## Abstract

This paper presents a novel application of intensity-based volume registration to manufacturing using voxel-based computer-aided manufacturing (CAM) models. The introduced techniques are presented in the context of machining irregularly shaped materials by integrating volumetric imaging feedback to computer numerical control (CNC) machine tools. This requires a comparison and alignment to be performed in the software to geometrically “fit” the source design model inside a rendered starting material. The requirements differ from many typical registration applications in that the workpiece will necessarily be larger (i.e., greater in volume) than the desired final computer-aided design (CAD) file. Therefore, models need to be aligned for toolpath generation to workpiece counterparts that have been either volumetrically offset or contain additional material/volume. Intensity-based registrations are unique in that they consider only the voxel values over the entire volume. Although advancements in medical imaging have produced efficient, robust voxel registration algorithms, these techniques have not yet been applied to manufacturing. This research introduces the use of maximization of mutual information (MMI) for voxel-based CAM to drive an alignment registration for systems integrating imaging technology. A simple but novel method, which the authors have named minimization of distance variance (MDV), is also introduced. This minimizes the variance between voxel intensities to demonstrate the design of a similarity metric for a simple case in machining rough castings.

## 1 Introduction

### 1.1 Problem Context.

The machining of parts from irregularly shaped starting stock presents a challenge to manufacturing engineers. The typical *computer-aided manufacturing* (CAM) software requires that the *computer numerical control* (CNC) programmer defines a simple cylinder or block stock of known dimensions as a starting material. This is completely adequate for the majority of manufacturing operations since extruded materials can be ordered within an acceptable tolerance. However, a number of situations exist where the starting volume cannot be easily defined or is unknown. These cases require that a part be machined from an irregularly shaped or *near-net-shape* (NNS) starting workpiece.

Research introducing optical scanning and *computer tomographic* (CT) technologies into manufacturing [1–3] presents new possibilities for analysis through production. As part of a larger *digital twin* (DTw) representation [4,5], this volumetric data can be leveraged in CAM systems to align target *computer-aided design* (CAD) models to their respective NNS starting material. Not only would this registration alignment reduce the need for tedious, manual positioning of the part model inside a rendered, irregularly shaped stock, but it could prove a valuable tool in the continued advancement of adaptive manufacturing operations through DTw feedback mechanisms.

Numerous and robust distance-based registration techniques exist for point cloud data sets. However, these are intended to align matching structures taken at different times or vantages. Such techniques are not designed to explicitly handle alignments for offset geometries. Those that are intended for non-identical point cloud registrations focus on feature extraction. Such approaches have been shown to work well [6] but require dominate planar features for the segmentation and fit. The current body of work will benefit from additional methods for cases when correlating feature extraction cannot be easily applied.

The medical imaging field has produced a rich body of research describing different registration techniques for aligning CT and *magnetic resonant imaging* (MRI) files of growing or changing subjects. Popular methods used the field are voxel intensity-based [7]. Although these representations are memory intensive compared with parametric surfaces, research in voxel-based CAM [8–14] has shown some advantages over classical approaches in five-axis machining. Specifically, voxels begin to be profitable as part complexity increases and collision avoidance becomes difficult. This work focuses on exploring the challenges of registering NNS structures in the context of a voxel environment that lends itself to point sets and image stacks. The application of intensity-based registration is introduced as a new approach to adaptive machining for manufacturing systems implementing volume renderings from optical scanners, CTs, and voxel CAM.

### 1.2 Areas of Application.

The integration of imaging feedback and registration may be applied to numerous manufacturing setups when an uncertainty in the description of the starting volume exists. However, several specific cases involving irregularly shaped workpieces will benefit from a new method to achieve part/stock registration.

Subtractive machining operations that finish rough castings must contend with some level of uncertainty in a starting workpiece. For example, many such rough castings are complex and can vary in their dimensional compliance. Additional measurements are typically necessary to ensure that all regions allow for adequate material to be removed. If the casting is in tolerance, the CAD file still must be manually placed inside the casting model. Figure 1 shows this scenario for an impeller. The engineer must also determine how to position the physical casting in the CNC’s coordinate frame to ensure the pre-programmed toolpaths that engage the part properly. Current practice assesses the casting prior to machining in a process called “marking-out,” which typically involves a manual leveling procedure on a special plate [15]. Optical scanners and computer vision programs have been applied in research to calculate adjustments necessary for alignment or if the casting should be scrapped. This is typically achieved through fiducial markers, matching datums, and fixture schemes [16]. However, the processes of marking out and any subsequent part-fixture adjustment can be time-consuming. The alignment of a CAD model to a rendered version of the rough casting prior to the toolpath generation would ensure a centrally positioned part minimizing the variance of the material removed. This in turn will eliminate discrepancies in depth of cut, tool engagement, and feedrates.

The use of micro X-ray CT in an industrial environment has been introduced as a non-destructive method for analyzing a workpiece’s internal structure. Unlike cameras and point cloud scanners, CT imaging fully renders the volume’s density characteristics as a grayscale image stack. Thus, information describing both the part’s external and internal characteristics are available. Subtractive and additive (re)manufacturing have utilized CT imaging to assess defects and structural integrity [1–3]. Internal structures, such as cooling channels, can be leveraged from these image stacks to drive registration alignments. Such metrics have been shown to excel in similar problem sets and can now be applied to parts with internal supports or integrated features.

Recent advances in hybrid machining [17,18] present exciting possibilities for additive–subtractive manufacturing. However, successive iterations of these operations that require a new starting volume should be used for each subtractive step. The entire hybrid manufacturing process can be generalized to pairs of additive–subtractive operations: {(*A*_{0}, *S*_{0}), (*A*_{1}, *S*_{1}), …, (*A*_{N}, *S*_{N})}. At the end of each additive phase *A*_{i}, the corresponding subtractive phase *S*_{i} must sculpt away excess volume to particular regions of interest. Material deposition requires advanced simulation to model placement of the additional volume for subsequent subtractive toolpaths [19]. As a result, these operations execute without any adaptive corrections. Instead, the volume at the end of the additive phase *A*_{i} needs to be described by a real-time DTw rendering. [4,5]. A generalized schematic of such an operation is shown in Fig. 2. Here, new G-code (ISO 6983) is adaptively generated at the completion of each additive step *A*_{i}. These workpiece scans serve as the new starting points for each subtractive step *S*_{i}. However, an automated means to align the rendering with the target model will be needed.

### 1.3 Overview.

A popular technique widely used in medical imaging called *maximization of mutual information* (MMI) [20,21] is described and briefly compared with the existing registration techniques. An additional registration method, referred to as *minimization of distance variance* (MDV), is a simple but novel approach included to demonstrate the design of a custom intensity-based metric.

It should be noted that voxel representations in this research were conducted over a uniform grid structure without any space-saving schemes. This naive approach simplified registrations by allowing direct indexing for joint histogram calculations and avoiding rendering conversions. However, this necessitates large amounts of memory and incurs considerable computational expense. In practice, an optimized scheme must be used, a voxel-based approach to be viable. Registrations will need to ideally be completed in less than a minute, not in less than an hour, which is the case for high-resolution data sets on a uniform grid. These initial simulations further assume an idealized, noise-free rendering. Prior works [21,22] in mutual information indicate that this assumption is fairly reasonable for the current stage of this research. Filtering schemes and robust MMI variants [23,24] deserve separate consideration. Instead, this article is focused on the introduction and functional application of intensity-based registration as a novel approach to specific manufacturing problems. A more complete validation addressing important practical issues will be left to a future study and discussed in the conclusion of this article. All simulations were performed in matlab^{®} R2017a using mathworks^{®}optimization toolbox™. Voxel operations were performed in the voxel-based CAM software sculptprint^{©}. The resulting alignment orientations are shown in sculptprint^{©} or as a point cloud for clarity. A standard impeller model was selected to demonstrate the intensity-based registrations. The model was chosen for its complicated free-form structure and practical use. A parallelepiped and cylinder with a rectangular base are also included to illustrate certain geometrical effects. Each of these stereolithography files was discretized in sculptprint^{©} and then exported to matlab^{®} as an image stack. Registrations were made on a 0.1–0.5 mm resolution (i.e., voxel length) with volume offsets of 1.5 mm or 2 mm for the reference volumes.

## 2 Voxels and Discrete Geometry Computer-Aided Manufacturing

Voxels can be most simply and concisely defined as 3D image pixels. Although pixels are the unit squares constituting an image in 2D, voxels are unit cubes that constitute a volume in 3D. Any data type can be tied to specific voxels indicating material, density, or distance. In the context of CAD/CAM modeling, a voxel typically indicates a solid or empty space as a boolean. Typical CAM software packages rely on geometric modeling kernels using analytic representations such as triangular meshes, *boundary representation* (B-rep), and *non-uniform rational basis spline* (NURBS) surfaces. Although these are much more spatially efficient when compared with voxel models, computations for the automated toolpath generation on higher-order surfaces (e.g., surface offsetting and set unions) become very complex and potentially unstable [12,25,26]. By contrast, voxel models are composed of independent constituents in a 3D array. They are not governed by mathematical relations between control points to maintain certain curvatures. Modifying a part’s features becomes a simple task of removing “blocks” of volume to achieve a desired shape. Although voxels permit fast boolean operations, their drawback is the considerable load that accompanies processing large amounts of discrete data [10,27]. The decision to pursue a voxel-based CAM solution should be made after considering a part’s complexity, importance of tool simulation and collision avoidance, and availability of modern GPU hardware for parallel processing.

Various structures have been developed to represent and store voxel data.^{1} A uniform grid [28] is by far the simplest. In such a grid, a bounding box subdivides space into cells (i.e., voxels) of equal size. This is simply an ordered 3D array with its matrix lattices aligned to a particular coordinate system. Volumes contained inside the bounding box are represented by voxel values at discrete locations. Regular, uniform spacing in a grid structure allows for easy implementation and indexing. However, these structures are impractical when attempting to represent high-resolution volumes since refining the resolution increases the number of voxels by a cubic order of magnitude. The memory allocation required can quickly become too cumbersome for CPUs/GPUs to store and process. Uniform grids are therefore only practical when the target volume can be represented at relatively low resolutions [12].

The shortcomings of a direct, single-level uniform grid can be overcome through more efficient schemes. Sparce voxel octrees [29,30] are a hierarchical data structure that allows for practical use of high-resolution renderings. Instead of allowing the grid size to dictate computational efficiency, octrees recursively subdivide a space into octants. These are further subdivided in the next level of the tree based on whether the information is contained that requires smaller voxel sizes to effectively render. For example, a cube containing a full solid or empty space can remain at a higher level of the tree; no finer resolution is necessary. This corresponds to constructing a larger “box” instead of several smaller ones describing the same information. Octants are continually subdivided to a desired extent about some boundary layer. However, octrees require more computational overhead for parallel offsetting operations, which is one of the benefits of voxels over traditional parametric CAM. To address this issue, hybrid data structures [13,31–33] have been developed to balance the storage and efficiency advantages between grids and octrees. As mentioned previously, this paper demonstrates intensity-based voxel registrations on a uniform grid for simplicity. Future research should employ a hybrid structure for efficiency and practicality. This article examines intensity-based registrations through the context of previous works [8,12–14,32,34] demonstrating the practical application of voxel processing in CAM.

## 3 Image/Volume Registration

The general goal of image registration is to obtain a transformation of a floating image $F$ that best aligns it with a reference image $R$. The term “image registration” can apply to alignment techniques for both 2D and 3D subjects. Since this paper is concerned with the registration of 3D models, the terms “image” and “volume” will be used interchangeably. The process of registration can be simplified to the interaction of four main components: a transformation function, a cost function, a similarity metric, and an optimization algorithm. Figure 3 shows the generalized interaction between these four structures. The transformation function iteratively takes the parameter(s) to be optimized and uses the current value(s) to describe a particular geometric transformation.

An image registration typically results in a partial or complete superimposition of $F$ over $R$. However, this is not always the case, as when attempting to align partial point cloud scans through a distance-based algorithm. The selection of a particular registration technique must be done with the knowledge of the criteria necessary to achieve the desired alignment. Since no optimization algorithm outperforms all problem sets, the selection of a particular registration technique must be done with the knowledge of the criteria necessary for a particular situation. In practice, different techniques can be used in series to leverage the relative strengths of different schemes depending on initial conditions.

Various classification criteria have been proposed to categorize published registration methods. In general, all techniques can be categorized as either distance-based or feature-based. Distance-based registrations involve the direct minimization of distances between the constituents of the floating and reference images. Feature-based techniques focus on the correspondence of extracted, differential properties contained in the two images. These feature-based approaches can be further subdivided into geometric-based (or surface-based) and intensity-based techniques. Geometric-based methods typically involve shape parameterization or segmentation to identify regions of interest. By contrast, intensity-based registrations are created with an expectation of how $F$’s voxel intensities should relate to the intensities of $R$ when properly aligned [35].

### 3.1 Intensity-Based Registration.

A significant portion of the published research in image registration is in medical imaging [36,37]. Development of intensity-based metrics and techniques is driven by challenges such as noise, soft tissue deformation, and variation in structure from one scan to another. Although fiducial markers can be employed for closest point techniques in medical applications, it is invasive and undesirable. Intensity-based registrations instead compare the pixel/voxel values between $F$ and $R$ with no assumptions made regarding geometric structure. This makes the approach very powerful. Intensity-based registrations are also easily applied to CT/MRI files representing collections of grayscale voxel values. Although values can be evaluated at a single resolution level, most effective intensity-based registration techniques employ a pyramidal multi-resolution approach to increase efficiency.

### 3.2 Brief Comparison to Distance and Geometric-Based Registrations.

*iterative closest point*(ICP). This flexible and popular algorithm is widely used in geometric processing and image registration problems [38,39]. Software designed for point cloud data acquisition and registration commonly use an ICP variant to join partial scans. The basic method matches two surfaces by minimizing the distance between corresponding points based on an initial displacement. In general, a point set $P1$ is transformed through a rotation

**and translation**

*R***so that each point $\rho 1i\u2208P1$ will be closest to its corresponding point $\rho 2i\u2208P2$. This is achieved through a squared distance minimization, written as**

*t**ρ*

_{1i}for each

*ρ*

_{2i}and determines the likelihood of correct convergence. Various modifications and improvements [40,41] have been proposed to ICP that make it a robust and efficient solution for aligning point clouds. However, the technique assumes one image is a subset of the other. In other words, ICP and other purely distance-based approaches are designed to register identical shapes. They are not intended for NNS parts. ICP should be applied when the mean-squared error between associated points can be driven to zero (i.e., $\u2225\rho iR\u2212\rho iF\u2225\u22480$). Such an error convergence is only true when the structures are (very nearly) identical but misaligned. In contrast, the distances between surface points of an NNS and a CAD model will be a finite number (i.e., $\u2225\rho iR\u2212\rho iF\u2225=di$). It is possible for a transformation to cause enough points in $F$ to overlap with $R$ to minimize the overall mean-squared difference between the images but allow a subset of points to be completely misaligned. To demonstrate this, Fig. 4 shows the result of applying ICP to a floating and offset NNS impeller models.

^{2}Note that the resulting registration leaves sections of the $F$ completely outside of the larger offset volume $R$.

Geometric-based registrations rely on the extraction and matching of surface information from a rendered model. The technique’s success relies on the presence and identification scheme for a surface/line(s) of interest. Segmentation and/or shape matching of shared key features is typically leveraged. Algorithms like chamfer matching minimize the distance between corresponding curves and patterned surfaces. Other advanced methods such as the “spash” are designed for matching curvatures in free-form structures. However, these too are designed to align parts based on shared features. These features can degrade or may not be present in the case of NNS parts. However, matching plane angles between corresponding features have been shown to achieve precise NNS registrations for rough castings [6]. This works very well for the cases where machined parts have clearly defined flat surfaces. However, free-form geometries that are not easily decomposed to geometric primitives and parts with no dominate planar features can still pose a challenge to geometric-based registrations. As with any optimization method, the selection of a registration technique depends on the application, and various approaches can be implemented in series to achieve a robust registration sequence. Table 1 summarizes some general advantages and disadvantages tied to the categorical methods.

Subcategory | Technique | Advantages | Disadvantages |
---|---|---|---|

Closest point | ICP | Popular and fast with many variants available for different robustness requirements | Not intended for NNS alignments without feature segmentation; susceptible to initial conditions |

Geometric | Chamfer matching pattern search | Precise matching for most non-free-form parts and is a good choice to augment ICP | Requires dominate planes/lines for matching; only plane angle alignments applicable to NNS |

Intensity | MMI, MDV | No structural assumptions; flexible and considers the full part volume | Large memory overhead; aliasing errors from discrete sub-voxel transformations |

Subcategory | Technique | Advantages | Disadvantages |
---|---|---|---|

Closest point | ICP | Popular and fast with many variants available for different robustness requirements | Not intended for NNS alignments without feature segmentation; susceptible to initial conditions |

Geometric | Chamfer matching pattern search | Precise matching for most non-free-form parts and is a good choice to augment ICP | Requires dominate planes/lines for matching; only plane angle alignments applicable to NNS |

Intensity | MMI, MDV | No structural assumptions; flexible and considers the full part volume | Large memory overhead; aliasing errors from discrete sub-voxel transformations |

## 4 Maximization of Mutual Information

*mutual information*(MI) between two files [20]. Therefore, MI acts as the similarity metric criteria in Fig. 3. In the context of voxels, MI indicates how much information one volume contains about the other based on the intensities of corresponding voxels. Mathematically, the MI,

*I*(

*X*,

*Y*), between two random variables across the discrete domain can be written as

*p*(

*z*) and

*p*(

*x*,

*y*) are the probability mass and the joint probability mass functions, respectively.

*H*(

*X*) and

*H*(

*Y*) are the two marginal entropies for

*X*and

*Y*, and and

*H*(

*X*,

*Y*) is the bivariate entropy contained between the two random variables. Histograms are constructed to approximate the probability mass functions of

*X*and

*Y*individually and a joint histogram is used to approximate the joint entropy. A bivariate, or joint, histogram is analogous to traditional histogram plots but with an added dimension. It can therefore be visualized as a surface over a 2D plane. Qualitatively, these entropy terms represent the amount of “uncertainty” or “unpredictability” contained in an information source. From Eq. (2), MI depends on the joint entropy of the two variables taken together as well as their marginal entropies. These individual entropy measures indicate the amount of information contributed from overlapping regions of the volumes

*X*and

*Y*. Rewriting Eq. (2) in terms of a translated, floating volume $F$ and reference volume $R$,

*T*(

*X*). Finding a position of $F$ that most closely aligns it with $R$ is equivalent to maximizing the MI between the two objects. Minimizing the final term in Eq. (5) directly decreases the overall amount of uncertainty between the two images. The absolute value of the joint entropy is decreased with transformations that result in complex regions of $R$ superimposed over complex regions of $F$ [20]. In other words, minimizing the joint entropy term is analogous to maximizing how well $F$ “describes” $R$. The amount of joint entropy contained between the two images is evaluated based on the occurrence of filled voxel space at various grid points. For example, all voxels with value 2 in $R$ should also be overlaid with voxels of value 2 in $F$, and so on. If a number of corresponding voxels in each volume space are of the same or similar intensity, the joint entropy (i.e., uncertainty) is decreased and the MI is maximized.

However, it should be noted that minimizing the joint entropy alone is not equivalent to maximizing mutual information. In fact, this is what makes MMI a robust choice for registering images with slight variations in voxel/pixel value. The marginal entropies $H(T(F))$ and $H(R)$ are not invariant to the amount of overlap between the two images [42]. These terms are capable of changing through each registration iteration. In some situations, this overlap dependence is mitigated through normalization schemes. Since MI depends on both the marginal and joint entropy, this causes the similarity metric to not assume a specific linear dependence on mapped voxel intensities [20]. However, an intuitive understanding of MMI can still be obtained by considering the minimizing of joint entropy by aligning the two sources to match voxels/pixels of “close” value.

## 5 Application of Maximization of Mutual Information to Voxel Models

The success of MMI in various clinical registrations is promising for applications in the voxel-based CAM. All published studies and applications of image/volume registration researched for this paper derived the pixel/voxel intensity values from the measurement device. However, rendered CAD models do not contain any innate intensity information. Therefore, voxel intensity values must be assigned for DTw volume registration. Otherwise, the boolean values used to only represent solid and empty space will not contain enough information to properly direct an alignment. The registration will terminate on any complete superimposed $F$ over $R$.

This research generates voxel intensities through an exact Euclidean distance transform [43]. This results in voxels located on the surface of the part being assigned a value of 1. These values increase for more central regions of the volume to reflect the shortest distance to empty space. Figure 5 shows a simple example of how pixel values are assigned on a 2D image through a distance transform. Here, higher weighted pixels are located near the central regions of the square. Figure 6 shows the successful registration of a basic impeller model to an offset structure using an MMI. A larger reference volume $R$ was generated by a convolution offset of 1.5 mm. The two part models can be seen in Fig. 6(a). After the registration converged to a placement (Fig. 6(b)), a minimal distance to the NNS surface of approximately 2 voxel lengths (1 mm) was calculated from the outer surface of the reference volume $R$.

Although uniformly offset volumes serve as a good initial testing paradigm, such structures are not necessarily representative of a practical NNS workpiece. For example, a rough casting would not be machined under the assumption that all surfaces (including inaccessible regions on the bottom surface) need to be machined. In practice, only some regions on the starting material will be modified when compared with the final CAD model. Recall that each subtractive phase *S*_{i} must sculpt away volume previously added in excess to regions of interest. However, the volume at the end of the additive phase *A*_{i} will usually have some features or parts identical to the desired final geometry. Figures 7(a) and 7(b) can be taken as simplified versions of the “before” and “after” models of a hybrid operation. Unlike the uniformly offset parts, only the lower portions of the models are nearly identical.

The results of applying MMI to this test case is shown in Figs. 8(b) and 8(d). A misaligned result of a non-MI technique is included for comparison in Fig. 8(c). The registration correctly terminates at a position with the part aligned to the corresponding, “non-modified” region of Fig. 7(a). This simple demonstration appears to indicate that MMI can be a viable alternative to geometric-based registration techniques.

However, it should be remembered that intensity-based similarity metrics are governed completely by the voxel values of the volumes. Although MMI does not require the use of geometric primatives, it does necessitate a scheme for assigning intensity values. Since an exact Euclidean distance transform was used to generate these values, the similarity between the overall shapes dictate how well the registration performs. In other words, MI is linked to how much information about $F$ is contained in $R$. MMI will fail as parts to be registered become more dissimilar without modification to the intensity assignment scheme. As the volume $R$ grows or distorts to become more dissimilar to $F$, the common information to be reflected decreases. An extreme case is shown in Fig. 9 where the impeller model is registered to a cylindrical stock. The alignment is clearly a failure since the parts are so dissimilar they do not contain enough correlating information to converge properly. A more realistic case is presented in Fig. 10 for analysis. This shows how a larger offset can begin to affect the alignment when the parts are geometrically simple (i.e., no complex regions of $R$ to overlap with $F$). The figure shows the outer surface of two parallelepiped models before and after an MMI registration. These are shown as point clouds for clarity despite being computed as solid voxel models. Unlike the impeller model, the parallelepiped was aligned (Fig. 10(b)) to its 2 mm NNS offset model with a global maximum MI measure of 0.3344. Note that there is an approximate 5 deg rotational error from a perfectly centered position despite the registration converging to a global maximum.

These two cases demonstrate that MMI should be applied only to cases where either geometries are closely identical or when voxel intensity values can be generated to reflect a close resemblance. It should be also be noted that without a steep penalty function supplementing the metric, the overlap invariance of MMI can lead to erroneous convergences with segments of $F$ outside of $R$. If $R$ is too dissimilar to $F$, not enough shared information will be present for the registration to succeed. The performance of MMI will change based on the intensity values so different assignment schemes can augment the metric for specific applications. In short, these results confirm that MMI should only be used when enough information is reflected between the two geometries but is effective when complex, free-form geometries need to be aligned. However, this information is only reflected in the voxel values.

## 6 Minimization of Distance Variance

In light of the advantages and disadvantages of MMI, this research also investigated how a simple similarity metric could be constructed for a generalized application of registration for rough casting machining. The method is intended to direct the alignment of two uniformly offset structures with minimal variance across all to-be-machined surfaces. Although it is not intended for more complicated alignments, the minimization of voxel variance is a novel similarity metric to drive the registration. Specifically, this method focuses on minimizing the variance among select voxel intensity values assigned through a Euclidean distance transform and is referred to as *minimization of distance variance* (MDV). Although a proper analysis of the computational efficiency should be conducted on a grid-tree structure, it is interesting to note how the simplicity of MDV would be advantageous as resolution increases for CNC applications. Instead of constructing a joint histogram from two sets of intensity values, MDV only references the floating point intensities of $R$ in a lookup table after a set operation and index. This analysis is left to a future study which implements MDV in a practical hybrid octree structure so that the information will be more meaningful.

### Minimization of distance variance

The use of a non-implication index works fairly well for moderately offset structures. However, the engineer interested in exploring or improving on the MDV metric is advised to index interpolated intensity values of $R$ based on the overlap of select surface voxels of $F$ using an octree data structure. This research elected to determine the overlapping/non-overlapping regions after the voxel center points of $F$ had been interpolated into the space of $R$. This was done to simplify the process in working with a uniform grid structure.

Tables 2 and 3 show selected test results for registering the parallelepipend and impeller, respectively. The reference models were generated with 2 mm and 1.5 mm convolution offsets and the floating volumes $F$ were translated and rotated to a starting misalignment. A genetic algorithm was selected to drive the MDV optimization of alignments with a voxel resolution of 0.1 mm. Every test converged to a position of $F$ entirely inside the NNS stock. Each resulting alignment did not appreciably differ from the MMI result in Fig. 6 despite minor discrepancies in rotation and translation. Unlike the previous alignment using MMI, the direct measure of variance values allowed for minimal rotational error in aligning the simple shapes.

0.1 mm Voxel resolution | Rotation error (deg) | Distance to NNS surface (mm) | ||||||
---|---|---|---|---|---|---|---|---|

R.Z | R.Y | R.X | Min | Max | Mean | Mode | SD | |

Parallelepiped (1.5 mm offset) | 0.06 | 0.01 | 0.03 | 1.10 | 1.80 | 1.62 | 1.60 | 0.78 |

0.36 | 0.03 | 0.06 | 0.75 | 2.14 | 1.62 | 1.70 | 2.12 | |

0.18 | 0.07 | 0.17 | 0.88 | 1.96 | 1.62 | 1.70 | 1.23 | |

0.02 | 0.00 | 0.18 | 0.91 | 1.90 | 1.62 | 1.40 | 1.66 | |

0.02 | 0.00 | 0.18 | 0.91 | 1.90 | 1.62 | 1.40 | 1.66 | |

Parallelepiped (2.0 mm offset) | 1.78 | 1.47 | 1.77 | 1.30 | 2.5 | 2.22 | 2.30 | 1.61 |

1.97 | 0.08 | 1.82 | 1.27 | 2.80 | 2.22 | 2.20 | 1.74 |

0.1 mm Voxel resolution | Rotation error (deg) | Distance to NNS surface (mm) | ||||||
---|---|---|---|---|---|---|---|---|

R.Z | R.Y | R.X | Min | Max | Mean | Mode | SD | |

Parallelepiped (1.5 mm offset) | 0.06 | 0.01 | 0.03 | 1.10 | 1.80 | 1.62 | 1.60 | 0.78 |

0.36 | 0.03 | 0.06 | 0.75 | 2.14 | 1.62 | 1.70 | 2.12 | |

0.18 | 0.07 | 0.17 | 0.88 | 1.96 | 1.62 | 1.70 | 1.23 | |

0.02 | 0.00 | 0.18 | 0.91 | 1.90 | 1.62 | 1.40 | 1.66 | |

0.02 | 0.00 | 0.18 | 0.91 | 1.90 | 1.62 | 1.40 | 1.66 | |

Parallelepiped (2.0 mm offset) | 1.78 | 1.47 | 1.77 | 1.30 | 2.5 | 2.22 | 2.30 | 1.61 |

1.97 | 0.08 | 1.82 | 1.27 | 2.80 | 2.22 | 2.20 | 1.74 |

0.1 mm Voxel resolution | Rotation error (deg) | Distance to NNS surface (mm) | ||||||
---|---|---|---|---|---|---|---|---|

R.Z | R.Y | R.X | Min | Max | Mean | Mode | SD | |

Impeller (1.5 mm offset) | 0.15 | 0.07 | 0.01 | 1.00 | 2.67 | 1.66 | 1.80 | 2.03 |

0.03 | 0.03 | 0.08 | 1.70 | 3.00 | 1.59 | 1.70 | 3.28 | |

0.71 | 0.13 | 0.03 | 0.77 | 2.57 | 1.66 | 2.10 | 3.07 | |

0.02 | 0.04 | 0.18 | 1.04 | 2.69 | 1.66 | 1.70 | 1.93 | |

0.00 | 0.03 | 0.02 | 1.12 | 2.60 | 1.66 | 1.70 | 1.70 | |

Impeller (2.0 mm offset) | 1.50 | 1.41 | 1.58 | 1.55 | 3.49 | 2.30 | 2.40 | 2.13 |

1.39 | 1.40 | 1.80 | 1.63 | 3.55 | 2.30 | 2.30 | 2.16 |

0.1 mm Voxel resolution | Rotation error (deg) | Distance to NNS surface (mm) | ||||||
---|---|---|---|---|---|---|---|---|

R.Z | R.Y | R.X | Min | Max | Mean | Mode | SD | |

Impeller (1.5 mm offset) | 0.15 | 0.07 | 0.01 | 1.00 | 2.67 | 1.66 | 1.80 | 2.03 |

0.03 | 0.03 | 0.08 | 1.70 | 3.00 | 1.59 | 1.70 | 3.28 | |

0.71 | 0.13 | 0.03 | 0.77 | 2.57 | 1.66 | 2.10 | 3.07 | |

0.02 | 0.04 | 0.18 | 1.04 | 2.69 | 1.66 | 1.70 | 1.93 | |

0.00 | 0.03 | 0.02 | 1.12 | 2.60 | 1.66 | 1.70 | 1.70 | |

Impeller (2.0 mm offset) | 1.50 | 1.41 | 1.58 | 1.55 | 3.49 | 2.30 | 2.40 | 2.13 |

1.39 | 1.40 | 1.80 | 1.63 | 3.55 | 2.30 | 2.30 | 2.16 |

After investigating prior work in the field of image registration, this is the only voxel similarity metric designed to directly minimize the variance of distance measurements. The method’s simplicity, however, comes at the cost of limited use. MDV performs well for the very simple scenario of uniformly offset volumes, but more advanced situations obviously require further development. It should also be noted that this basic version of MDV is not directly useful for machining applications since a Z-axis offset will pervade every registration alignment. That being said, the described MDV technique demonstrates that an effective intensity-based registration program can be written with minimal development for special cases as they are encountered in industry. The manufacturing engineer can direct a similar metric to converge by directly weighting voxels differently or by applying a more sophisticated assignment scheme to augment/improve the presented process.

## 7 Conclusions and Future Work

This paper has introduced the application of intensity-based registration techniques in a manufacturing environment and in the context of voxel-based CAM. The use of such methods provides a new perspective for select problems pertaining to NNS volume alignments. Specifically, a direct application of MMI shows that it is effective in registering more complex free-form structures and smaller offsets. Initial applications also appear to indicate that it will perform well in situations where some features are shared between the two volumes. Unlike geometry-based techniques, MMI does not require that these shared features be on the surface or be segmented as geometric primitives. MDV was introduced as a very simple but novel metric that can be augmented with more advanced techniques (e.g., ray casting or incorporated surface matching).

However, future work is needed to fully validate the use of intensity-based registrations in manufacturing beyond this introduction. First, the utility of intensity-based metrics still needs to be confirmed at resolutions suitable for precision machining. While studies utilizing MMI for CT/MRI registrations operate on bounded image stack volumes of 250^{3}–450^{3} voxels, precision machining applications require much higher resolutions. A complete study leveraging an efficient hybrid octree scheme will be the next step in continued research. Sensitivity and robustness analyses should be performed once MMI and/or MDV have been altered to operate on such a structure. Second, problems regarding real-world volume acquisition should be researched if intensity metrics are further validated on a refined data structure. If point cloud data is used, a means of extrapolating points to yield a “water tight” model for machining will need to be explored. The effects of noise and foreign bodies in a machining environment should also be addressed. However, studies [21,23,24] have already validated MMI and its variants are reasonably robust to added noise. Applying filtering schemes to eliminate noisy data should also improve rendered voxel data in future research conducted outside a simulation environment.

Finally, care needs to be taken when interpolating alignments for larger scale use cases. Voxel registrations can suffer from aliasing errors due to fluctuations in the total voxel number under sub-voxel transformations. Future research will need to confirm that the variation from interpolation does not appreciably affect the overall results for MMI or any improved MDV variant.

The main challenge of NNS registrations is that many techniques are designed to align identical geometries or match geometric primitives. However, the smaller number of cases registering dissimilar, free-form structures can benefit from intensity-based metrics in a voxel CAM environment. By operating on discrete volumetric data, as opposed to surface data defined by B-rep and NURBS surfaces, intensity-based registrations can provide engineers with another solution to address the manufacturing challenges of NNS and adaptive machining.

## Footnotes

Conversion from a voxel model to a point cloud was performed by indexing all surface voxels following a distance transform.

## Acknowledgment

This research is funded through NSF grant fund CMMI-1329742 (Funder ID: 10.13039/100000001). Staff and students in the Georgia Tech Precision Machining Research Consortium (PMRC) and Tucker Innovations Inc. also contributed greatly to the completion of this work.