minimum and maximum structure sizes as specified by minimum and maximum nonzero thresholds ti and t_{2}, respectively, as disclosed herein.
[0073] At block 1906, the method 1900 may include converting, according to the normalization specification, each of the rank values to a corresponding threshold value to generate the density threshold matrix 110 by specifying higher and lower rank clipping values (e.g., r_{2} and r_{?}, respectively, as disclosed herein with respect to Figure 13). Further, in response to a determination that a rank value of the rank values is greater than the lower rank clipping value and less than the higher clipping value, the method 1900 may include specifying the corresponding threshold value according to a function (e.g., H(r) = f(r) for r_{?} < r < r_{2}).
[0074] Figure 20 shows a computer system 2000 that may be used with the examples described herein. The computer system 2000 may represent an operational platform that includes components that may be in a server or another computer system. The computer system 2000 may be used as a platform for the apparatus 100. The computer system 2000 may execute, by a processor (e.g., a single or multiple processors) or other hardware processing circuit, the methods, functions and other processes described herein. These methods, functions and other processes may be embodied as machine readable instructions stored on a computer readable medium, which may be nontransitory, such as hardware storage devices (e.g., RAM (random access memory), ROM (read only memory), EPROM (erasable, programmable ROM), EEPROM (electrically erasable, programmable ROM), hard drives, and flash memory).
[0075] The computer system 2000 may include the processor 2002 that may implement or execute machine readable instructions performing some or all of the methods, functions and other processes described herein. Commands and data from the processor 2002 may be communicated over a communication bus 2004. The computer system may also include the main memory 2006, such as a random access memory (RAM), where the machine readable instructions and data for the processor 2002 may reside during runtime, and a secondary data storage 2008, which may be nonvolatile and stores machine readable instructions and data. The memory and data storage are examples of computer readable mediums. The memory 2006 may include a density rank matrix normalization for 3D printing module 2020 including machine readable instructions residing in the memory 2006 during runtime and executed by the processor 2002. The density rank matrix normalization for 3D printing module 2020 may include the modules of the apparatus 100 shown in Figure 1 .
[0076] The computer system 2000 may include an I/O device 2010, such as a keyboard, a mouse, a display, etc. The computer system may include a network interface 2012 for connecting to a network. Other known electronic components may be added or substituted in the computer system.
[0077] What has been described and illustrated herein is an example along with some of its variations. The terms, descriptions and figures used herein are set forth by way of illustration and are not meant as limitations. Many variations are possible within the spirit and scope of the subject matter, which is intended to be defined by the following claims  and their equivalents  in which all terms are meant in their broadest reasonable sense unless otherwise indicated.
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