1 Introduction

In the field of cultural heritage conservation and diagnostics, artworks and archaeological finds are often associated with digital imaging. Digital imaging techniques provide, in fact, detailed documentation and enable the utilization of image processing, analysis and computer graphics techniques to enrich the data and offer new opportunities both for study and enjoyment [1].

More recently, multiband imaging techniques have increasingly become part of the analytical methods used to examine cultural heritage objects. These techniques, which include both luminescence imaging methods, such as ultraviolet-induced luminescence (UVL), visible-induced infrared luminescence (VIL) and visible-induced visible luminescence (VIVL), as well as reflectance imaging methods, namely visible reflectance (VIS), infrared reflectance (IRR) and ultraviolet reflectance (UVR), are used by scientists, conservators, archaeologists and curators in a variety of fields [2]. Digital imaging techniques play an important role in the representation and characterization of the object. They enable the enhancement of visible features, dynamic visualization of the object, simulation of grazing light and representation of invisible features with the strict requirement of portability and non-invasiveness [3]. In this context, several techniques based on the acquisition of multiple pictures of a single item have been developed in recent years, and because of their affordability and dependability, they have nowadays become standard in preserving records of archaeological finds [4,5,6]. One of these techniques is reflectance transformation imaging (RTI), a computational photographic method that captures a subject’s surface shape and colour of an object’s surface and allows interactive re-lighting of the object from any direction. RTI images are created using multiple digital photographs of an object taken from a stationary camera position. In each shot, light is projected from a different known direction. This process creates a series of images of the same subject with different highlights and shadows. The illumination information from the images is processed to create a mathematical model of the surface, which allows the user to interactively re-illuminate the RTI image and examine its surface on a screen [7, 8]. RTI imaging has quickly transformed into a widely utilized approach for recording and scrutinizing cultural heritage items, enabling for further image enhancements to highlight surface details and features typical of the execution technique or conservative issues, i.e. engravings, reliefs, cracks or textures [9,10,11]. RTI is based on an algorithm called polynomial texture map, which was developed in 2001 at HP Labs [12] with the goal of representing the surface of objects. The algorithm allows the assessment of surface normals based on the captured photographs. It defines for each pixel a function that contains information about how the RGB channels change under different lighting conditions. This method was then implemented in 2008 thanks to the use of hemispherical harmonics (HSH) [13]. The outcome of employing these techniques for mathematically synthesizing illumination information is the creation of a 2D pseudocolour image called a normal map, where the RGB value of each pixel represents the orientation in the space of surface normals. RTI has garnered recognition as a well-established technique over numerous years and has been employed in the examination of various archaeological objects [5, 14, 15]. Several researches have been conducted to highlight its potential and limits [16, 17], as well as to optimize the illumination conditions [18], new frameworks and new dedicated software tools for processing multi-light image collections [9]. In addition, comparison studies with other techniques, such as photogrammetry, have been performed to verify its potential as a complementary technique [19, 20]. One of the most interesting applications of the RTI technique, also used in the field of cultural heritage, is certainly the possibility of reconstructing the profile of semi-flat objects by the method of normal map integration with different algorithms, starting from the 2D image [13, 21, 22]. The normal map integration technique has a long-standing history too and has gained considerable popularity over the years, particularly in 3D applications like graphical modelling for video games [23, 24]. Therefore, it proved to be efficient in most situations, but unlike the integration conducted on synthetic maps, RTI-based normal maps integration showed some limitations, particularly in representing low spatial frequencies [25,26,27,28]. RTI-based normal maps might display altered pseudocolour scale and/or artefacts, not due to ideal lighting and shooting conditions. This is why it is very important to have an efficient integration method and a high-quality normal map image to obtain the most accurate rendering of an object, which is of vital importance when it comes to the digitalization of heritage assets. It is crucial to carry out quantitative studies on the object rendering efficiency of RTI with normal map integration. These were partially performed in [26], but without a systematic method. In the context of this research, we present a quantitative evaluation of the ability of the RTI technique followed by the integration of the normal map for profilometric applications on semi-three-dimensional objects. The results obtained by RTI technique were compared with those resulting from the application of other diffused 3D techniques with the aim of comparing the accuracy in reproducing the objects under examination. In this regard, ad hoc profilometric targets were designed and realized [29], acquired with the RTI technique to obtain the normal map that is processed to create a 3D model. The same targets were also studied using diverse standardized techniques: micro-photogrammetry, laser scanning and optical profilometry. Several systems for object illumination have been developed, and these include a system with a free moving flash source and a system with dome lights [28]. In addition, in [26] it was shown that when the illumination source is changed from a dome light to a flashlight, the results are not distorted if the user is careful to keep the luminance the same in each photograph.

2 Materials and methods

The reference targets, used to test the considered techniques, were implemented to simulate semi-three-dimensional objects, i.e. objects where the gradient on the z-axis varies only slightly. This was specifically opted considering the intrinsic limitations of the RTI technique: the field of view is fixed above the subject; therefore, it cannot represent all features with a strong variation in height, see Fig. 1b and c. The techniques employed and their specifications are summarized in Table 1. In this table, the specifications indicated for each technique refer to the configuration used in the specific acquisitions of this study. The resolution, in fact, for some techniques depends on the sensor, the lens and the field of view, and therefore, the theoretical pixel size for each technique has been indicated, without considering, for example, geometric distortions, chromatic aberrations and diffraction limits.

Fig. 1
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a Light patterns produced by a dome (left) and a handheld flashlight (right) on the black spheres. b Graphic representation and c real-life picture of the layout of the experimental set-up

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Table 1 Technical specification of the techniques considered in this study for the specific set-up utilized
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2.1 Profilometric targets

Two types of targets were realized: an embossed target and a debossed target. The first type, namely the embossed target, has a base area of 2 \(\times\) 5 cm and a thickness of 0.5 cm, and it displays on the wider side a set of 20 extrusions of different known heights and widths. The second, namely the debossed target, has a base area of 3 \(\times\) 6 cm and a thickness of 0.5 cm, and it displays a set of 20 intrusions of different known depths and widths. Both the embossed target and the debossed target were made in three different shades, i.e. white, grey and black, to evaluate the influence of their ability to absorb light on the final result.

2.1.1 Embossed reference target

The design of the target was implemented with CAD Inventor software [30], and the objects were fabricated using the Formlabs Form2 high-resolution stereolithographic 3D printer [31]. Each target (2 cm \(\times\) 5 cm \(\times\) 0.5 cm) has a set of 20 parallelepiped-shaped extrusions, each with variable height or width. Sets 1 and 2 have extrusions of constant width but different height. Sets 3 and 4 have extrusions with different width but constant height, see Table 2 and Fig. 2. The original printing colour is grey. After printing, two more copies were painted with white-and-black matt varnish.

Table 2 Theoretical values of the heights and widths of the extrusions for the embossed target
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Fig. 2
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Pictures of the white, grey and black (from left to right) embossed reference target

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2.1.2 Debossed reference target

The debossed targets were obtained from the imprint of the embossed targets. In fact, the original 3D printed embossed grey target was used as a model. Additional walls were added to form a shell into which the liquid white silicone rubber (Liquid Mold, Resin Pro brand [32]) was poured. Carbon black pigment was mixed with the rubber in different ratios to obtain grey and black targets. Each target (3 cm \(\times\) 6 cm \(\times\) 0.5 cm) has 20 different intrusions, each with variable depth or width. Sets 1 and 2 have intrusions of the same width but different depth. Sets 3 and 4 series have intrusions with different width but same depth, see Table 3 and Fig. 3.

Table 3 Theoretical values of the depths and widths of the intrusions for the debossed target
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Fig. 3
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Pictures of the white, grey and black (from left to right) debossed reference target

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2.2 RTI and normal integration

A RTI file is a file consisting of a series of photographs taken under a specific lighting direction. To create the RTI file, a Nikon D810 camera (7360 \(\times\) 4912 pixels, 14-bit RAW files) with a Nikkor 105 mm f/2.8 D Macro and a remote control were used to collect forty pictures of the object of interest having a field of view of 50 \(\times\) 35 mm each. The camera was held on a tripod, with the camera lens pointing downward and the object placed on a support below. A Godox Witstro AD360 flash unit with a peak power of 360 W served as the light source. The highlight method was used in this study for building the RTI object, which involves manually moving the light source around the object in a hemispherical pattern while maintaining a constant distance. To track the position of the light direction in each photograph, a black reflective reference sphere with a diameter of 12 mm was placed next to the object. As shown in Fig. 1a, it is possible to retrieve the direction of the incident light from the position of the light on the black sphere. All images were processed using Adobe Camera Raw to apply colour management, with adjustments performed using X-Rite Color-Checker Target and Color-Checker Camera Calibration software. The images were then exported as jpeg files and processed using RTI Builder software. The resulting RTI files can be viewed using RTI Viewer [33], and this software provides interactive rendering of images and allows the user to view and export the normal map image of the object [7]. MATLAB [34] was used to integrate the normal maps to obtain the three-dimensional model of the object to be analysed. The code for the integration of the normal map leading to the construction of the three-dimensional model was written by Quéau [13]. The integration process is very efficient for high spatial frequencies, but not very efficient for low spatial frequencies, as seen in [26]. Therefore, the code [25] was implemented to correct the curvature of the base plane.

2.3 Micro-photogrammetry

Micro-photogrammetry relies on conventional close-range digital photogrammetry principles, enabling the creation of a computable three-dimensional model of a small object with significantly enhanced magnification of its features. The equipment set-up employs the same camera and illumination system used previously in RTI imaging, with the important difference that each photograph must be taken at a different angle of view, allowing triangulation of the object based on pixel location on the image. In addition, due to the shallow depth of field, focus stacking is required to obtain sharp details at all object sizes. A recently released open-source software Structure from Motion (Agisoft Metashape [35]) was used to process the images.

2.4 Laser scanner

The laser scanner is a measuring device that relies on a laser travel time sensor to compute the distance between the device and a surface by measuring the round-trip time of a light pulse emitted from a diode [36]. The instrument used in this study belongs to the Arvedi Laboratory of Non-Invasive Diagnostics of the University of Pavia. The scanning system used employs a RS3 Integrated Scanner, a linear laser scanner with a reported axial resolution up to 30 µm and 46 µm minimum point spacing. The scanner is affixed on a mobile arm, featuring 7 degrees of freedom (Romer Absolute Arm 7-Axis “SI”), both manufactured by Hexagon Metrology (Stockholm, Sweden). The primary software employed to acquire and edit the point cloud and then to obtain the profile of the objects under consideration was the PolyWorks Suite [37].

2.5 Optical profilometer

Optical profilometry measures the topography of a surface with submicrometer vertical resolution by scanning the surface with a white light probe and measuring the reflected light [38]. At the INRiM Institute, the SensoFar high-resolution confocal optical profilometer giving an image with a pixel size of 1.67 µm. The single profile measurement is limited both in the x-y plane and along the z-axis to a corresponding volume of approximately 1 mm3, requiring horizontal and vertical repositioning to extend the measurement to the cover the entire sample.

3 Results

Profilometric analyses were conducted on the white, grey and black reference objects using reflectance transformation imaging (RTI), micro-photogrammetry and laser scanning techniques. At this stage, optical profilometry was applied to assess the maximum resolution on designated targets and served as a benchmark for measuring the actual dimensions of the marks after the printing and modelling process. For generating a three-dimensional model, both RTI and laser scanning methodologies proved effective, whereas micro-photogrammetry did not fulfil the purpose because recognizable patterns were lacking, given the entirely monochromatic nature of the target. With the point cloud obtained, it was possible to measure the height and width of the extrusions and intrusions using Cloud Compare software. Results are shown in the following sections divided by technique and type of target.

3.1 Results obtained by RTI and normal integration

After the photographic acquisitions of all the targets, the RAW images were processed with Adobe Camera Raw, converted to JPEG and subsequently processed with RTI Builder software. The resulting normal map was then processed with MATLAB using the normal map integration algorithm, further corrected for the intrinsic deformations of the process by normalizing the target surface to a plane. The acquisitions and post-processing are extremely fast, allowing for the profilometric reconstruction to be obtained in a few minutes. The ending results are illustrated in Fig. 4.

Fig. 4
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Three-dimensional reconstruction via RTI and normal integration of: a white embossed; b grey embossed; c black embossed; d white debossed; e grey debossed; f black debossed

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3.1.1 RTI and normal integration on embossed target

The data obtained with the RTI and the normal integration of the 3D models of the embossed targets are shown in Table 4. As shown in Fig. 5a, the black target is best represented with the RTI and the normal integration technique, especially with respect to extrusions with a height of up to 0.5 mm. The extrusions with height more than 0.5 mm and all extrusions of the white and grey targets are underestimated. Figure 5b shows that the black target is best represented, also in terms of the width of the extrusions. The width of the extrusions of the white and grey targets tends to be overestimated.

3.1.2 RTI and normal integration on debossed target

As for the debossed targets, the results are very inaccurate. The results obtained for the grey target, which are considered representative, are summarized in Table 5. The depths are all flattened to a maximum of 0.1 mm, while the widths are overestimated but still follow the trend of the theoretical values. This is well illustrated in Fig. 6.

Table 4 Experimental values obtained with the RTI and the normal integration of the heights and widths (mm) of the extrusions for the embossed targets
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Fig. 5
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a Comparison of the heights values obtained for extrusions set 2 (white, grey, black targets); b widths for extrusions set 4 (white, grey, black targets)

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Table 5 Experimental values obtained with the RTI and the normal integration of the depths and widths of the intrusions for the grey debossed targets
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Fig. 6
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Comparison of the values of the depths obtained for the set 1 of intrusions of the grey target and comparison of the values of the widths obtained for the set 2 of intrusions of the grey target

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A first obvious result is that there are many distortions on brighter targets, probably due to the greater scattering of light. In fact, the specular and diffuse components of the reflected light are often difficult to distinguish on grey and white targets. The black target, on the other hand, provides a more accurate representation of the distribution of normal vectors. This trend applies to both embossed and debossed targets, as it is due solely to the colour of the target and the exposure of the photograph. Figure 7 clearly shows that for the embossed targets, the footprints are clearly reproduced, while for the debossed targets, the representation of footprint depths appears distorted. These results clearly show that the RTI method is unable to capture the effective geometry of the surface producing the artefact when a depression deeper than 0.1 mm is present. In these situations, the light cannot create shadows and reflections that are used in normal integration to determine the three-dimensional features of the objects.

Fig. 7
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Pseudocolour image, where the colours represent a relative height scale of: a the debossed target, b the embossed target

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In addition, the height of the object is almost completely absent. The target is flattened almost to the height of the base because, as mentioned earlier, reconstruction of the profile is possible only for semi-three-dimensional objects.

3.2 Results obtained by micro-photogrammetry

It was not possible to obtain results with this technique. The software was unable to process the images to produce sparse or dense clouds, probably because of the excessive homogeneity of the colours and shapes of the object. As shown in Fig. 8, several markers were placed on the target and background to try to correct the homogeneity problem, but this was not helpful in creating the 3D model. These results show the limitations of this technique in creating a 3D model of a monocromatic object with no particular features that can be detected by the structure for estimating three-dimensional structures from two-dimensional image sequences.

Fig. 8
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Photographs acquired for the implementation of photogrammetry

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3.3 Results obtained by laser scanner

3.3.1 Laser scanner on embossed target

The data of the 3D models of the embossed targets obtained with the laser scanner are summarized in Table 6. As shown in Fig. 9a, the heights of the extrusions for the three targets are underestimated. In particular, the 1 and 2 mm high protrusions of the white target are underestimated. The measurements for the grey and black targets follow the trend of the theoretical data. In contrast, Fig. 9b shows that the widths tend to be overestimated, although they are quite well represented for the black target.

Table 6 Experimental values obtained with the laser scanner of the heights and widths (mm) of the extrusions for the embossed targets
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Fig. 9
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a Comparison of the values of the heights obtained for the set 2 of extrusions of the white, grey and black target; b comparison of the values of the widths obtained for the set 4 of extrusions of the white, grey and black target

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3.3.2 Laser scanner on debossed target

The results for the grey target, which is considered representative, are summarized in Table 7. The depth of the intrusions up to 0.3 mm is reproduced quite well, but intrusions beyond that are flattened. Widths are overestimated, but the representation is more accurate compared to results obtained with RTI and normal integration. This is well illustrated in Fig. 10.

Table 7 Experimental values obtained with the laser scanner of the depths and widths of the intrusions for the grey debossed targets
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Fig. 10
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Values of the depths for intrusions set 1 (grey target) and widths for intrusions set 2 (grey target)

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As shown in Fig. 11, this technique provides better accuracy in the representation of the object. In fact, unlike the integration of normals, it is possible to see the full height of the sides of the target objects. As in the case of RTI, it has been shown that the deepest impressions are incorrectly displayed.

Fig. 11
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Three-dimensional reconstruction obtained with laser scanner of: a white embossed target; b grey embossed target; c black embossed target; d white debossed target; e grey debossed target; f black debossed target

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3.4 Results obtained by optical profilometry

The surface profile measurements were taken as shown in Fig. 12 using the confocal optical profilometer and some of the final results are presented in Fig. 13. These measurements required several hours for scanning for two reasons: the first is due to the system’s intrinsically low speed, and the second is the limited scanning range both vertically and horizontally corresponding to a field of view of 1.27 \(\times\) 0.95 mm2, necessitating multiple scans that are then combined to provide the complete surface profile. Although the profilometer is an instrument that potentially offers highly precise measurements, it must be considered that, since light is our probe, narrow and deep elements do not allow for the acquisition of accurate profilometric information. This applies for some intrusions in the debossed targets.

Fig. 12
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Confocal optical profilometry acquisition on a debossed grey target

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Fig. 13
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Three-dimensional reconstruction obtained with optical profilometer of: a on the left extrusion of height = 2 mm and width = 0.3 mm; on the right extrusion of height = 0.3 mm and width = 0.1 mm; b on the left intrusions of depth = 0.5 mm and width = 2 mm; on the right intrusion of depth = 1 mm and width = 0.1 mm

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3.4.1 Optical profilometry on embossed target

The data obtained with the optical profilometry of the embossed targets are summarized in Table 8. It can be seen that the heights are well reproduced, with a slight underestimation.

Table 8 Experimental values of the heights of the extrusions for the embossed targets
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3.4.2 Optical profilometry on debossed target

The results for the debossed targets are summarized in Table 9. In this case, the technique also shows an excellent representation, but with a slight overestimation. Since this is the most accurate and well-studied technique [38], it has been used as the gold standard. It was possible to confirm that the 3D printer used is very powerful, as the theoretical values are very similar to those measured with the profilometer. Images and data were collected only from the most important imprints of the grey embossed and debossed targets, since the technique is particularly time-consuming. Some results of the 3D reproduction of the targets are shown in Fig. 13.

Table 9 Experimental values of the depths of the intrusions for the debossed targets
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4 Discussion

There is, of course, an error in presentation associated with all the profilometric methods considered in this study. For instance, the size of the assessed grey target marks was plotted and is presented in Fig. 14, comparing them with the theoretical values established in the 3D digital model and subsequently replicated through 3D printing. In general, the traditional methods, i.e. laser scanning and optical profilometry, properly represented all elements pertaining to the embossed target. Quantitative analysis of the results revealed that RTI combined with normal integration could replicate heights with an error of less than 10% for extrusions up to 0.5 mm. However, it has significant limitations in representing higher extrusions. Given the number of photographs taken for each object, the intrinsic faults in the highlight acquisition method, often due to the lack of precise homogeneity in illumination with the manually pointed flashlight source (see Fig. 1a), can be considered negligible. This is certainly plausible when the field of view is small compared to the distance from the light source, as in our case. However, if the field of view becomes comparable or larger to the distance from the source, the non-uniformity of illumination becomes significant, but it is possible to correct this effect with appropriate normalizations using reference surfaces.

In [39] notably, reproduction accuracy was affected by the target’s colour, with a growth in error for lighter shades. However, RTI demonstrated excellent resolution for high spatial frequencies, effectively reproducing smaller extrusions. Nonetheless, low spatial frequencies displayed considerable distortion, necessitating correction through a shape-based reference. The laser scanner technique demonstrated better accuracy in reproducing low spatial frequencies, but it generally underestimated the height of all extrusions. This technique also showed an increase in error for brighter targets: the higher extrusions in the white target are poorly represented and show errors and artefacts. As expected, the most positive outcomes for the assessed marks were consistently exhibited by the optical profilometer. Lastly, concerning the debossed target, none of the three techniques succeeded in replicating the depth of the intrusions deeper than 0.3 mm. In most cases, the measurement error was of the order of the intrusions depth. Micro-photogrammetry, on the other hand, did not provide any result.

Fig. 14
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Comparison of the results obtained for the heights of the embossed target with the different techniques

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Fig. 15
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Profile obtained with a optical profilometry and b RTI of a selected element c of the black embossed target (width = 300 µm and height = 300 µm). d Detail of the footprint obtained with the three different techniques

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Fig. 16
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Profile obtained with a optical profilometry and b RTI of a selected element c of the black debossed target (width = 2 mm and depth = 500 µm). d Detail of the footprint obtained with the three different techniques

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Figures 15 and 16 show the 3D representation and the drawn profile of the two footprints highlighted by the red rectangle in Figs. 15c and 16c. In Fig. 14b, the profile of the selected embossed footprint obtained with RTI and normal integration is plotted, so that the distortions caused by this technique can be clearly seen. This is particularly clear when compared to the profile shown in Fig. 15a, which was obtained using optical profilometry. The profile line plotted for RTI profilometric data in the Y-axis shows that the surface irregularities are amplified with high frequencies and the small gap in the centre is clearly visible. The low frequencies, on the other hand, are distorted, which can also be seen for a plot of the data along the X-axis. Looking at the line profile plotted in Fig. 16b from RTI profilometric data, we can see that the edges of the 3D model of the debossed footprint appear smoothed compared to the optical profilometry shown in Fig. 16a. The profile obtained with optical profilometry is almost a step-like line. Although the result could have been predicted considering that the pixel size in the images obtained with the profilometer is an order of magnitude smaller than that of the RTI images, the error introduced in the normal map integration increases the errors in the reconstruction of profile. In fact, even with the same pixel size between the profilometer and RTI, for example, when used in microscopy mode, the latter would still provide a less precise profile reconstruction, especially for sharp edges where the surface normal is difficult to determine as highlighted in the next section.

4.1 RTI and its potential in resolving details

A noteworthy finding was the ability of RTI and the normal integration technique to meticulously capture high spatial frequencies, such as the texture resulting from surface irregularities, including those produced by 3D printing.  These small irregularities and defects, measured to be between 5 and 10 µm using optical profilometry, are shown in Fig. 17. The printer’s footprint was successfully captured using RTI when a high-magnification lens was mounted on the camera. However, as shown in Fig. 18, the 3D model fails to maintain the correct proportions.

Fig. 17
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Reproduction of the texture on the surface of the black embossed target by RTI and normal integration

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Fig. 18
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Footprint of the printer reproduced with a optical profilometry and b RTI and normal integration. c Profile of the normal integration reconstruction

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5 Conclusions

As expected, optical profilometry proved to be the most adept technique in faithfully replicating the profile. However, it comes with the drawback of being time-consuming in measurement and lacking portability. In comparison with the one introduced with the RTI technique, the laser scanner exhibited less deformation, but it requires higher measurement times and demonstrated scarce proficiency in reproducing high spatial frequencies. Micro-photogrammetry requires more discernible patterns to generate a reliable 3D model for the targets. RTI and normal map integration demonstrated outstanding axial resolution, reaching up to 5 µm. This method performs well for elements with heights up to 5 µm, where the difference of measurement between real and theoretical values does not exceed 10% of the total height. For what concerns, the widths the best performance seems to be for footprints with a width of 3 µm to 1 mm, where again the difference of measurement between real and theoretical values does not exceed 10% of the total width. Nevertheless, it exhibited substantial inaccuracies in representing low spatial frequencies, which is easily fixed by applying a reference hump during post-processing. Since the RTI and normal integration give result with relative dimension, it is always needed a proper scaling of the size of the object considered. It is an economically efficient methodology, necessitating solely a camera and a flashlight. Approximately ten minutes is required for each measurement. It is a portable technique for on-site assessments, and the subsequent post-processing is rapid, establishing RTI as an exceptional approach for evaluating and representing the surfaces of cultural heritage objects. The intrinsic errors contained in the acquisition method, in many cases realized without a precise uniformity in illumination with the flashlight source manually pointed, are negligible with the number of images shot in the cases considered. Though the accuracy of the reproduction is influenced by the colour of the target, it often involves the need for calibration and flattening [25, 26].