1 Introduction

Video projectors are now in a widespread use for various purposes beyond just image presentation on a white screen. Because of their recent technological progress and cost efficiency, they are also useful for such applications as projection mapping on complicated shapes and 3D reconstruction of objects. Among these, 3D scanning systems using a projector and a camera have been researched for a long time.

Previously, because such a system usually relied on just a single depth cue, 3D shapes were reconstructed by standard stereo imaging, depth from defocus (DfD), or photometric stereo techniques. By using such traditional methods, it is difficult to incorporate multiple depth cues into a single captured image, which would result in more stable and more accurate depth measurement. Recently, several techniques have been proposed to improve the 3D quality by installing coded apertures into the projectors or by increasing the number of projectors, which would provide multiple depth cues for reconstruction. In the first case, the coded aperture installed in the projector increases depth-dependent information and avoids defocus blur. However, since patterns projected on a target surface can be represented by convolution of the aperture and the projection pattern, it is difficult to analyze such convoluted information. In the second case, the number of depth cues is increased by simply using more projectors, a technique that is similar to the multi-view stereo technique. However, contrary to a multi-camera system, multiple patterns projected onto the same object’s surface interfere with each other, making it difficult to separate features. A practical solution is to use a different wavelength (color) for each projector to be able to decompose them as from independent projectors. However, with this method, only three wavelengths (red-green-blue, or RGB) can be used with common (off-the-shelf) projectors, and crosstalk phenomena between different color channels remains an open problem.

Both the abovementioned techniques can be understood as a construction of a unique light field in 3D and extraction of depth cues from a part of it. Because the structure of a light filed composed by multiple projectors is complicated, analysis of such a synthetic light filed in practice is quite difficult. In this paper, instead of separating a light field from different depth cues, we propose a learning-based technique to extract depth information directly from the light field.

Figure 1 shows the overview of the proposed shape reconstruction technique. The key to our technique is based on the fact that such a projector-sourced light field is locally smooth and consists of similar blocks in 3D space; the multiple projectors project some regular patterns that coalesce to form a continuous light field in the space [1, 2]. Our solution is to sample a large number of small image areas or patches from the light field and apply a machine learning technique to reduce the size of the dataset as well as shorten the depth calculation time. The method can be divided into two phases: a learning phase and a shape reconstruction phase. In the learning phase, because there is no medium to reflect the light rays in the air, the patterns are invisible and unobservable, and therefore, we put a planar board in the scene to capture the patterns. Further, we conduct a principal component analysis (PCA) to eliminate redundant dimension and thus compact the data. In the shape reconstruction phase, to reduce both the dataset size and a calculation time, we use an approximated nearest neighbor (ANN) search algorithm combined with a belief propagation (BP) algorithm, which efficiently removes noise thereby increasing robustness and quality in the final output.

Fig. 1
figure 1

Overview of the proposed shape reconstruction technique

The main contributions of this paper are as follows.

  1. 1

    Multiple depth cues (e.g., disparity and defocus) can be superposed into a single captured image to make depth measurements stable and accurate. Also, multiple projectors can be used simultaneously. Note that the number of the projectors need not be limited to the number of color channels because our technique is not based on pattern separation approach, which is common in previous techniques.

  2. 2

    Capturing all possible sample images is not always necessary; virtually generated images can be used for supervisory signals in the learning phase.

  3. 3

    Our proposed system can incorporate various feature extraction and learning algorithms. We venture to propose a simple method, which uses PCA and ANN as it does not require hyperparameters except for the number of dimensions, whereas deep learning techniques [3] require many fine-tuned hyperparameters and a long processing time for learning.

2 Related works

In an active stereo system, a video projector is often used as a light source to measure a wide area in a short period of time, and the history of those techniques was summarized in [4]. Such systems usually rely on just a single depth cue, and therefore, 3D shapes are reconstructed by standard stereo imaging, by DfD or by photometric stereo techniques. Recently, several techniques that use multiple depth cues for active stereo systems have been proposed. A method devised by Masuyama et al. relies on both stereo and DfD cues by projecting multiple patterns along the same optical axis but using different focal lengths [5]. However, sharing the same optical axis complicates the system, and overlapping multiple patterns severely lowers the contrast. Zhang et al. proposed a method for projecting different patterns and successfully reconstructed a high-density depth map by analyzing the captured defocused image set [6]. Achar proposed a method projecting a pattern with different foci to enlarge the possible depth range [7]. However, because those approaches require multiple images for reconstruction, they are complicated and have only a limit range of application. Our technique is based on a single image and free from these problems.

Another approach to increase the depth cues using a video projector is to attach a coded aperture mask to the projector. Girod et al. used an asymmetric aperture to distinguish the forward and backward blur for depth from defocus (DfD) [8]. Moreno-Noguer et al. installed a small circular aperture to use the DfD technique [9], and Kawasaki et al. put a coded aperture on a video projector to improve accuracy and density while applying the DfD method [10, 11]. However, for all those techniques, depth range is limited and reconstruction is unstable and inaccurate because the analysis of defocus blur is still an unsolved problem and difficult to apply.

Because patterns formed by video projectors construct a synthetic light field, the above techniques can be considered extractions of depth cues from a sampled light field projected onto an object surface. However, as mentioned earlier, because of its complicated nature, only limited research has been done to analyze of a synthetic light field created by projectors. Kawasaki et al. proposed a technique to capture the entire light field by a special sampling machine [12], but the dataset size proved to be very large and calculation time high. In contrast, we propose two approaches to create a blur-free light field and to recover 3D information using compact data representation and low computational time.

Sagawa et al. proposed a depth measurement method based on a convolutional neural network (CNN) [3]. Although purposes of both Sagawa’s and our methods are the same, such as dimension reduction of patch feature, the approaches of our and Sagawa’s methods are totally different, i.e., Sagawa et al. tried to explicitly separate overlapped patches into each projector’s pattern and trained a CNN to reduce the effects of color crosstalks caused by using multiple projectors, whereas our method learns to use the overlapping light fields as a whole. Also, the CNN requires many hyperparameters and a long processing time for training, whereas our method using PCA does not require hyperparameters (except for specifying the number of dimensions of the reduced data space), nor a long computational time for learning.

Fanero et al. [13] proposed to use a camera with infrared illumination for close-range human 3D capturing. Their method assume close-range measurement and limited kinds of materials (e.g., limiting to human hands or faces), where illuminated intensity can be a range cue, whereas our method can deal with much wider object classes, object shapes, and working ranges.

3 Overview

3.1 System configuration

The learning-based 3D shape reconstruction method proposed in this paper is characterized by integrating multiple types of depth cues. Our technique can be applied to various system configurations of active 3D measurement, particularly where conventional analytical reconstruction methods are difficult to apply. Examples of our system configuration are shown in Figs. 2 and 3.

Fig. 2
figure 2

System configuration and process flow of system 1. a Learning with real capturing: captured by changing the depth of a planar board with known position on which the specially designed pattern is projected. b Depth measurement

Fig. 3
figure 3

System configuration and process flow of system 2. a Learning with virtual capturing: geometric relationships between each projector and the reference object is calibrated individually. b Depth measurement

Figure 2 shows a configuration using a single or multiple video projectors incorporating a coded aperture [12]. In this system, both the aperture mask and projected pattern involve mutually parallel lines. Figure 4 shows an example of how a projector with a coded aperture constructs the light field in the space. As the distance from the projector to the target surface changes, the projected patterns change. Because the aperture mask has many small slits, the projector preserves both high-frequency patterns and the total amount of light energy. By projecting a pattern that varies according to the depth, it is possible to perform depth measurements based on depth cues from defocus blur. In theory, if there are multiple projectors, shape can be recovered with depth cues by disparity. However, because in practice, those depth cues are intermixed, common techniques cannot be applied.

Fig. 4
figure 4

Pattern projection with coded aperture for different target depths

Figure 3 shows an example system configuration that consists of one charged coupled device (CCD) camera and multiple common (off-the-shelf) projectors. In general, it has been necessary to project patterns with different wavelengths in order to separate depth cues after measurement. However, because our proposed method has the advantage that single color patterns from multiple projectors can be differentiated by a machine learning approach, simpler 3D shape reconstruction systems are possible without needing different wavelengths.

In addition to the example system configuration, Figs. 2 and 3 show the processes involved in our proposed method, which has two main stages: the image database learning phase and the depth measurement phase. In the learning phase, the database is constructed from the actual or virtually captured images using data compression. Then, in the measurement phase, a 3D shape of the target scene is reconstructed from a captured image using the database.

3.2 Algorithm overview

Figure 5 outlines the algorithm of our proposed method. When constructing an image database in the learning phase, a large number of small image patches are sampled from the light field and stored in the database (see Section 4.1). However, this means that the size of the image dataset as sampled from the light field becomes extremely large. In particular, because complicated projection patterns with different types of disparities and defocus effects are projected from multiple projectors, spatial frequency becomes high. For this reason, we propose feature extraction using PCA to reduce the size of the image dataset and thereby shorten the processing time for distance estimation [14, 15]. The same process is performed not only when constructing the database but also when matching the input image with those already in the database.

Fig. 5
figure 5

Algorithm overview of learning based 3D reconstruction

Next, distance is estimated by matching each input image patch with image patches already in the database in the depth shape reconstruction phase (Section 4.3). Originally, this matching process required full search, and thus, the calculation cost was high. In the proposed method, ANN processing is used to perform high-speed matching (see Section 4.2). Regardless of the projection patterns and/or the number of projectors, learning-based reconstruction processing as described above can perform 3D measurement.

4 Learning-based 3D shape reconstruction

4.1 Database creation

In the learning phase, we capture an image for each depth and build a database consisting of image patches with depths as labels. In order to reduce the size of the samples dataset, we then perform feature extraction based on the eigenimages obtained through PCA. The image patches then are represented by feature vectors and stored in the database.

In the proposed method, image databases are created from either actual images or virtually captured images. Figure 2a shows an example of a case requiring actual image capturing. When a coded aperture is installed in a projector, interference between the light source and the lens makes virtual capture of sample images difficult [12]. Therefore, actual image capturing is necessary. In this case, patterns from the projector are imaged on a white planar board placed on a motorized stage, are captured, and are stored in the database.

When using off-the-shelf projectors that do not have a coded aperture or similar modification, the virtual image capture approach can be applied. Figure 3a shows an example of a configuration and procedure for creating a pattern using multiple projectors, which can be simulated easily by computer graphics (CG). In practice, external parameters for each projector and a camera as well as internal parameters of them are acquired by geometric calibration of the real system. Then, a pattern is simultaneously projected from all virtual projectors onto a virtual planar board, and sample images are captured by a virtual camera while moving the planar board.

4.2 Low-dimensional representation by PCA

When the proposed method estimates pixel depth, an image patch around the pixel is compared with the previously obtained sample images. By repeating this process for all pixels in the captured image, the 3D shape of the target object is reconstructed. As this is a time-consuming process, in practice, we use PCA to obtain low-dimensional feature representation (also known as eigenspace representation) of the image patches to reduce the calculation time for data matching.

PCA has long been used in computer vision studies to reduce the dimensionality of image data. Examples of PCA has been used for such processing including facial image recognition or analysis to represent or separate changes of illumination or facial expressions [14, 15], object recognition across various images captured from different 3D viewpoints [16], and fast image matching achieved by dimension reduction of raw image data vectors [17].

Figure 6 shows the process of extracting features from projected patterns. To apply PCA, we first collect the training set from the sample images to calculate the eigenspace of the dataset. In the proposed method, the sample images are obtained in the calibration process, where the images of the fronto-parallel planes are captured by an actual setup of a projector camera system or are generated by a virtual projector simulation. From the sample images, image patches of the same size are extracted.

Fig. 6
figure 6

Extracting image features of sample images

The image patches are first represented as column vectors of p1,p2,⋯,pN by simple rasterization. If the image patches are M by M pixels, the dimension of the column vectors is M2. Here, we use L=M2 for simplicity. The average image patch is calculated by \(\bar {\mathbf {p}}=\frac {1}{N}\sum \nolimits _{k=1}^{N}\mathbf {p}_{k}\), and deviation from the average data by \(\mathbf {q}_{k}=\mathbf {p}_{k}- \bar {\mathbf {p}}\). A set of orthonormal bases for representing qk can be calculated using PCA.

In normal PCA, eigenvectors u1,u2,⋯,uL of the L×L covariance matrix:

$$ \mathbf{C} =\frac{1}{N}\sum\limits_{k=1}^{N}\mathbf{q}_{k} \mathbf{q}_{k}^{\top}=\mathbf{A}~\mathbf{A}^{\top}, $$

where A=[q1 q2qN], are used for the orthogonal basis set. However, in computer vision problems, often N<L, and then, we can use eigenvectors v1,v2,⋯,vN of the N×N matrix L=A A for forming the basis set to save the computational cost of eigenvector calculation [14]. The basis vectors, then, can be calculated by \( \mathbf {u}_{i} =\sum \nolimits _{k=1}^{N} (\mathbf {v}_{i})_{k}~ (\mathbf {q}_{k}) \) for i=1,⋯,N, where (vi)k is the kth element of vector vi.

From the obtained basis, the representation of a new image patch r is (w1 w2wN), where \(w_{i}=\mathbf {u}_{i}^{\top } (\mathbf {r}-\bar {\mathbf {p}}) \). Let eigenvectors v1,v2,⋯,vN be sorted by the descending order of the associated eigenvalues. Then, u1,u2,⋯,uN are aligned in the order of optimal representation of the training set {q1,⋯,qN} for minimizing the sum of errors of l2 norm. Thus, if the image patch r is similar to the training set, \(\phantom {\dot {i}\!}(w_{1}~w_{2}~\cdots ~w_{L'})^{\top }\) where LN, is a good L’-dimensional representation of r. The process to determine the basis set using PCA can be regarded as a process of learning the image features for representing the pattern training set.

Figure 7 shows the eigenvector basis for different pattern projections in the system shown in Fig. 3a. The top row images are extracted from grid patterns created by a single projector, the middle row images are from grid patterns made using two projectors, and the bottom row images are from grid patterns made using three projectors. These results show that the basis represents the features of the training image set for different types of light field.

Fig. 7
figure 7

Samples of eigenvectors visualized in image format (eigenimages) for grid patterns. The top row images are eigenimages for a grid pattern to use one projector, the middle row images are for grid patterns by two projectors, and the bottom row images are for grid patterns by three projectors. From left columns to right, the 2nd, 3rd, 11th, and 12th eigenimages in the order of eigenvalues

In the database construction step, we calculate the low-dimensional (L’-D) representations for the sample patch images. In the 3D measurement step, we calculate the L’-D vectors for the image patches around each pixel. These patches are matched with the images sampled for each depth in the L’-D vector space.

4.3 Efficient depth estimation by ANN and MRF

In the depth estimation phase, we use the proposed technique to capture a target object as shown in Figs. 2b or 3b. Then, we convert the input images to low-dimensional representations by calculating the coefficients of eigenimages. Next, we search for the sample image patch most similar to each input image pixel. Although the proposed method requires only 10 to 40 dimensions, computational cost to find such patches for all the depths is still high. To reduce the calculation cost, we use an ANN search [18], where input data are stored in a k-d tree structure, which reduces the processing time with minimal sacrifice in accuracy.

Although a reconstructed 3D shape inevitably contains some amount of noise cased by wrong depth estimation, such wrong depths are efficiently removed or corrected by a Markov random field (MRF) approach. For MRF, cost volume is usually required; however, ANN originally returns just a single cost for maximum similarity. Because top 10% depth candidate values ordered by similarity includes the correct depth with a 90% probability based on our survey, we modified ANN to output that top 10% with cost (the reciprocal of similarity). We use BP [19] to solve the MRF.

5 Experiments

In order to verify the effectiveness of the proposed method, we implemented two kinds of systems. The first system uses one camera and two projectors fitted with coded apertures. The second system uses three unaltered (off-the-shelf) projectors with one camera. The depth cues for each system are summarized in Table 1. Experimental results using each system are described sequentially in the following Sections 5.1 and 5.2, respectively.

Table 1 The depth cues for the two proposed methods and previous method

5.1 System 1 using projectors fitted with coded apertures

Generally, the shallow depth of field of an off-the-shelf projector limits the depth measurement range of a projector camera system. Kawasaki et al. proposed a method to install a coded aperture to extend the depth range [12]. In the proposed system, we extend the technique by increasing the number of projectors, as shown in Fig. 8a. With this setup, because the optical phenomenon is complicated, we construct a database from the learning phase by capturing actual images. Figure 9 shows the actually observed patterns generated using a projector with a coded slit pattern aperture installed and those created by a normal (off-the-shelf) projector with a circular aperture for comparison. As shown in the figure, high-frequency patterns for all ranges are preserved by the coded aperture, whereas patterns are rapidly blurred out by the circular aperture.

Fig. 8
figure 8

Actual optical system. a Setup with two projectors. b Coded aperture installed on the right projector lens

Fig. 9
figure 9

Projected patterns with high frequency aperture and circular aperture

5.1.1 Eigenimages created by PCA

We installed one of this projector and a cameraFootnote 1 and acquired images for depths in the 250 to 550 mm range at sampling intervals of 0.5 mm. Because of the limitation of the length of the motorized stage, we put a close-up lens to change the scale as to be 1/3 of real length. As a result of a dimensional compression by PCA, the eigenimages obtained are as in Fig. 10, showing vertical, horizontal, and combined vertical-horizontal patterns.

Fig. 10
figure 10

Eigenimages created by PCA. Eigenimages with high contribution ratios are arranged in order from the top left

5.1.2 Accuracy evaluation

In order to verify the depth estimation accuracy of the proposed method, we conducted a depth measurement experiment on a flat board. In this experiment, vertical and horizontal patterns were projected at focusing distances of 250 mm and 350 mm, respectively. The target screen is placed on the motorized stage and moved to a depth of 250 to 550 mm while capturing at 10 mm intervals. The depth value was estimated using the proposed method. In addition to the case using two projectors, the case using only one projector that projects the vertical pattern was also tested. The matching window sizes were set to 16×16 or 12×12 pixels, and the number of feature dimensions was reduced to 12 by PCA. Note that the window size and the number of feature dimensions were set relatively small to verify the effectiveness of increasing the depth cue, i.e., the number of projectors. Other two methods, Microsoft Kinect 2 and an active stereo method using a random dot pattern, were also evaluated for comparison.

Figure 11 shows the the root mean square error (RMSE) of the estimated depth value. The graph demonstrates that, even with smaller window, the proposed method using two projectors could recover the correct depth for all the tested ranges with almost the same accuracy as Kinect 2Footnote 2. Accuracy of the case using one projector was deteriorated when using smaller matching window; even when using larger window, its accuracy was degraded at depths of 480 mm and 550 mm.

Fig. 11
figure 11

Depth estimation accuracy

5.1.3 Evaluation of learning-based feature extraction

In the next experiment, the robustness against the textures on a target object was evaluated; a vertical pattern was projected onto a flat board at a focusing distance of 250 mm. For comparison, the shapes were reconstructed by NCC without dimension reduction, by NCC with dimension reduction by hand-crafted feature [12], and by using Microsoft Kinect. In the experiments, the size of the image patch was set to 24×24 pixels for all techniques. Therefore, the length of the feature vector before dimensional compression is 24×24×3=1,728. Then, it is reduced to 20 or 48 dimensions by PCA and hand-crafted feature technique. The experimental results are shown in Fig. 12. The proposed method, combining PCA and ANN, achieved a lower RMSE than the hand-crafted feature technique, although it is inferior to NCC.

Fig. 12
figure 12

RMSE of planes with texture. Manual decode means a hand-crafted feature proposed in [12]. Our method performed better than the previous technique except Kinect

The same experiment was also carried out on different textures shown in Fig. 13, and the experimental results are shown in Fig. 12. When we measured these variously textured objects, reconstruction accuracy was equal to or higher than NCC. In addition, some textures (e.g., dappled, milk drop, and newspaper) show that the accuracy improves when more dimensions are used. Furthermore, if a checkered or dappled pattern image was measured, the proposed technique can perform with almost the same accuracy as Kinect, whereas it gets worse if the texture is strong (e.g., milk drop, newspaper, or wood board).

Fig. 13
figure 13

Textures used in the experiments. a Checker pattern. b Dappled texture. c Milk drop texture. d Newspaper. e Wooden board

5.1.4 Evaluation by 3D object reconstruction

Next, we reconstructed the shape of a real object. As with the previous section, we compared our reconstruction method with NCC and hand-crafted feature [12]. The experimental results are shown as depth images in Fig. 14 and as point clouds in Fig. 15. Hand-crafted feature provides good reconstruction results for 48 dimensions. However, when the reduction is performed for 12 dimensions, the reconstruction quality is significantly degraded. On the other hand, the proposed method can estimate stable depths even when the feature dimensions are compressed to 12 dimensions because of the efficient dimensional reduction by PCA.

Fig. 14
figure 14

Reconstruction results (depth images) of the real object using system 1. a Input image. b Result by NCC. c Result by the proposed method (PCA+ANN, 12D). d Result by [12] with hand-crafted features (12D). e Result by the proposed method (PCA+ANN, 48D). f Result by [12] with hand-crafted features (48D)

Fig. 15
figure 15

Reconstruction results (point clouds) of the real object using system 1. a NCC (48D). b Proposed (PCA+ANN, 48D)

5.2 System 2 using multiple projectors

This system consists of three video projectors and a single CCD camera. Because three projectors are off-the-shelf products and just a static pattern is projected from each device, virtual data creation by CG simulation is possible, as shown in Fig. 16a.

Fig. 16
figure 16

System 2 implementation and projected patterns. a Experimental system with a camera and a video projector. b Several patterns for one-shot scan. (i) Random grids, (ii) Wave grids [20]. (iii) Wave grids2 [21]

Our proposed technique does not depend on projection patterns. To demonstrate this advantage, we used several well-known patterns (Fig. 16) for one-shot scans (spatial encoding patterns) in our experiment: a random grid pattern, a wave grid pattern [20], and a pattern (wave 2) that is commonly used for single color one-shot scans [21].

5.2.1 Effect of increasing the number of projectors

We first tested the effectiveness of increasing the number of projectors while using NCC and the unicolor wave pattern, which has relatively high reconstruction accuracy. The unicolor wave pattern was used in each projector. Red and green patterns were assigned to two projectors; in the case of three projectors, blue was added to the two-projector setup. In the learning phase, we used CG simulation to create sample images in the 400 to 700 mm range at sampling intervals of 1 mm. For NCC, when comparing a captured image to sample images, the window size was 32×32 for the coarse process and 24×24 for the fine process.

In this experiment, four objects with arbitrary shape and texture were measured and reconstructed: a pair of boots, a buffalo skull (buffalo), a hand, and a monkey doll (Fig. 17). Measurement results using a conventional gray-code method were referenced as ground truth. Figure 18 shows the results. As the number of projectors increased, the accuracy improved for all four objects. This is because more projectors could make more complicated patterns, which produce better depth cue information.

Fig. 17
figure 17

Objects used in the experiments. a Hand. b Buffalo skull. c Boots. d Monkey

Fig. 18
figure 18

Comparison on RMSE with varying the number of projectors. The larger number of projectors results in better accuracy

5.2.2 Reconstruction of textured and curved surface objects with various patterns

To evaluate the dimension reduction of feature values using PCA, the following two methods were compared: NCC without dimension reduction and the proposed method involving dimension reduction by PCA. In this experiment, three projectors were used. For the proposed method with PCA, as the same condition with the experiment of Section 5.1, the patch size was set to 24×24. For the PCA, principal components up to the 30th level were applied.

As a result of dimensional compression by PCA, eigenimages were obtained (Fig. 19). A combination of line patterns was observed in the images.

Fig. 19
figure 19

Eigenimages created by PCA. Eigenimages with a high contribution rate are arranged in order from the top left

Depth maps and the reconstruction results are shown in Figs. 20, 21, and 22, and point clouds and 3D surfaces reconstructed from wave 2 pattern are shown in Fig. 23. We found that the proposed method could reconstruct object shapes with curved surfaces and non-uniform texture using any tested projection patterns. However, we observed some individual pattern tendencies, such as our PCA-based reconstruction method was affected by the complicated curve and texture while using the grid pattern (Fig. 20).

Fig. 20
figure 20

Reconstruction results of textured and curved surface object with grid pattern. a Input. b Depth map (PCA). c Depth map (NCC)

Fig. 21
figure 21

Reconstruction results of textured and curved surface object with wave pattern. a Input. b Depth map (PCA). c Depth map (NCC)

Fig. 22
figure 22

Reconstruction results of textured and curved surface object with wave pattern 2 proposed by [21]. a Input. b depth map (PCA). c Depth map (NCC)

Fig. 23
figure 23

Point clouds and reconstructed shapes of textured and curved surface objects with wave pattern 2 proposed by [21]. a Point cloud (ANN). b Point cloud (NCC). c Reconstructed shape (ANN). d Reconstructed shape (NCC)

5.2.3 Precision evaluation

We validated the 3D reconstruction precision of NCC and the proposed PCA-based method by comparing the results of the experiments described in Section 5.2.2. Figure 24 shows the comparison results on RMSE, which show no significant difference for RMSE between NCC and PCA methods. This means that the dimension reduction by PCA did not degrade the reconstruction accuracy.

Fig. 24
figure 24

Comparison on RMSE with varying reconstruction method (NCC and PCA) and projection pattern. Note that accuracy is almost the same between NCC and PCA (our method) where the data size and computational cost of PCA are much lower than NCC

Next, we discuss the difference between CNN and PCA for dimension reduction by reviewing the results of earlier work [3] using three of the same objects (hand, buffalo, and boots). In that earlier work, RMSE values for the CNN-based method for the three objects were 1.03, 1.41, and 1.22, and those for NCC were 1.22, 1.74, and 1.55Footnote 3, respectively. This is naturally understood that CNN outperforms other methods if the condition is the same in recent vast studies. Since the feature extraction processes are almost the same for both Sagawa’s and our techniques, if CNN is used for our method for feature extraction process, it is expected to produce better results. Incorporating deep learning techniques into the proposed method is our important future work; however, we believe that it does not reduce the contribution of our paper.

5.2.4 Processing time for reconstruction

The results comparing the reconstruction time (Fig. 25) clearly shows that PCA reduced the reconstruction timeFootnote 4, despite the combination of a coarse-to-fine approach for NCC. This is because PCA requires just 30 dimensional features, whereas NCC requires 24×24×3=1,728 dimensions. The computational complexity of reconstruction is O(n) for NCC and O(logn) for PCA, where n is the number of sample images.

Fig. 25
figure 25

The reconstruction time by PCA and NCC

6 Conclusions

We propose a learning-based 3D reconstruction technique for active stereo systems. With our technique, a light field is formed by attaching a coded aperture to the projector and/or by using multiple projectors that project arbitrary patterns and are in arbitrary poses. The resulting light field forms patterns on 3D surfaces with rich depth cue information. Because these cues are not easily extracted by existing analysis, we propose a learning-based approach that can be applied universally for various arbitrary types of image cues. The dataset of sample images for different depths is generated by real (actual) scan or by CG simulation. To realize efficient matching to sample data when processing the depth measurement, the dimensionality of the raw data of image patches in the sample image dataset is reduced by PCA so that the the image patches in the captured image can be compared with the sample image dataset in low-dimensional space. Experimental results prove that our technique is stable irrespective of target object materials, sensor noise, and projection patterns. In the future, deep learning techniques and real-time parallel processing will be applied.