ApplicationsofWaveletTransformsinStylisticArtAnalysisJananiChinnam,DavidSchmidt,MatthewSims,AlyssaSopanarat

IntroductionA wavelet is a wave-like oscillation applied in computational image compression andthe digital technology of art identification. The Haar wavelet is a sequence of rescaledfunctions which form a basis, with analysis similar to Fourier analysis in that it allowsa target function over an interval to be represented with an orthonormal basis. Wecontrast the two analysis computations to display the advantages of a wavelettransform for this context. In the visual art industry, statistics produced from thewavelet decomposition provide a digital signature unique to artists that assist inauthentication and stylistic analysis through pattern recognition.

Abstract

Methods

Applications

Wavelet transform computations can be utilized to assist in the stylistic analysis andauthentication of artworks. In the analysis of paintings, the Haar Wavelet Transformperforms at a higher precision than many other computational methods such asFourier transforms, as shown in our computations. Our results display the advantagesof wavelet applications in authentication and forgery analysis in the art industry.

Determining an artist’s style based solely on technical analysis of materials and visualinspection often proves inconclusive results. Wavelet transforms and computational toolsof machine learning provide an additional form of insight, on the assumption that thebrush strokes can be characterized by an artist’s signature movements. These transformsseparate the details of an image into different scales to capture local differences. Thedecomposition allows for the extraction of high resolution differences not perceivable invisual inspection by the human eye (Jafapour).

Authenticating a painting consists of four sequential phases: image acquisition,segmentation, feature extraction, and classification. The segmentation steps selects areasfrom the painting where it is expected to find the author’s identity or signature,specifically locations with low hue variation. The texture of these areas’ brushstrokes aremeasured on the intensity component using a wavelet based approach. Texture attributesare obtained by applying a pyramidal algorithm recursively to compute variance,standard deviation, entropy for horizontal, vertical, and diagonal texture characteristics.The texture of a block of the image is then represented by a ten-dimensional texturevector, and the vector for a single brushstroke will be given by the average of its blockvectors. The next step is assigning a class to each brushstroke, which is then combinedto produce a classification result for the entire painting (Tiexeira).

Analysis Algorithm

Pattern Recognition

Forgery DetectionThe decomposition of the wavelet transform produceslocal orientated spatial frequency data that captures theunique signature of an artist’s brushstrokes. Similar tohandwriting analysis, this process characterizes ‘aestheticsignatures’ within the style of the artist (Polatkan). Subtledifferences in these styles can reveal a forgery. The vectorof coefficients produced from the wavelet transform,along with error statistics for each matrix (subimage).Authentic works will visually resemble a ‘point cloud’,where all works by an artist will reside close together,while imitations will be relatively far, as shown in Figure6. This is called the K-Nearest Neighbor (KNN) algorithm,which measures the distance between the patterns of anunknown class and every training pattern (Teixeira).Computing the distances between pairs of works will resultin a distance matrix that reveals forgeries where thedistances fall outside the range of authentic works (Lyu).

ResultsandConclusion

ReferencesCabeen, Ken, and Peter Gent. “Image Compression and the Discrete Cosine Transform.” College of theRedwoods.“Haar Wavelet Image Compression.” Ohio State University.Jafapour, Sina, et al. “Stylistic Analysis of Paintings Using Wavelets and Machine Learning.” Princeton University.Lyu, Siwei, et al. “A Digital Technique for Art Authentication.” Proceedings of the National Academy of Sciences ofthe United States of America, National Academy of Sciences, 7 Dec. 2004.Lyu, Siwei, et al. “Wavelet Analysis for Authentication.” Wavelet Analysis for Authentication, Dartmouth College,2005.Marcus, Matt. “JPEG Image Compression.” Dartmouth College, 1 June 2014.Polatkan, Gungor, et al.“Detection of Forgery in Paintings Using Supervised Learning.” Duke University, Mar. 2009.Stork, David G. “Computer Image Analysis of Paintings and Drawings: An Introduction to the Literature.”Teixeira, Guilherme N., et al. “Pattern Recognition Applied in Fine Art Authentification.” Catholic University of RioDe Janeiro.

Computing a full digital signature of an artist for any given piece of art is acomputationally extensive procedure; however, Haar transform and Fourier transformcomputations can be exemplified in a simpler image compression application. In thisproject, an artwork by Van Gogh was compressed, using the Haar Wavelet Transformand the Discrete Cosine Transform, a form of Fourier transform. In order to quicklytransmit images, the data of an image is frequently compressed from an original pixel-by-pixel matrix to a simpler form that still optimally conveys the features of theoriginal image (Marcus).

Image CompressionIn this project, the Haar Wavelet Transform was used for image compression. In thisproject, the image was divided up into 8x8 pixel matrices and each one wastransformed separately to make computations simpler. The Haar Wavelet Transformfocuses on taking the average of two values next to each other in a row along with thehalf of the difference of the two values (Cabeen and Gent). This first transformation isalso done to the columns of the image matrix. Once the first transformation isperformed, the averages are grouped together, and the averages of these values arecomputed along with half of the differences of the pairs. Lastly, the average of thesecond transformation’s resulting values (only two values) is found along with halfthe difference of these two values. These three transformation matrices can bemultiplied together to get the transformation matrix W.

As evidenced by Figures 3-5 above, the Haar Wavelet Transform is widely regarded as asubstantially more precise method of image compression than the DCT. The imagecompressed through the DCT is incredibly ‘blocky’: most of the 8x8 pixel blocks can beclearly discerned from one another. On the other hand, the Haar transform compressionproduced a very smooth image. The difference in quality stems from the fact that thediscrete cosine transform, which approximates pixel color values through sums of smoothcosine functions, is unable to adequately represent sharp contrasts in color; on the otherhand, since Haar transforms fit to pixel data rather than fixing pixel data to a specifiedcurve, these transforms produce clear images. Additionally, the DCT computation favorslow-wavelength colors; to produce images with colors that reflect sums of cosine curves,high-frequency pixels are often dimmed to create a more homogeneous image. Thesecomputations and contrast exemplify the smooth precision of the wavelet transform andthe advantages it provides in stylistic analysis.

Figure 3: Original Image

Figure 4: Haar Wavelet Compression

Figure 5: Discrete Cosine Compression

The Discrete Cosine Transform largely follows the same procedure, as it deals with8x8 pixel blocks. The main idea behind the Discrete Cosine Transform is that an 8X8image’s grayscale values can be compressed by merely adding function values along astandard cosine function (Cabeen and Gent). The transformation matrix, U, containssamples of the standard cosine function in multiples of pi/16. The values of this matrix’stop row are a constant, while the remaining rows (i = 1 to i=7) represent the cosinevalues of the 8 points that evenly divide the function when its length is restricted to pi*i(Cabeen and Gent). All matrix values are halved so that the two cosine function valuescombine to produce an average, not a summation of the values.

Figure 1: Haar Wavelet Transform Matrix W

Figure 2: Discrete Cosine Transform Matrix U

Image processing in the fine arts field allows quantifiable analysis of paintings.Algorithms for digital analysis provide art theorists details on a painting’s features, suchas brushstrokes and texture, which can be used for further classification on identificationor forgery. The wavelets applied to paintings are also useful in image compression andcan make it simple to store an image with large amounts of data by determining whichpieces of data are important. Using wavelets for image compression results in lessquality loss than other image compression techniques, such as using a discrete cosinetransform. Wavelets are utilized in image analysis and the art industry due to their strongability to recognize abrupt changes in wave patterns and provide statistical data.

Wavelets are short duration waves with zero mean value, and they can be scaled andtranslated to break two-dimensional images into different scales to obtain information onthe image. Wavelets enable the objective analysis of artwork by quantifying informationregarding orientation and density of linear elements such as brushstrokes (Lyu). Theprocess involves first dividing images into matrices (with dimensions of multiples of 8)and performing the transform to measure textures and using it as feature to describe thebrushstrokes . The complex coefficients of each matrix are computed, along withwavelet norms (Jafapour). This decomposition changes the basis of the digital image toone where the functions are localized in space, orientation, and scale (Lyu). Theseextracted statistics provide a digital signature that can then be classified .The Haar Wavelet Transformation Matrix (H) and the Discrete Cosine Transformation

Matrix (U) are shown in Figures 1 and 2, respectively. Note that W should be made intoan orthogonal matrix H in order to make the transformation computations simpler. Thegoal of the transformations HAHT and UAUT is to get the original image into a form thatwill allow repetitive data to be removed (get repetitive values close to 0 in magnitude).The maximum values of these transformation matrices are determined, and then dividedby the maximum values respectively. In order to remove these repetitive values, abenchmark value (α) that is between 0 and 1 must be assigned, and it determines the levelto which values close to 0 are removed. (Cabeen and Gent) For example, if α= 0.1 and theabsolute value of an entry in the transformation is <0.1, this value will be eliminated. Asα decreases, the number of deletions for a given image will decrease, resulting in aclearer,yet less compressed, image. Once values are tested in comparison to α, the imagemust be converted back into a standard image format through the followingmultiplication HT(HAHT)H and UT(UAUT)U (Marcphaus). This is performed on eachcolor channel (red, green, and blue), and then these matrices are recombined to form thecompressed image. The amount of image compression performed can be estimated bytaking one minus the quotient of the compressed file size and the uncompressed file size.Image compression of 92% for each type of transformation is presented below along withthe original image: