Chemometrics and Intelligent Laboratory Systems 108 (2011) 2–12

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Chemometrics and Intelligent Laboratory Systems

j ourna l homepage: www.e lsev ie r.com/ locate /chemolab

Extracting useful information from images

Sergey KucheryavskiAalborg University Esbjerg, ACABS research group, Niels Bohrs vej, 8, 6700 Esbjerg, Denmark

E-mail address: svkucheryavski@gmail.com.

0169-7439/$ – see front matter © 2010 Elsevier B.V. Adoi:10.1016/j.chemolab.2010.12.002

a b s t r a c t

a r t i c l e i n f o

Article history:Received 5 October 2010Received in revised form 27 November 2010Accepted 14 December 2010Available online 22 December 2010

Keywords:Image processingImage analysisTexturesFractal analysisAngle measure techniqueWaveletsImage morphology

The paper presents an overview of methods for extracting useful information from digital images. It coversvarious approaches that utilized different properties of images, like intensity distribution, spatial frequencycontent and several others. A few case studies including isotropic and heterogeneous, congruent and non-congruent images are used to illustrate how the described methods work and to compare some of them.

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© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Image analysis is used widely and successfully in many areas ofscience and industry. It is an essential and non-invasive tool forgetting quantitative information about structure of investigatedobjects as well as for establishing a link between this informationand various non-visual properties of the objects.

Any image contains a large amount of data. Just an ordinary colorpicture taken with a chip digital camera consists of several millionpixels with three numeric values for each — color channels. Most ofthe basic mathematical methods and algorithms for processing andanalysis of digital images were developed in 70–80th of the lastcentury when first computers became available. Nowadays themodern methods allow to solve quite sophisticated problems, likerecognition of face and emotions, analysis of moving objects andmany others. Image analysis also has found its success in industry, forexample for analysis of particles, for evaluation of homogeneity ofmixings and in many other applications. The non-invasivity, simplic-ity and high speed of data acquisition procedure as well as relativelylow price of the acquisition hardware make this approach quiteattractive for analysis of industrial processes.

When someone mentions image analysis and chemometrics at thesame time, in most of the cases thematter is MIA –Multivariate ImageAnalysis – an efficient and popular tool for processing and analysis ofmultichannel and hyperspectral images, invented by Geladi et al. [1].Multivariate image analysis treats image pixels as objects and thecorresponding color channels (or wavelength in case of multichannel

and hyperspectral images) as variables. Being applied to a hyper-spectral image it allows to clusterize it, find outliers and extremepixels, and make classification and regression models for discoveringparts with certain properties (for example evaluating quality of beefsamples [2], detection disease marks in body tissues [3], estimation ofhardness of maize kernels [4], and much more).

However a lot of information can be extracted even from simplegrayscale images. Such images can be acquired by a camera with CCDor CMOS sensor, transformed from color or hyperspectral images (likescore image, for example) or be a result of a simulation process. It isalso possible to process color images by analyzing their color channelsseparately, so each channel is considered as a grayscale image. In anycase all these images are represented by a matrix with intensityvalues.

This paper describes methods and algorithms for extracting usefulinformation from grayscale images. The analytical procedure for suchimages is pretty straightforward and includes several steps besidesthe image acquisition.

The first step is image processing. It usually consists of twosubsteps— image enhancement and post-processing. Image enhance-ment aims to improve quality of an acquired image. Usually it includescorrection of lightning conditions (brightness and contrast) andgeometrical distortions if any as well as noise reduction. Imageenhancement methods are very well described in various textbookson image processing (for example [5,6]) and will be barely touchedhere.

Post-processing is mainly about how to prepare images for furtheranalysis. The most used post-processing technique is segmentation —

selection of image parts that are most important for the analysis. Inthe simplest case, segmentation just removes background pixels, but

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there are also cases where implementing of complex segmentationalgorithms is necessary (for example detection of cells withpredefined properties: color, shape and so on).

The second step is devoted to extracting features from images andthis paper mainly focuses on this one. This step implies estimation ofvarious quantitative properties of an image, like, for example, spatialfrequencies content, roughness, geometrical properties of depictedobjects and many others. The extracted features can be used as a finalresult, like, for example, distribution of particle size, or treated as aninput for further analysis, including regression and classification.

This paper reviews this common approach for extracting relevantinformation from images. Several feature extraction methods arediscussed. Most of these methods can be used widely for differentpurposes — since they are quite universal. Most of them also extractmultivariate information from images and can be naturally integratedwith chemometrics methods for data analysis. Each method isdescribed briefly using one or two study cases and the description issupplemented with information and references about where and howthe particular method has been used. It is important to underline thatthe paper does not claim to be exhaustive — the amount of differenttechniques for image features extracting is way larger. However theauthor hopes that the selected methods will give a decent illustrationof main principles for extracting information from images.

2. Case studies

To show how the chosen methods work and to compare some ofthem, four case studies will be used, with both isotropic and non-isotropic images.

2.1. Segmentation of biochips

Biochip is a rectangular substrate with a number of chemicalmicroreactors — typically small spots of gel with different reagentsinside. If one deposits a solution with another reagent (conjugatedwith fluorescent label) over a chip surface, reactions will happen insome cells and further, having a proper post-processing, it is possibleto see these cells as light spots on the chip. Biochips allow to analyzereactions among hundreds of reagents at the same time and are usedwidely, for example, in medicine for discovering different mutationsof a disease.

For automatic analysis a biochip image has to be segmented toindividual cells. That could be done easily since the cells are orderedon a chip. However due to different reasons, a segmented cell can beshifted on a resulted image. In Fig. 1 a result of such segmentationwith some correctly and incorrectly segmented cells is shown. The

Fig. 1. Digitized image of segmented biochip and exam

main problem in this case study is to find how the segmentationerrors can be corrected.

2.2. Recognition of plastic particle concentrations

This case study is mostly devoted to illustrate how to extract anduse textural features. Texture recognition and analysis is a largeseparate part of image analysis. Textures have no exact definitions butcan be considered as images that contain some stochastically orregularly repeated patterns. Most of stochastic textures are isotropicand invariant to rotations.

In this case study 32 images of dark and light plastic particles,mixed in different proportions: 10:1 (10 volumes of dark and 1volume of light particles), 10:2, 10:3 and 10:4 were prepared as anexample of stochastic textures. The particles had the same size andshape. For each of the four cases eight pictures were taken. Themixtures were shaken for about 1 min before each shot. Finally theimages were cropped and downsampled to the resolution of512×512 pixels.

Examples of the images from each set are shown in Fig. 2. Themaintask for this case is to discriminate images with different concentra-tions of white pellets. It can be noticed that the last two sets givealmost similar images, so some problems can be expected.

2.3. Morphological analysis of pharmaceutical pellets

Analysis of morphological properties of particles is also a quitecommon problem. It can be used, for instance, in microscopy forrecognition and analysis of different cells and bacteria, for monitoringprocesses where particles are involved (like controlling of size andshape of pellets) and in many other areas.

In this case study an image with about 1000 small pellets ofsaccharine with coated layer of paracetamol will be used to showmain steps of morphological analysis. The pellets have physical sizefrom 30 to 40 μm. The original image has a resolution of2048×2048 pixels. The part of the image is shown in Fig. 3.

2.4. Classification of brain tomograms

In this case study a set of images corresponds to one sample. Theimages are scans of magnetic resonance imaging (MRI) tomograms ofhuman brains with early stage of Alzheimer disease (AD) and withoutit. So features must be calculated for all images from a particular setand combined to a vector.

The MRI images were taken from OASIS public database (http://www.oasis-brains.org). This dataset is a part of project “Open Access

ples of correctly and incorrectly segmented cells.

Fig. 2. Images of dark and light plastic particles mixed with different proportions: a) 10:1; b) 10:2; c) 10:3; and d) 10:4.

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Series of Imaging Studies (OASIS)” aimed at making magneticresonance imaging (MRI) data sets freely available to the scientificcommunity.

All images have been received from patients in the age from 18 to95 years and divided into several groups depending on the stage ofthe disease. In this case study 40 MRI samples were used, 20 frompatients without AD and 20 from patients with early stage of thedisease. Each sample consists of 175 sagittal scans of a brain. Everyscan has a spatial resolution of 256×178 pixels. In Fig. 4 differentscans of the same brain are shown as an example.

The main objective for this study case is to find features that willallow to distinguish patients with and without Alzheimer disease byMRI images.

3. Methods for extracting image features

So, how to extract useful information from images? The interestingthing is that within some constrains there is no need to extract anyspecial features at all. The intensity values and its spatial distributionon an image are quite informative themselves. This is true, however,mostly for congruent images — where position of each pixel is

Fig. 3. Fragment of image of saccharine pellets with coated paracetamol layer.

important, since it describes the quality of the same part of thedepicted object [7] on all images.

Human faces are one of the good examples of congruent images,but only if they are cropped in the same way and resampled, so theimages with different faces have the same size. For such images,position and properties (like intensity or color) of each pixel areimportant since they reflect individual features of the face (e.g.distance between eyes, shape of leaps, etc.). This is utilized in a well-known face recognition method — eigenfaces — that uses a sort ofSIMCA algorithm for recognition and classification [8].

Another instance of congruent images is cells of biochips,described in the previous chapter (case study one). The problemwith incorrect segmentation of biochip cells can be solved using aquite straightforward and efficientmethod, based on principal compo-nent analysis (PCA), which has been proposed by A. Pomerantsev andO. Rodionova [9]. The main idea was to unfold the segmented imagesto one-dimensional profiles by concatenating their rows and toanalyze the shape of the profiles. In Fig. 5 several unfolded profiles

Fig. 4. Selected scans of brain MRI tomography image.

Fig. 5. Profiles of correctly and incorrectly segmented biochip cells.

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from different cells are shown. It is obvious that the correctlysegmented cells (labeled as “good” on the plot) have profiles withalmost normally distributed peaks (high peaks at the middle,decreasing to the both tails) whereas the profiles for incorrectlysegmented cells behave not so smoothly.

By applying PCA to the profiles and analyzing the correspondingscore plots simultaneously with the cell images, a solution has beenfound. Thus first principal component explained an average bright-ness of the cell, as shown in Fig. 6a. The second principal componentwas responsible for cell shifting in the direction NW–SE (maindiagonal) and the third component— in the direction W–SE (Fig. 6b).The correctly classified cells lie very close to the center of thecoordinates on the PC2 vs. PC3 score plot. So the score values from the

Fig. 6. PCA results for unfolded cell images: a) PC1 score values vs. average intensity;b) PC2 vs. PC3 score plot.

second and third components can be straightly used for segmentationcorrection.

There aremany other examples where intensity values of an imageare used directly for analysis or classification. However in many cases,even for congruent images, extracting specific integral or localfeatures allows to analyze them more efficiently. It is especiallyimportant for the images where objects are distributed stochastically,like textures. Depending on the particular case features can reflectprobabilistic properties of intensity distribution (intensity histogramand its statistics, statistics of gray level co-occurrence matrix)distribution of spatial frequencies on an image (like Fourier andwavelet transform), complexity of image on different scales (e.g.angle measure technique and fractal analysis), morphological prop-erties of depicted objects andmany others. In this chapter some of thementioned methods will be discussed.

3.1. Image statistics

Statistical analysis of intensity distribution is one of the simplestways to get some quantitative information about images. The mainidea is to consider an image as a result of statistical process, so eachintensity value has some probability. Therefore the objective is toestimate a probability density function and to calculate its properties,like minimum and maximum values, statistical moments (includingaverage and standard deviation), skewness, kurtosis, entropy and soon.

The main object for investigation here is intensity histogram h(a)that shows how often a pixel with intensity a appears on an image.Intensity histogram is used widely in image processing for automaticcorrection of brightness and contrast of images, since it shows theinformation about range of used intensity levels and how frequenteach level appears. It is also quite usable for segmentation purposes insome special cases.

In order to calculate statistics all h values have to be normalized bydivision to a total number of pixels of an image. This gives anapproximation of probability density function — p. Then for a given p,the ordinary mk and central uk moments of order k can be calculatedas follows:

mk = ∑L−1

l=0l kpl ð1Þ

uk = ∑L−1

l=0l−m1ð Þkpl: ð2Þ

Here L is a number of intensity levels (256 for ordinary grayscaleimage) and m1 is the first ordinary moment — average value. Thesecond central moment, u2, is variance of intensity, so the standarddeviation can be computed as σ =

ffiffiffiffiffiu2

p.

Central and ordinary moments have the following relations:

u2 = m2−m21

u3 = m3−3m1m2 + 2m21

u4 = m4−4m1m3 + 6m21m2−3m4

1 :

ð3Þ

Using the moments one can calculate an asymmetry coefficient, g1,showing skewness of the distribution (if the coefficient is 0 thedistribution is symmetrical relative to mean value), and excesscoefficient, g2, that shows the kurtosis (0 for normal distribution, N0if the distribution is more narrow, and b0 if the distribution is wider):

g1 =u3

σ 3 ð4Þ

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g2 =u4

σ 4 −3: ð5Þ

Another useful statistic is entropy, E; it has its maximum if thedistribution is uniform:

E = − ∑L−1

l=0pllog2 plð Þ: ð6Þ

The intensity distribution statistics as well as histogram itself canbe quite efficient when used as features for image classification,especially when any changes in color or brightness are important todiscriminate samples from different classes. Case study two (imageswith plastic pellets) is one of such cases, since the concentration ofwhite pellets has a direct influence on intensity distribution. In Fig. 7the score (left) and loading (right) plots are shown for two cases —

where image histogram (top) and several intensity distributionstatistics (bottom) were used as predictors. As we can see in bothcases a decent separation between classes is achieved. In the case ofstatistics the first principal component describes the concentrationwhereas for histograms the concentration is described by linearcombination of first and second components.

It is obvious that statistical features depend on lighting conditionsvery much. This must be taken into account when acquiring images—constant lighting is a must condition when taking images from thesame study case. It is also important to say that image statistics areintegral features and do not depend on shapes or other geometricalproperties presented on an image. If 50% of pixels are black and 50%are white, it does not matter how they are distributed on an imageplane — as a chessboard or randomly — statistically these two imagesare equivalent.

Fig. 7. PCA results for plastic particles images using intensity histogram

3.2. Morphological analysis of particles

Particle analysis implies first of all determination of morphologicalproperties, such as area, perimeter, center of mass, roundness, fittingellipses or rectangles, and many others. Particle here means a set ofconnected pixels that shared some common properties. It could be areal particle or a cell as well as any part of image that corresponds tothis definition.

The morphological properties are useful themselves and often theonly information needed. However they can also be used for furtheranalysis, like recognition and classification of different types ofparticles for instance [10].

First of all, particles should be segmented from the rest part ofimage. As a result of segmentation a binary image is obtained whereall particles' pixels have a value of 1 and all others are set to 0. Thereare many segmentation algorithms available [5]; their description isbeyond the scope of this paper. However in some simple cases whereparticles just have to be separated from background the most trivialapproach, based on manual or automatic threshold gives acceptableresults.

Sometimes segmentation also needs an additional step to disjointagglomerated particles. It can be done with the help of binarymorphology methods, like erosion and opening [6], or with quiteefficient algorithm of watershed segmentation [11]. The last one,however, is effective only for rounded or ellipsoid particles.

In Fig. 8 an example that includes all these steps is shown. Theoriginal image with pellets (a) was binarized with automatic thresh-olding (b), then watershed algorithm was applied to disjoint stackedpellets (c). Finally all objects that touching image borders weredeleted (d).

As soon as image is binarized and all particles are disjoined theanalysis is pretty straightforward. Thus perimeter can be found asnumber of object pixels that have at least one background pixel

as features (top) and statistics of intensity distribution (bottom).

Fig. 8. Segmentation of a pellets: a) original image; b) binarization with threshold; c) disjoint pellets with watershed algorithm; d) removing border pellets.

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connected. Center of mass can be calculated by averaging x and ycoordinates of all particle pixels (first moment). Area can becalculated directly by counting number of pixels for each particle.Thus Fig. 9 shows size distribution of pellets from the case study 3image. Many other properties, such as for instance roundness andorientation can be found from bounded ellipse and rectangleparameters.

3.3. Wavelet transformation

Wavelets are functions that satisfy a set of requirements; inparticular, they have a good localization both in frequency and spatial(time) domain [12]. Examples of some wavelets are shown in Fig. 10.Wavelet transformation (WT) implies decomposing of a signal using aset of scaled and shifted versions of a particular wavelet function as abasis. This transformation not only gives a signal frequency content(as Fourier transform does as well) but also shows in which parts ofthe signal one or another frequency dominates. For images it means,first of all, an ability to underline edges and contours of differentorientations and to distinguish objects with various sizes andstructures.

Currently wavelet transformation is widely used in signal proces-sing, mostly for denoising and compression of various signals [13](including images, thus relatively new image format JPEG 2000 useswavelet based method for image compression [14]; another exampleis a file format DjVu for storing scanned documents [15]). However, inthe case of image analysis, WT may also be very useful as a featureextracting method. Thus wavelet texture analysis — an approach forclassification and recognition of different textures (as well as anyother images with isotropic structure)— is one of the most developedapplications of wavelet analysis.

It is known, that wavelet transformation of a digital signal can becarried out as a linear filtering procedure with two filters — H: {h1,h2, …, hn} (low-pass) and G: {g1, g2, …, gn} (high-pass) [13]. Filter Hsmoothes the signal whereas filter G gives details — difference

Fig. 9. Distribution of pellet area size.

between the original signal and its approximation given by H. Byapplying these filters to a raw signal y the first level of transformationcan be found:

s1n = Hy = ∑k ynh2n−k

d1n = Gy = ∑k yng2n−k :ð7Þ

Here {sn} — the approximation (smoothed version) of the signal yand {dn} — the details. Finally s and d are also downsampled by two(because half of the frequencies were removed), so their length is halfof the original signal's length.

By repeating this procedure successively to all approximations sthe full transformation, consisted of the details from different levels,can be obtained. Thus the first level shows information about highestfrequencies content whereas the last levels give information aboutpresence and localization of low frequencies in the signal. Coefficientsof the filters are calculated depending on a chosen wavelet function.

To performwavelet transformation of a digital image the filters areapplied successively to image rows (r) and columns (c), as shownschematically in Fig. 11. Therefore for each level, the transformationprocedure produces three types of details: horizontal (HcGr), vertical(GcHr) and diagonal (GrGc). The smoothed version of image isobtained by applying filter H first to rows and then to columns ofthe image. The example of the two steps applied to an image withdark and white plastic particles is also shown in Fig. 11.

So wavelet transformation details give the information aboutspatial frequencies presented in an image, their amplitude, localiza-tion and orientation. How can this information be used to calculatequantitative image features for further analysis or modeling? One ofthe most usable and straightforward approaches, which is alsoutilized in texture analysis, is to calculate statistics of the details'coefficients from different levels. Thus using just three statistics (forexample, mean value, energy and entropy) with three types of detailsfrom first five transformation levels gives a vector with 45 imagefeatures.

In order to show how this approach works let us apply it to plasticparticle images from case study two. For each of the images a vectorwith 36 wavelet features was calculated. To do that, first, the wavelettransformation was performed (two different wavelet functions —

Haar and Coiflet of order five, coif5, were applied) and three types ofdetails from first six levels of transformation were stored. That gave6×3=18 matrices of details' coefficients. For each of the matrices,two statistics were computed: energy (sum of squares) and entropy.

On the final step thematrix of featureswas analyzedwith principalcomponent analysis (PCA), the corresponding score and loading plotsfor the first two components are shown in Fig. 12. First of all it shall benoticed that the score plots give quite decent separation between thegroups of images. However the choice of proper wavelet function isimportant. Thus the simplest Haar wavelet gives better separationbetween two last classes, than the function from the Coiflets family.

Fig. 10. Examples of wavelet functions: a) Haar, b) Coifflet, c) Morlet, and d) Daubechies.

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The joint analysis of score and loading plots shows that both theenergy and entropy are important for discrimination, however itsinfluence depends on the wavelet used.

The simplicity and fast calculation speed inherent in the describedapproach as well as efficiency for texture recognition allow it to bespreadwidely as for analysis of textures and for other image analyticaltasks as well. Thus in [16], Barati, Liu and MacGregor applied severalmethods for textural features extraction to analysis of quality of rolledsteel sheets and found out wavelet based method as the mosteffective. The paper is also giving a comprehensive overview of mostpopular texture analysis methods.

Besides the mentioned cases wavelets are also widely used forinstance for classification of aerial and satellite images [17], forcontent-based image indexing and searching [18], and in many otherapplications.

3.4. Angle Measure Technique

Robert Andrle has developed Angle Measure Technique (AMT) forquantitative description of curves complexity (originally — riverchannels and other geomorphic lines), as an alternative to fractalanalysis [19]. It has been later introduced as an image analysis tool inthe paper written by Huang and Esbensen [20]. Now it is used first ofall for classification of textures and in related tasks, like quantitativecharacterization of powders [20,21] and porous space [22], analysis ofparticles [23] and many others.

To use Angle Measure Technique for getting features from animage, the image has to be unfolded to a one-dimensional signal sinceAMT deals with curves. The unfolded signal is usually called imageprofile, P. The simplest way to get P is to use row-by-row unfoldingwhere an image matrix is reshaped to a vector by concatenation of itsrows (as it was used in problem with biochips). In Fig. 13 an imagewith plastic particles and part of the corresponding unfolded profileare shown as an example. Usually for textures the unfolding way hasno significant influence for further analysis but in some cases (mostlyfor analysis of non-isotropic images) it might be quite important howto unfold images, so there are some more sophisticated unfoldingmethods also available.

Fig. 11. Scheme and example of one level wavelet transformation of a digital image.

On the next step a number of basic points (8–10% of the totalnumber of pixels on an image) are chosen uniformly along the imageprofile. These points are used to analyze how the image profile ischanging on different distances. To get this a set of scales [smin, smax]is defined. Usually a minimum scale equals to 1 pixel, whereas amaximum corresponds to the size of the biggest element on an image.If there is no a priori information about depicted objects' size, thewidth of the image is a good value for smax. A distance between scalesis 1 pixel, so if there is an image with 256 pixel width, the scale vectoris [1, 2, 3, …, 255, 256].

After that, for each scale s a circle with radius equal to s is builtaround every basic point (as shown in Fig. 14a) and points where thecircle intersects with the profile (B and C in Fig. 14a) are found. Finallya complement to the angle BAC is calculated for all basic points andaverage value is gotten and stored. Computing the average angle foreach scale value from the range results in a vector usually called asAMT-spectrum or AMT-features.

In Fig. 15a, a plot with AMT-spectra, calculated for the images withdark and light plastic pellets (the same that were used in the previouschapter) are shownusing semi-log scale. Obviously there is a difference incurve trends for different image classes, thus for images with lowestconcentration of light pellets the spectra lie below and for the higherconcentration above the others. However the spectra are highly over-lapped, especially for the imageswith concentrations 10:3 and 10:4 (lightgray lines). By applying principal component analysis it is possible todiscriminate first three classes, as shown in Fig. 15b, but points for thementioned classes are overlapped. In general the discrimination is slightlybetter than for the wavelet features case (Fig. 12).

Angle Measure Technique has a significant drawback — it is quiteslow, especially for large images due to the loops with floating pointcalculations needed to find circles and intersection points. To tacklethis problem a lightweight version of the algorithm, called “LinearAMT” has been proposed. The main idea is to use pixel–pixel relationsinstead of making circles and finding their intersections with profileapproximation. The scheme of the algorithm is presented in Fig. 14b. Ifscale, for example, is equal to n, then for a point x a complement to theangle between P(x), P(x−n), P(x+n) is calculated. This makes theprocedure much faster and keeps the necessary information aboutsignal scale complexity. For analysis of textures it can give even betterresult in terms of discrimination and classification.

Thus in Fig. 15c the AMT-spectra for the same images with plasticparticles but obtained using linear AMT are shown. Obviously the firstpart of the spectra is quite different from the computed with theclassic algorithm. However the peaks, showing the profiles complex-ity, have the same location and shape and the spectra are moredistinguishable. PCA score plot that was obtained using the calculatedspectra as X data shows good discrimination between samples fromdifferent classes; this time even last two cases (10:3 and 10:4) areseparable (Fig. 15d).

3.5. Fractal analysis

Fractal analysis became very popular in the 80th after a set ofpublications of Benoit Mandelbrot [24,25]. The main concept of fractal

Fig. 12. Score (left) and loading (right) plots for PCA analysis of wavelet textural features, calculated for images with plastic particles, using different wavelets.

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analysis is fractal dimension that can be considered as a quantitativemeasure of objects' self-similarity or self-affinity. Mandelbrot definedthe fractal dimension as follows [25].

Let us consider a set of points J in the space and choose somefunction, h(δ)=γ(d)δd, for example, square, cube, sphere, or segment.Here, γ(d) is a geometric coefficient, which is equal to 1 for thesegments and the squares, γ=π/4 for the circles, and γ=π/6 for thespheres. If one covers the set J with the chosen functions, it forms ameasure Md=∑h(δ) of this set. When δ→0, the measure Md, tendsto zero, or to infinity, depending on the measure dimension, d. Thecritical value of d, for which Md changes its limiting value from 0 to ∞is called Hausdorff dimension:

Md = ∑γ dð Þδd = γ dð ÞN δð Þδd→δ→0

0;dbD∞; d N D

:

�ð8Þ

The Hausdorff dimension is a local property of a set. It can beshown, that for such objects as lines and planes, D is equal to 1 and 2,

Fig. 13. Image with plastic particles (left) a

accordingly. However, for some objects, the value of D is not integer.In this case the objects have a fractal dimension.

Typically, objects with fractal Hausdorff dimension look veryirregular or fractured. If it is a figure on a plane, for instance, it has veryindented perimeter. If Hausdorff dimension is close to two (dimen-sion of normal plane figures) — the irregularity is not very obvious. Ifthe dimension is close to one (dimension of lines or smoothedcurves), the perimeter is very indented, so the figure is considered tobe close to a curve. Therefore, one of the practical applications offractal dimension could be a measure of heterogeneity of objects.

This is widely utilized in different ways. Fractal analysis was founduseful for quantitative description of porosity, deformation structuresand fracture surfaces.

There are many algorithms for calculating fractal dimension ofobjects from digital images. The most simple and popular is the box-counting algorithm [25], where an image is divided into segmentswith size equal to r pixels and number of segments N, having at leastone object pixel inside are calculated. By repeating this procedure forfrom 1 pixel to half of width of the image and linear approximation of

nd part of its unfolded profile (right).

Fig. 14. Scheme of AMT spectrum calculation using classic (left) and linear (right) algorithms.

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relations between log(r) and log(N) a Hausdorff dimension can befound.

Fractal analysis gives just one number — a dimension of depictedobject. However in some cases it could be multivariate. These casesare when a sample is described with set of images instead of just oneimage or part of it. Thus A. Dmitriev purposed [26] a method for

Fig. 15. Results of AMT analysis: mean angle spectra (left) and corresponding PCA sco

detection of early stages of Alzheimer disease based on fractal analysisof MRI tomograms of patients.

The idea of the method is that for each scan two binary images aresegmented — one for gray matter of brain and the other for whitematter. The example is shown in Fig. 16. Calculation of fractaldimension of each segmented part for all scans gives vector of

re plots (right) obtained using classic (top) and linear (bottom) AMT algorithms.

Fig. 16. Segmented gray (top) and white (bottom) matter of a brain from a particular MRI scan and corresponding plots for calculation of fractal dimension.

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features, describing morphological complexity of brain matters on thewhole MRI image.

A feasibility study showed that by applying PLS discriminantanalysis with cross-validation to the fractal dimension vector a goodseparation between samples with and without AD can be obtained, asshown in Fig. 17.

4. Software and implementation

There are at least several computer programs that allow toimplement and use the described methods for extraction imagefeatures. Perhaps the most flexible and efficient is MATLAB withseveral toolboxes, particularly Image Processing Toolbox andWaveletToolbox if wavelet transformation is planned to be used. All examplesshown in this paper were made using this bundle, thus the author'simplementation of AMT can be found in: http://www.acabs.dk/files/amt.zip. There are also many third-part toolboxes, for example,MIA_Toolbox from Eigenvector Research, or several toolboxes forfractal and wavelet analysis.

The other reasonable choice is free and open-source softwareImageJ written in Java. ImageJ has all basic functions for imageprocessing and analysis, including analysis of particles and image

Fig. 17. PLS DA score plot for MRI images.

statistics. It also can be extended with numerous plug-ins available.There is an easy application program interfaces (API) that allow towrite user plug-ins in Java. It is also possible to automatize actionswith macros.

There are several image analysis toolboxes for R; however, fromthe author's point of view, they are not very stable. There is also apossibility to analyze images with Python (using NumPy and SciPylibraries). A well-known development environment for visualprogramming, LabVIEW, has a set of functions for image analysisand processing.

5. Conclusions

Image analysis is one of the non-invasive techniques that givepretty wide possibilities for indirect measurements, especially ifinvestigated properties of depicted objects relate to their structuralfeatures. The most important part of a chain from image acquisition toprediction of non-visual properties is extraction of relevant featuresfrom images.

In the present paper a general approach to extraction of imagefeatures is described. Several methods that utilized different proper-ties of images are discussed and compared.

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