Fractal and statistical characterization of brushstroke on paintings

Abstract

Identification of an individual artist’s touch on paintings is studied using surface metrology. Paintings’ topographies were measured using focus variation and stitching, creating 13 x 13 mm maps with 1 µm sampling intervals, and 169 megapixels, with a 10X objective lens. Topographic characterization parameters were analyzed for their ability to differentiate different painters’ renderings. Statistical treatments from data mining were used to discriminate, by optimization, multiscale topographic signatures characterized by a multitude of areal texture parameters. It appears that a fractal dimension can define 3 characteristic scale ranges. One from 3 to 70 µm corresponds to brushstroke details. Another, from 70 to 700 µm, corresponds to the topography of the material of the canvas fabric. Finally, scales greater than 700 µm correspond to undulations of the canvas. For scales less than 50 µm, the fractal structure of the topography left by brushstrokes follows a power law characterized by the slopes of the topography. The topography of the clouds painted on the canvas has an Sdq (topographic slopes) increasing with the clarity of the clouds at scales of 3-500 µm. According to the Torrance-Sparrow theory, the higher the Sdq, the more diffuse the light on the surface. The painter therefore wanted to show, by his brushstroke, that the light clouds diffuse more light giving an impression of local brightness. This study is confirmed by the analysis of the painting of Max Savy, a French painter from Carcassonne (1918-2009), which was measured with a white light interferometer Zygo NewView 7300, a X100 objective lens giving a 517 µm x 517 µm stitched surface, with a sampling interval of 0.109 µm. The box-counting method for estimating the fractal dimension of the topography of an oil painting appears optimal by the fact that it morphologically integrates scale variations of the local slopes of the surface morphology. This method thus characterizes the multiscale aspects, as well as the scale changes, of the topography.

A New RTI Portable Instrument for Surface Morphological Characterization

“…Moreover, as the brushstrokes of the sea are coarser than the sky ones, their curvatures are lower (Figure 12b). This result is contrary to the results obtained by Bigerelle et al [38]: Sdq is higher for the sea part. This can be explained by the fact that a high-resolution white light interferometer with a 100× lens was used to measure 100 zones of 517 × 517 µm.…”

“…MorphoLight is therefore used to measure topographies and curvatures from the re flectance method. Bigerelle et al [38] highlighted the idea of the painter to reflect the sky on the sea. The objective is therefore to show the ability of MorphoLight and its morpho logical analysis method, based on standards from metrology, to topographically identify the signatures of the painter and to understand how the investigated sea part represent the reflection of the sky (same color palette, close roughness but a slightly different paint ing technique allowing a different visual rendering).…”

“…These parameters are calculated according to ISO and EUR standards from metrology. SDQ is especially calculated as Bigerelle et al [38] highlighted the relevance of this parameter to characterize the brushstrokes of both parts (sky and sea) of this painting using a multiscale analysis from topographical data obtained by white light interferometry (100× lens). Sum of the mean height of the five highest peaks and the mean height of the five deepest valleys 1 A summit is a surface point whose height is higher than its eight nearest neighbors (including peaks).…”

Fractal contours: Order, chaos, and art

“…Since this early work, many other authors explored and further developed similar approaches leading to a singular fractal dimension characterizing Pollock's abstractions 11,12 and paintings by other artists. 7,8,[13][14][15][16] Other methods for pattern recognition and categorizing images are based on contour curvature statistics, 17 topological invariants, 18 detrended fluctuation analysis, and multifractal index, 19 various measures of complexity such as Hurst exponent, 20 various measures of self-similarity, 21 information entropy, 14,22 and hidden Markov models, 23 to name a few approaches. Numerous attempts to use such singular parameters to date artworks or at least assign them to a particular epoch or style have been undertaken.…”

Optical surface metrology for heritage science: proof of concept and critical-constructive discussion

“…[6][7][8][9][10] Among common 3D profilometry techniques are laser triangulation, 11 structured light projection, 12 focus variation microscopy, 13 white light interferometry. 14 In this work, surface data are obtained by laser profilometry based on conoscopic holography interferometric sensors. 15 This technique is well accepted in the heritage network, allowing versatile applications from archaeology 16 to paintings, 9,17 and more recently to book heritage.…”