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Fractals are geometric shapes that can display complex and self-similar patterns found in nature (e.g., clouds and plants). Recent works in visual recognition have leveraged this property to create random fractal images for model pre-training. In this paper, we study the inverse problem --- given a target image (not necessarily a fractal), we aim to generate a fractal image that looks like it. We propose a novel approach that learns the parameters underlying a fractal image via gradient descent. We show that our approach can find fractal parameters of high visual quality and be compatible with different loss functions, opening up several potentials, e.g., learning fractals for downstream tasks, scientific understanding, etc.
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Forensic science and fractal nature analysis
Forensic photography, also referred to as crime scene photography, is an activity that records the initial appearance of the crime scene and physical evidence in order to provide a permanent record for the court. Nowadays, we cannot imagine a crime scene investigation without photographic evidence. Crime or accident scene photographs can often be reanalyzed in cold cases or when the images need to be enlarged to show critical details. Fractals are rough or fragmented geometric shapes that can be subdivided into parts, each of which is a reduced copy of the whole. Fractal dimension (FD) is an important fractal geometry feature. There are many applications of fractals in various forensic fields, including image processing, image analysis, texture segmentation, shape classification, and identifying the image features such as roughness and smoothness of an image. Fractal analysis is applicable in forensic archeology and paleontology, as well. The damaged image can be reviewed, analyzed, and reconstructed by fractal nature analysis.
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- PAR ID:
- 10327782
- Date Published:
- Journal Name:
- Modern Physics Letters B
- Volume:
- 35
- Issue:
- 32
- ISSN:
- 0217-9849
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Over the recent decades, a variety of indices, such as the fractal dimension, Hurst exponent, or Betti numbers, have been used to characterize structural or topological properties of art via a singular parameter, which could then help to classify artworks. A single fractal dimension, in particular, has been commonly interpreted as characteristic of the entire image, such as an abstract painting, whether binary, gray-scale, or in color, and whether self-similar or not. There is now ample evidence, however, that fractal exponents obtained using the standard box-counting are strongly dependent on the details of the method adopted, and on fitting straight lines to the entire scaling plots, which are typically nonlinear. Here, we propose a more discriminating approach with the aim of obtaining robust scaling plots and extracting relevant information encoded in them without any fitting routines. To this goal, we carefully average over all possible grid locations at each scale, rendering scaling plots independent of any particular choice of grids and, crucially, of the orientation of images. We then calculate the derivatives of the scaling plots, so that an image is described by a continuous function, its fractal contour, rather than a single scaling exponent valid over a limited range of scales. We test this method on synthetic examples, ordered and random, then on images of algorithmically defined fractals, and finally, examine selected abstract paintings and prints by acknowledged masters of modern art.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1142/S0217984921504935
