**4. X-ray Diffraction characterization of Films**

172 Recent Advances in Crystallography

**Ortho-**

**25-46**

**System Space**

**System Space**

**System Space**

**Cubic**

**Group**

**(hexagonal**

**Group**

**149-155**

**Point**

**Table 1.** Symmetry of the planes of a point group.  The chiral orientations are highlighted in yellow.

**Point**

X-ray diffraction (XRD) is a non-destructive technique that reveals crystallography of an unknown material using monochromatic X-rays. X-rays are generated by an X-ray tube that uses a high voltage to accelerate the electrons released by a cathode to a high velocity. The so generated electrons collide with a metal target, the anode, creating the Xrays. [36] Different X-ray sources are used based on the need. Tungsten or a crackresistant alloy of rhenium (5%) and tungsten (95%) are generally used in the medical field. When soft X-rays are needed for special applications like mammography, a molybdenum source is used. In crystallography, a copper target is most common, with cobalt often being used when fluorescence from iron content in the sample might otherwise present a problem. For a copper target, X-ray emissions commonly contains a continuous white radiation and two characteristic x-rays, Kα (λ = 0.15418 nm) and Kβ (λ = 0.13922 nm) leading from 2p 1s and 3p 1s transitions, respectively. In general, the Kα transition is more intense than Kβ, and is a combination of Kα1 and Kα2. This is because of the slight difference between two possible spin states of 2p electrons. Monochromatic Kα X-rays can be obtained by using suitable filters that absorb the unneeded white radiation and Kβ. For example, a Ni foil is commonly used for radiation of copper target. [37]

**Figure 1.** Stereographic projections of the A) (001) and B) (00-1) orientations of CuO. These two orientations are superimposable mirror images of each other; they are achiral. The radial grid lines on the stereographic projections correspond to 30o increments of the tilt angle, χ. C) the interface model of (00-1) CuO (front, blue Cu atoms) on (001) CuO (back, brown Cu atoms) with a common [010] directions. These two orientations are superimposed onto each other, indicating that they are achiral.

174 Recent Advances in Crystallography

**(C)**

**(A) (B)**

**Figure 1.** Stereographic projections of the A) (001) and B) (00-1) orientations of CuO. These two orientations are superimposable mirror images of each other; they are achiral. The radial grid lines on the stereographic projections correspond to 30o increments of the tilt angle, χ. C) the interface model of

(00-1) CuO (front, blue Cu atoms) on (001) CuO (back, brown Cu atoms) with a common [010] directions. These two orientations are superimposed onto each other, indicating that they are achiral.

**Figure 2.** Stereographic projections of A) (111) and B) (-1-1-1) orientation of CuO. These two orientations are nonsuperimposable mirror images of each other; they are chiral. The radial grid lines on the stereographic projections correspond to 30o increments of the tilt angle, χ. C) the interface model of (-1-1-1) CuO (red oxygen atoms) on (111) CuO (violet oxygen atoms) with a common [1-10] direction. These two surfaces are nonsuperimposable; therefore, they are chiral.

X-ray diffraction works on the principle of Bragg's law, nλ = 2d sin θ, where λ is the x-ray wavelength, d is the lattice spacing and θ is the Bragg angle. The layers of a crystal act like weak reflecting mirrors for the X-rays. Only if the path difference of the reflected X-rays is a whole number of wavelengths does constructive interference occurs, as shown in Figure 3.

In general, X-ray diffraction patterns are plots of intensity versus 2θ angle. Different planes in a crystal diffract at different angles giving a pattern which is unique to a crystal. The intensities of the reflections are determined by the distribution of the electrons in the unit cell. X-rays going through areas with high electron density will reflect strongly and areas with low electron density will give weak intensities. Therefore, every crystalline material has a unique X-ray diffraction pattern that can be used to determine the crystallinity and phase of the deposited material.

Thin films deposited on polycrystalline substrates can be analyzed by running two types of experiments. In the first case, a symmetric scan (gonio scan) can be used to evaluate the outof-plane texture of the film. It is a conventional θ-2θ scan for the Bragg-Brentano geometry. Figure 4 (A) shows a schematic representation of a gonio scan. Where, ω is the incident angle and θ is the diffracted angle.

**Figure 3.** Schematic representation of Bragg's law.

In a gonio scan, ω = θ and ω + θ = 2θ. Another way to characterize thin films is through glancing angle measurements, as shown in Figure 4 (B). In this case, ω + θ = 2θ and ω is fixed but θ varies. Glancing angle measurement is a good technique to measure polycrystalline grains. As the w is fixed and the θ is varied, all the planes in the material are brought into the Bragg condition. However, this technique cannot be used for highly oriented films or epitaxial films.

Epitaxial films deposited on single crystals are oriented both out-of-plane as well as inplane. In this case, X-ray characterization such as diffraction patterns, pole figures, azimuthal scans, and rocking curves are performed.

Unlike polycrystalline films, epitaxial films grow with one orientation and show only a family of planes in the pattern. In Figure 4, an epitaxial magnetite film deposited on a Ni(111) substrate shows only the {111} family peaks. [38] The experimental setup to determine out-of-plane orientation is similar to the gonio scan but instead a 2θ-omega scan is run. The difference between this and the gonio scan is that there can be an offset between 2θ and omega, so that omega = ½ 2θ + offset. This is useful when collecting a diffraction scan from an epitaxial film, when the tilt of the film is compensated by the offset. Identification of the pattern is done by comparing the pattern with the existing patterns in the database. For example, a magnetite film is identified by JCPDS#19-0629 pattern. However, this does not provide any information about in-plane orientation of the film. To determine the in-plane orientation of the film, X-ray pole figures and azimuthal scans are run. Pole figures can be used to probe planes which are not parallel with the geometric surface of the sample. The sample is moved through a series of tilt angles, χ, and at each tilt angle the sample is rotated through azimuthal angle, φ, of 0 to 360o. Peaks occur in the pole figure when the Bragg condition is satisfied. During the experiment 2θ is fixed, which is normally the highest intensity peak of a randomly oriented powder diffraction pattern of the material. Figure 6 is a schematic representation of a pole figure measurement. Azimuthal scans can be considered as a cross-section of a pole figure. They are obtained when the measurement is carried out at a specific 2θ for only specific tilt angle, χ, and rotated azimuthally, φ, from 0 to 360o. Comparing the azimuthal scans at specific tilt angle for a substrate and a film, we can obtain the epitaxial relationships.

176 Recent Advances in Crystallography

phase of the deposited material.

angle and θ is the diffracted angle.

**Figure 3.** Schematic representation of Bragg's law.

azimuthal scans, and rocking curves are performed.

oriented films or epitaxial films.

with low electron density will give weak intensities. Therefore, every crystalline material has a unique X-ray diffraction pattern that can be used to determine the crystallinity and

Thin films deposited on polycrystalline substrates can be analyzed by running two types of experiments. In the first case, a symmetric scan (gonio scan) can be used to evaluate the outof-plane texture of the film. It is a conventional θ-2θ scan for the Bragg-Brentano geometry. Figure 4 (A) shows a schematic representation of a gonio scan. Where, ω is the incident

In a gonio scan, ω = θ and ω + θ = 2θ. Another way to characterize thin films is through glancing angle measurements, as shown in Figure 4 (B). In this case, ω + θ = 2θ and ω is fixed but θ varies. Glancing angle measurement is a good technique to measure polycrystalline grains. As the w is fixed and the θ is varied, all the planes in the material are brought into the Bragg condition. However, this technique cannot be used for highly

Epitaxial films deposited on single crystals are oriented both out-of-plane as well as inplane. In this case, X-ray characterization such as diffraction patterns, pole figures,

Unlike polycrystalline films, epitaxial films grow with one orientation and show only a family of planes in the pattern. In Figure 4, an epitaxial magnetite film deposited on a Ni(111) substrate shows only the {111} family peaks. [38] The experimental setup to determine out-of-plane orientation is similar to the gonio scan but instead a 2θ-omega scan

**Figure 4.** Schematic representation of (A) gonio scan used for textured films and (B) glancing angle used for polycrystalline grains.

To determine the quality of epitaxy, X-ray rocking curves are run. The rocking curves indicate the mosaic spread of the film relative to the substrate. The larger the full width at half maximum (FWHM), the larger the mosaic spread. In the experimental setup for a rocking curve, only the omega axis is scanned as data are collected. All other axes, such as 2θ, are fixed at specific angles. In a perfect single crystal, the FWHM is small indicating that all domains are aligned. The width of the rocking curve is a direct measurement of the range of orientation of the sample. In general, the rocking curves are performed for substrate as

#### 178 Recent Advances in Crystallography

well as the film for comparison of mosaic spread. If the mosaic spread of the film is low and comparable to the substrate, the peaks in the pole figure become sharper and more intense. Rocking curves have been used to understand the in-plane misorientation of ZnO, AlN, and GaN on sapphire and MgO films grown on GaAs.

**Figure 5.** X-ray diffraction pattern of an epitaxial magnetite (Fe3O4) film on Ni(111) single crystal.


**6. Long-term outcomes**

Complications are a critical component of RTSA. Patients need to be counseled on intraoperative, perioperative, and notably, short-term complications. The most common intraoperative complications are glenoid or humeral fractures along with poor screw fixation [12]. The consequence of poor screw fixation is its lasting impact and conversion to long-term complications such as glenoid screw lucency. Zhou and colleagues [13] discussed the prevention techniques, such as hand reaming the humerus and preserving glenoid bone stock by avoiding reaming beyond subchondral bone margin. Postoperative short-term complications dominate the outcomes of RTSA and range from scapular notching, infection, dislocation, revision, nerve

Scapular notching is by far the most common complication during the first 24 months postoperative. Scapular notching has an incidence of 38%, 57%, 55% and 73% in four recent studies respectively [12, 15, 24, 25]. These findings oblige additional research to review instrumentation and confirm incidence levels on scapular notching in RTSA. In 2014, Feeley and colleagues [12] found that decreasing the neck-shaft angle or a higher inclination angle and 3 mm lateral offset of the glenosphere prosthesis decreased the rate of scapular notching by 16%. Furthermore, Zhou and colleagues [13] assert that continued complication management, by adding inferior placement of the glenosphere, is "the most important factor in the avoidance of inferior impingement." The next step is to investigate whether scapular notching will evolve, both de novo and from early to late stage scapular notching during short-term follow-up. An important question to consider is will the patient be free from scapular notching for the remainder of the prosthesis? [12] Feeley and colleagues [12] observed that of all the patients who did not experience scapular notching during the first 12 months (84% of patients), showed no new evidence of scapular notching during follow-ups up to 30 months. Conversely, Bacle and colleagues [15] found that after early scapular notching diagnoses were made, there was a 39% increase in the rate of notching beyond the 2-year follow-up period.

Another common short term complication deals with postoperative stability resulting in shoulder dislocations. Wierks [12] and colleagues as well as Bacle [15] and colleagues found 10% and 22% of dislocations occurred during the short term period. Of note, Bacle and colleagues [15] published that of the 15 dislocations documented in the sample size of 67, no cases were reported after the 2 year follow-up period. Zhou and colleagues [26] reviewed the most common and serious complications associated with RTSA and concluded that instability was a result of "lack of soft tissue tension, mechanical impingement, mismatch of the glenosphere and humeral socket size and improper version of the prosthesis". Therefore, to obtain the best outcome for patients, extensive knowledge of the prosthesis is imperative, along with understanding how to achieve soft tissue tension using vertical offset of your acromion—greater tuberosity distance and lateral offset of the tuberosity-glenoid distance [26]. Conversely, controversy exists within the RTSA literature regarding the decision to repair the subscapularis. Friedman and colleagues showed that subscapularis repair proclaimed no statistical significance over no repair [27]. In the study, 340 patients with RTSA plus repair had 0% dislocation rate, versus 251 patients with RTSA without repair showing a 1.2% dislocation rate; stating the claim that RTSA plus subscapularis repair is not indicated due to the absence of increase in overall complication rates [27]. Lastly, infections can have serious ramifications on patient satisfaction as well as the overall outcomes of the RTSA, resulting in one or two-stage revisions. A study conducted by Wierks

palsy, or even heterotopic ossification.

92 Advances in Shoulder Surgery

Thus, superfluous research is essential to answer this question.

The patient's age at the time of the arthroplasty plays a critical role in the survival of the prosthesis; younger patients (i.e., < 55 years of age) are more likely to be active in the workforce, while older patients are less likely to participate in physically strenuous activities and be retired [16]. In younger patients, prosthesis survival can exceed 10 years. For example, Bacle and colleagues [15] observed a 93% implant survival rate at 10 years; whereas Ek and colleagues [28] observed an implant survival rate of 88% at 5 years and 76% at 10 years postoperatively, regardless of any complications that may have arisen.

The RTSA has been shown to improve pain, strength, range of motion in abduction, external rotation, and forward flexion; in addition to showing improvement in metrics, such as ASES, SST, CMS, SPADI, and UCLA Should and UCLA Shoulder Score [16, 23, 24, 29]. Outcomes beyond the 24-month mark may be impacted by multiple variables, some of which include, prosthesis sizes, involvement of fracture, primary versus revision RTSA, and the lifestyle or activity level of the patient. In regards to repair of proximal humeral fractures, RTSA was found to provide superior results to a hemiarthroplasty for at least 5 years, respectively [30]. Muller and colleagues [24] investigated the size of the glenosphere, 36 mm and 44 mm, on functional outcomes following RTSA and found that both groups exhibited the most substantial progress in the first 6–24 months, followed by a plateau. Patients' progress was monitored by measuring flexion, abduction, external rotation at 0° and 90° of abduction, internal rotation at 90° of abduction, CMS, SPADI, and strength (kg) in abduction. Interestingly, Anakwenze and colleagues [31] found that a higher body mass index (BMI) put a patient at risk for deep surgical site infection (SSI) up to 3 years following RTSA. Their study looked at the effects of increased BMI on postoperative outcomes following a RTSA and total shoulder arthroplasty (TSA). Every 5 kg/m [2] increase in BMI was associated with higher risk of 3-year deep SSI [31]. In addition to BMI, tobacco use influences the success of the prosthesis up to 12 years after an RTSA [32]. Hatta and colleagues [32] found that current smokers had an increased risk for infection, component loosening, and fractures compared to non-smokers. Specifically, they found that the percentage of patients with periprosthetic fractures jumped 20% at the 9 year mark after RTSA [32].

Although, as previously stated, this study implies 24 months as the short-term interval for outcomes following an RTSA, short-term complications have the potential to extend into the long-term if not addressed or managed appropriately—eventually affecting the longevity of the prosthesis. More commonly, long-term complications include glenoid and/or humeral component loosening, polyethylene component wear, and scapular notching. Less common long-term complications include deep SSI, dislocation, readmission, and fracture; which primarily occur within the first 2 years postoperatively. As mentioned above, glenoid or humeral loosening is the most common complication observed 2 years after RTSA, particularly with an increased risk following a previously failed shoulder arthroplasty and excess mechanical load related to increased BMI [15, 33]. Further, the incidence of component loosening doubles between the second and fifth year follow-up as reported by Alentorn-Geli and colleagues [29]. Particlization of the polyethylene component may be of concern with RTSA in younger patients due to necessary durability and lifespan of the implant. Riley and colleagues [34] investigated the outcomes following RTSA using a metal-on-metal design and concluded that it is not an acceptable alternative to RTSA in young patients; they maintain that the polyethylene component is the more suitable option. Ek and colleagues [28] conducted a study evaluating RTSA in patients younger than 65 years of age using two groups: revision RTSA and primary RTSA. This study observed an increased incidence of scapular notching at less than 12 months follow-up and greater than 10 years follow-up; with 56% of patients experiencing some degree of scapular notching overall [28]. Conversely, Mollon and colleagues [34] reported that only 10% of patients experienced scapular notching; noting that risk factors for scapular notching included lower body weight, lower BMI, and RTSA on the non-dominant upper extremity. It is also worth mentioning the correlation between longer-term follow-up and increasing incidence of scapular notching [34]; which may be attributed to variation in size and placement of the glenosphere, and center of rotation of the prosthesis.

**Author details**

Sydney C. Cryder1

Athens, United States

Medicine, Athens, United States

United States

**References**

2015;**35**:1-7

\*, Samuel E. Perry2

Elbow Surgeons…[et al.]. Jan 2015;**24**(1):150-160

American volume. Dec 2010;**92**(Suppl 2):23-35

\*Address all correspondence to<EMAIL_ADDRESS>
and Elizabeth A. Beverly3

Current Outcomes Following Reverse Total Shoulder Arthroplasty: A Composite

http://dx.doi.org/10.5772/intechopen.72545

95

1 Department of Medicine, Ohio University, Heritage College of Osteopathic Medicine,

2 Department of Graduate Medical Education, Adena Health System, Chillicothe,

3 Department of Family Medicine, Ohio University, Heritage College of Osteopathic

[1] Berliner JL, Regalado-Magdos A, Ma CB, Feeley BT. Biomechanics of reverse total shoulder arthroplasty. Journal Of Shoulder And Elbow Surgery/American Shoulder And

[2] Nam D, Kepler CK, Neviaser AS, et al. Reverse total shoulder arthroplasty: Current concepts, results, and component wear analysis. The Journal Of Bone And Joint Surgery.

[3] Schairer WW, Nwachukwu BU, Lyman S, Craig EV, Gulotta LV. National utilization of reverse total shoulder arthroplasty in the United States. Journal Of Shoulder And Elbow

[4] Westermann RW, Pugely AJ, Martin CT, Gao Y, Wolf BR, Hettrich CM. Reverse shoulder Arthroplasty in the United States: A comparison of National Volume, patient demographics, complications, and surgical indications. The Iowa Orthopaedic Journal.

[5] Wiater BP, Baker EA, Salisbury MR, et al. Elucidating trends in revision reverse total shoulder arthroplasty procedures: A retrieval study evaluating clinical, radiographic, and functional outcomes data. Journal Of Shoulder And Elbow Surgery/American

[6] Middernacht B, Van Tongel A, De Wilde LA. Critical review on prosthetic features available for reversed Total shoulder Arthroplasty. Biomed Research International. 2016;**2016**:3256931

[7] Jo SH, Kim JY, Cho NS, Rhee YG. Reverse Total shoulder Arthroplasty: Salvage procedure for failed prior Arthroplasty. Clinics In Orthopedic Surgery. Jun 2017;**9**(2):200-206

[8] Dilisio MF, Miller LR, Siegel EJ, Higgins LD.Conversion to Reverse Shoulder Arthroplasty: Humeral Stem Retention Versus Revision. Orthopedics. Sep 1 2015;**38**(9):e773-e779

Surgery/American Shoulder And Elbow Surgeon. Jan 2015;**24**(1):91-97

Shoulder And Elbow Surgeon. Jul 2015;**24**(12):1915-1925


5.1. Black and white photography

eigenvalues can indicate linear dependencies in the data and cause deformations in the inter-

In general, principal components are not invariant against changes of scale in the original variables, as has been mentioned when referring to the normalized population principal components. Normalizing, or standardizing, the variables consists of performing the following

> xpj�xp ffiffiffiffi spp p h it

matrix whose columns are zj, it can be shown that its sample mean vector is the null vector and that its correlation matrix is the sample correlation matrix, R, of the original variables.

Remark 4.1: Applying that the principal components of the normalized variables are those obtained for the sample observations but substituting the matrix S for R, we can establish that if z1, …, zn are the normalized observations, with covariance matrix R ¼ ð Þ rik , where rik is the sample correlation coefficient between observations xi and xk, and if the pairs of eigenvalues and eigenvectors of R are

i

6. The proportion of the total sample variance explained by the ith principal component is v^i

The eigenvalues and eigenvectors of the covariance matrix, or correlation matrix, are the essence of the analysis of principal components, since the eigenvalues indicate the directions of maximum variability and the eigenvectors determine the variances. If a few eigenvalues are much larger than the rest, most of the variance can be explained with less than p variables.

In practice, decisions about the number of components to be considered must be made in terms of the pairs of eigenvalues and eigenvectors of the covariance matrix, or correlation

values there is a decrease with a linear tendency of quite steep slope and that from a certain eigenvalue this decrease is stabilized. That is, there is a point from which the eigenvalues are very similar. The criterion consists of staying with the components that

� �

exclude the small eigenvalues and that are approximately equal.

, j ¼ 1, …, p. If the matrix Z is the p by n

z ¼ u^1iz1 þ ⋯ þ u^pizp, i ¼ 1, …, p.

ffiffiffiffi v^i

, it has been empirically verified that with the first

p , i, k ¼ 1, …, p.

p .

s11 p ;…;

pretations, calculations, and consequent analysis.

122 Statistics - Growing Data Sets and Growing Demand for Statistics

4.2. Standardized sample principal components

� �, with v^1 ≥ ⋯ ≥ v^p ≥ 0, then

3. The sample covariance of ω^ i ð Þ ; ω^ k , i 6¼ k, is equal to 0. 4. The total sample variance is trð Þ¼ R p ¼ v^1 þ ⋯ þ v^p.

5. The sample correlation coefficients between zkand ω^ i are rzk,ω^ i ¼ u^ki

1. The ith principal component is given by ω^ i ¼ u^t

2. The sample variance of ω^ k is v^k, k ¼ 1, …, p.

4.3. Criteria for reducing the dimension

matrix, and different rules have been suggested:

a. When performing the graph i; λ^i

transformation zj ¼ D xj � x � � ¼ x1j�x1 ffiffiffiffi

ð Þ v^1; u^1 ,…, v^p; u^p

The black and white photograph shown in Figure 6 was considered. First, the image in .jpg format was converted into the numerical matrix Image of dimension 512 by 512 (i.e., 29 x29 ). Second, to obtain the observation vectors, the matrix was divided into blocks of dimension 23x23, Aij, with which 4096 blocks were obtained, and each of them was a vector of observations.

Figure 6. Black and white photograph of Lena.

$$\mathbf{Image} = \begin{bmatrix} \mathbf{A}\_{1,1} & \dots & \mathbf{A}\_{1,64} \\ \vdots & \ddots & \vdots \\ \mathbf{A}\_{64,1} & \dots & \mathbf{A}\_{64,64} \end{bmatrix} \tag{17}$$

E^j ¼

canonical basis of the vectors of B0 were the columns of the matrix PC ¼ e^t

components are shown in Figure 8.

½ � x1;…; x64 t ¼ PC y1;…; y64 � �t

to explain 97% of the variability, because P5

Therefore, the dimension of yM ¼ y � TM was 4096 � 64.

want to explain 98% of the total variability.

vectors of B0

Third component.

¼ y � PCt

2 6 4

^e1,j ⋯ ^e8,j ⋮ ⋱⋮ ^e57,j ⋯ ^e64,j

Each of the 64 matrices E^j was converted into an image. The images of the first three principal

At this point, it is important to mention that the data matrix x has been assumed to be formed by 4096 vectors of ℜ64 expressed in the canonical base, B. Also, the base whose vectors were the eigenvectors of S, B0 ¼ f g e^1;…; e^64 , was considered. The coordinates with respect to the

given a vector v that with respect to the canonical base had coordinates ð Þ x1;…; x64 and with respect to the base B0 had coordinates y1;…; y64 � �, the relation between them was

Thus, the coordinates of the 4096 vectors that formed the observations matrix had as coordinates, with respect to the new base, the rows of the matrix of dimension 4096x64 given by y ¼ x � PC. Eight, in order to reduce the dimension, it was taken into consideration that if we keep all the

change rule, we can consider the first two principal components; five components if we want

In order to compress the image, the first vectors of the base B0 were used. Moreover, supposing

Ninth, to reconstruct the compressed image, each row of yM was regrouped in an 8x8matrix. The ith row of yM, denoted by yMi ¼ ½ � bi,1; …; bi, 8; bi,9;…; bi, 16;…; bi, 64 , was transformed into

Figure 8. Images of the matrices of the first three principal components. (a) First component. (b) Second component. (c)

0ð Þ 64�M xM 0ð Þ 64�M x 64 ð Þ �M " #

TM ¼ IMxM 0Mx 64 ð Þ �M

that we were left with M, M < 64, the matrix TM given by Eq. (19) was defined:

i¼1 λ^i=

, we can perfectly reconstruct our data matrix, because y ¼ x � PC ) x ¼ y � PC�1

P64

. Additionally, for the case under study, to reduce the dimension, if we use the slope

3 7

Application of Principal Component Analysis to Image Compression

. Also, as PC is an orthogonal matrix, y1;…; y64 � � ¼ ½ � x1;…; x64 PC.

5 (18)

http://dx.doi.org/10.5772/intechopen.75007

125

j¼1 λ^j ¼ 97%; or eight components if we

1;…; e^t 64 � �. Then,

(19)

Third, each matrix Aij was stored in a vector of dimension 64, x, which contained the elements of the matrix by rows, that is, x ¼ ½ � ai,1; …; ai, 8; aiþ1,1; …; aiþ1,8;…; aiþ8,8 . This way, we had the observations xk ∈ ℜ64� �k ¼ 1; …; 4096 � �, which were grouped in the observation matrix x ¼ xij � � ∈ Μ4096,64ð Þ ℜ .

Fourth, the average of each column, x ¼ ½ � x1; …; x64 , was calculated obtaining the vector of means, and from each observation xij, its corresponding mean xj was subtracted. Thus, the matrix of centered observations U was obtained. The covariance matrix of x was S ¼ Ut U ∈ Μ64,64ð Þ ℜ .

Fifth, the 64 pairs of eigenvalues and eigenvectors of S, λ^i; e^i � �, were found, and they were ordered according to the eigenvalues from highest to lowest. The 8 largest eigenvalues are drawn in Figure 7. As can be seen, the first eigenvalue is much larger than the rest. Thus, the first principal component completely dominates the total variability.

Sixth, with the theoretical results and the calculations previously made, the 64 principal components y^j ¼ e^t j x ¼ ^e1,jx1 þ ⋯ þ ^e64,jxp, j ¼ 1, …, p, were built. The first principal component was y^1 ¼ �0:1167x1 þ ⋯ � 0:1166x64. Therefore, an orthonormal basis of ℜ64 was built.

Seventh, each vector e^j ¼ ^e1,j;…; ^e64,j � �t was grouped by rows in a matrix Μ8,8:

Figure 7. Graph i; λbi � �, i ¼ 1,…, 8, with λbi being the eigenvalues ordered from highest to lowest.

$$
\hat{\mathbf{E}}\_{j} = \begin{bmatrix}
\hat{e}\_{1,j} & \cdots & \hat{e}\_{8,j} \\
\vdots & \ddots & \vdots \\
\hat{e}\_{57,j} & \cdots & \hat{e}\_{64,j}
\end{bmatrix} \tag{18}
$$

Each of the 64 matrices E^j was converted into an image. The images of the first three principal components are shown in Figure 8.

At this point, it is important to mention that the data matrix x has been assumed to be formed by 4096 vectors of ℜ64 expressed in the canonical base, B. Also, the base whose vectors were the eigenvectors of S, B0 ¼ f g e^1;…; e^64 , was considered. The coordinates with respect to the canonical basis of the vectors of B0 were the columns of the matrix PC ¼ e^t 1;…; e^t 64 � �. Then, given a vector v that with respect to the canonical base had coordinates ð Þ x1;…; x64 and with respect to the base B0 had coordinates y1;…; y64 � �, the relation between them was ½ � x1;…; x64 t ¼ PC y1;…; y64 � �t . Also, as PC is an orthogonal matrix, y1;…; y64 � � ¼ ½ � x1;…; x64 PC. Thus, the coordinates of the 4096 vectors that formed the observations matrix had as coordinates, with respect to the new base, the rows of the matrix of dimension 4096x64 given by y ¼ x � PC.

Eight, in order to reduce the dimension, it was taken into consideration that if we keep all the vectors of B0 , we can perfectly reconstruct our data matrix, because y ¼ x � PC ) x ¼ y � PC�1 ¼ y � PCt . Additionally, for the case under study, to reduce the dimension, if we use the slope change rule, we can consider the first two principal components; five components if we want to explain 97% of the variability, because P5 i¼1 λ^i= P64 j¼1 λ^j ¼ 97%; or eight components if we want to explain 98% of the total variability.

In order to compress the image, the first vectors of the base B0 were used. Moreover, supposing that we were left with M, M < 64, the matrix TM given by Eq. (19) was defined:

$$\mathbf{T}\_M = \begin{bmatrix} \mathbf{I}\_{M \ge M} & \mathbf{0}\_{M \ge (64-M)} \\ \mathbf{0}\_{(64-M) \ge M} & \mathbf{0}\_{(64-M) \ge (64-M)} \end{bmatrix} \tag{19}$$

Therefore, the dimension of yM ¼ y � TM was 4096 � 64.

Image ¼

124 Statistics - Growing Data Sets and Growing Demand for Statistics

observations xk ∈ ℜ64�

� � ∈ Μ4096,64ð Þ ℜ .

x ¼ xij

nents y^j ¼ e^t

Figure 7. Graph i; λbi

� �

j

Seventh, each vector e^j ¼ ^e1,j;…; ^e64,j

2 6 4

A1, 1 … A1,64 ⋮ ⋱⋮ A64,1 … A64, 64

�k ¼ 1; …; 4096 � �, which were grouped in the observation matrix

Third, each matrix Aij was stored in a vector of dimension 64, x, which contained the elements of the matrix by rows, that is, x ¼ ½ � ai,1; …; ai, 8; aiþ1,1; …; aiþ1,8;…; aiþ8,8 . This way, we had the

Fourth, the average of each column, x ¼ ½ � x1; …; x64 , was calculated obtaining the vector of means, and from each observation xij, its corresponding mean xj was subtracted. Thus, the matrix

ordered according to the eigenvalues from highest to lowest. The 8 largest eigenvalues are drawn in Figure 7. As can be seen, the first eigenvalue is much larger than the rest. Thus, the

Sixth, with the theoretical results and the calculations previously made, the 64 principal compo-

, i ¼ 1,…, 8, with λbi being the eigenvalues ordered from highest to lowest.

x ¼ ^e1,jx1 þ ⋯ þ ^e64,jxp, j ¼ 1, …, p, were built. The first principal component was

of centered observations U was obtained. The covariance matrix of x was S ¼ Ut

y^1 ¼ �0:1167x1 þ ⋯ � 0:1166x64. Therefore, an orthonormal basis of ℜ64 was built.

Fifth, the 64 pairs of eigenvalues and eigenvectors of S, λ^i; e^i

first principal component completely dominates the total variability.

� �t

3 7

� �

was grouped by rows in a matrix Μ8,8:

5 (17)

U ∈ Μ64,64ð Þ ℜ .

, were found, and they were

Ninth, to reconstruct the compressed image, each row of yM was regrouped in an 8x8matrix. The ith row of yM, denoted by yMi ¼ ½ � bi,1; …; bi, 8; bi,9;…; bi, 16;…; bi, 64 , was transformed into

Figure 8. Images of the matrices of the first three principal components. (a) First component. (b) Second component. (c) Third component.

the matrix Bi given by Eq. (20), and the matrix Compressed\_image given by Eq. (21) was built:

$$\mathbf{B}\_{i} = \begin{bmatrix} b\_{i,1} & \cdots & b\_{i,8} \\ b\_{i,9} & \cdots & b\_{i,16} \\ \vdots & \ddots & \vdots \\ b\_{i,57} & \cdots & b\_{i,64} \end{bmatrix} \quad i = 1, \ldots, 4096 \tag{20}$$

Tenth and finally, Eq. (21) was converted into a .jpg file. Figure 9 shows the original image and

By increasing the number of principal components, the percentage of the variability explained is increased by very small percentages, but, nevertheless, nuances are added to the photo sufficiently remarkable, since they make it sharper, smooth out the contours, and mark the

The two methods that we will use are the peak signal-to-noise ratio (PSNR) and the entropy of the error image. The PSNR measure evaluates the quality in terms of deviations between the processed and the original image, and the entropy of an image is a measure of the information

Definition 5.1: Let N be the number of rows by the number of columns in the image. Let

Definition 5.2: Let the images under study be the 8 bit images. The peak signal-to-noise ratio of the

Figure 10 (a) shows PSNR of the reconstructions of the image versus the number of principal components used for the reconstruction, together with the regression line that adjusts the said cloud of points. Figure 10 (b) shows the values of the PSNR when we use three quarters (black), half (red), quarter (blue), eighth (green), sixteenth (brown), and the thirty-second part (yellow) of the components, which means a corresponding reduction in compression. A behavior close to linearity with a slope of approximately 0:2 can be seen. With the reductions

If the entropy is high, the variability of the pixels is very high, and there is little redundancy. Thus, if we exceed a certain threshold in compression, the original image cannot be recovered exactly. If the entropy is small, then the variability will be smaller. Therefore, the information of a pixel with respect to the pixels of its surroundings is high and, therefore, randomness is lost.

Definition 5.3: Let I be an 8 bit image that can take the values f g 0;…; 255 .  The mean square error (MSE) is

�n ¼ 1;…; N � � be the set of

n (22)

MSE ! (23)

compressed images with two, five, and eight principal components.

f g xnjn ¼ 1;…; N be the set of pixels of the original image. Let yn

�

5.1.1. Objective measures of the quality of reconstructions

tones more precisely.

reconstruction is

content contained in that image.

reconstruction pixels. Let rn ¼ xn � yn

considered, the PSNR varies between 27 and 63.

with which the value i∈f g 0; …; 255 appears. Then, the entropy is

$$\mathbf{\color{red}{\textbf{Compressed\\_image}}} = \begin{bmatrix} \mathbf{B}\_1 & & \cdots & \mathbf{B}\_{64} \\ \mathbf{B}\_{65} & \cdots & \mathbf{B}\_{128} \\ \vdots & & \ddots & \vdots \\ \mathbf{B}\_{4033} & & \mathbf{B}\_{4096} \end{bmatrix} \tag{21}$$

Figure 9. Original and compressed image with two, five, and eight principal components. (a) Original image. (b) Compression with two components. (c) Compression with five components. (d) Compression with eight components.

Tenth and finally, Eq. (21) was converted into a .jpg file. Figure 9 shows the original image and compressed images with two, five, and eight principal components.

By increasing the number of principal components, the percentage of the variability explained is increased by very small percentages, but, nevertheless, nuances are added to the photo sufficiently remarkable, since they make it sharper, smooth out the contours, and mark the tones more precisely.

#### 5.1.1. Objective measures of the quality of reconstructions

the matrix Bi given by Eq. (20), and the matrix Compressed\_image given by Eq. (21) was

Figure 9. Original and compressed image with two, five, and eight principal components. (a) Original image. (b) Compression with two components. (c) Compression with five components. (d) Compression with eight components.

B1 ⋯ B64 B65 ⋯ B128 ⋮ ⋱⋮ B4033 B4096

i ¼ 1, …, 4096 (20)

(21)

bi, 1 ⋯ bi, 8 bi, 9 ⋯ bi, 16 ⋮ ⋱⋮ bi, 57 ⋯ bi, 64

Bi ¼

126 Statistics - Growing Data Sets and Growing Demand for Statistics

Compressed\_image ¼

built:

The two methods that we will use are the peak signal-to-noise ratio (PSNR) and the entropy of the error image. The PSNR measure evaluates the quality in terms of deviations between the processed and the original image, and the entropy of an image is a measure of the information content contained in that image.

Definition 5.1: Let N be the number of rows by the number of columns in the image. Let f g xnjn ¼ 1;…; N be the set of pixels of the original image. Let yn � �n ¼ 1;…; N � � be the set of reconstruction pixels. Let rn ¼ xn � yn � �n ¼ 1;…; N � � be the error. The mean square error (MSE) is

$$MSE = \frac{1}{N} \sum\_{n=1}^{N} r\_n^2 \tag{22}$$

Definition 5.2: Let the images under study be the 8 bit images. The peak signal-to-noise ratio of the reconstruction is

$$PSNR = 10 \log\_{10} \left( \frac{\left(2^8 - 1\right)^2}{MSE} \right) \tag{23}$$

Figure 10 (a) shows PSNR of the reconstructions of the image versus the number of principal components used for the reconstruction, together with the regression line that adjusts the said cloud of points. Figure 10 (b) shows the values of the PSNR when we use three quarters (black), half (red), quarter (blue), eighth (green), sixteenth (brown), and the thirty-second part (yellow) of the components, which means a corresponding reduction in compression. A behavior close to linearity with a slope of approximately 0:2 can be seen. With the reductions considered, the PSNR varies between 27 and 63.

If the entropy is high, the variability of the pixels is very high, and there is little redundancy. Thus, if we exceed a certain threshold in compression, the original image cannot be recovered exactly. If the entropy is small, then the variability will be smaller. Therefore, the information of a pixel with respect to the pixels of its surroundings is high and, therefore, randomness is lost.

Definition 5.3: Let I be an 8 bit image that can take the values f g 0;…; 255 . Let pi be the frequency with which the value i∈f g 0; …; 255 appears. Then, the entropy is

Figure 10. PSNR of the reconstructions according to the used principal components. (a) PSNRof 256 reconstructions. (b) PSNR of some reconstructions.

$$H(I) = -\sum\_{i=0}^{255} p\_i \log\_2(p\_i) \tag{24}$$

the reconstructions using 8 components (black), 16 components (brown), 32 components

Application of Principal Component Analysis to Image Compression

http://dx.doi.org/10.5772/intechopen.75007

129

Finally, we consider the entropy of the images of the errors. Given an image, I, the value of each of its pixels is an element of the set 0f g ;…; 255 , and if we have a reconstruction, ^I, and consider the error, E ¼ I � ^I, then the value of its pixels will be an element of the set f g �255;…; 255 . Therefore, E cannot be considered as an image. Since a pixel of value eij in E is an error of the same size as �eij, to consider images we denominate image of the error to

Figure 12 (a) shows the entropy of the error image versus the number of principal components used for the reconstruction, together with an adjusted line of slope � 0.02. Figure 12(b) shows the entropy when we use 8 components (black), 16 components (brown), 32 components (green), 64 components (blue), and 128 components (red), respectively. With more than 200 principal components, the entropy of the errors is zero, which means that the errors have very little variability, and with fewer components, the decrease seems linear with slope �0:02.

In this section, we will consider the coordinates of the first vectors that form the principal

vectors will have 64 coordinates. Figure 13 shows the coordinates of the first six principal

As can be seen from Figure 13, all coordinates seem to have some component with period 8. This suggests that there may be some relationship with the shape of the blocks chosen and that most vectors are close to being periodic with period 8, because when we consider each of the

Figure 12. Entropy of the errors of the reconstructions converted into images according to the used principal components.

x23 dimension blocks,

components. If we consider that the vectors have been obtained as 23

(green), 64 components (blue), and 128 components (red), respectively.

Imð Þ¼ E eij

, being E ¼ eij .

5.2. Coordinates of the first principal component

components with respect to the canonical base.

(a) Entropy of differences (b) Entropy of some differences.

Figure 11 (a) shows the entropy of the reconstructions from 1 to 256 components. As can be seen, the entropy is increasing until the first 10 components, and then it becomes damped tending asymptotically to the value of the entropy of the image (7:4452). It can be seen that the difference with more than 170 components is insignificant. Figure 11 (b) shows the entropy of

Figure 11. Entropy of reconstructions according to the used principal components. (a) Entropy of reconstructions. (b) Entropy of some reconstructions.

the reconstructions using 8 components (black), 16 components (brown), 32 components (green), 64 components (blue), and 128 components (red), respectively.

Finally, we consider the entropy of the images of the errors. Given an image, I, the value of each of its pixels is an element of the set 0f g ;…; 255 , and if we have a reconstruction, ^I, and consider the error, E ¼ I � ^I, then the value of its pixels will be an element of the set f g �255;…; 255 . Therefore, E cannot be considered as an image. Since a pixel of value eij in E is an error of the same size as �eij, to consider images we denominate image of the error to Imð Þ¼ E eij , being E ¼ eij .

Figure 12 (a) shows the entropy of the error image versus the number of principal components used for the reconstruction, together with an adjusted line of slope � 0.02. Figure 12(b) shows the entropy when we use 8 components (black), 16 components (brown), 32 components (green), 64 components (blue), and 128 components (red), respectively. With more than 200 principal components, the entropy of the errors is zero, which means that the errors have very little variability, and with fewer components, the decrease seems linear with slope �0:02.

#### 5.2. Coordinates of the first principal component

H IðÞ¼�X

(b) PSNR of some reconstructions.

128 Statistics - Growing Data Sets and Growing Demand for Statistics

(b) Entropy of some reconstructions.

255

Figure 10. PSNR of the reconstructions according to the used principal components. (a) PSNRof 256 reconstructions.

� � (24)

i¼0 pi log2 pi

Figure 11 (a) shows the entropy of the reconstructions from 1 to 256 components. As can be seen, the entropy is increasing until the first 10 components, and then it becomes damped tending asymptotically to the value of the entropy of the image (7:4452). It can be seen that the difference with more than 170 components is insignificant. Figure 11 (b) shows the entropy of

Figure 11. Entropy of reconstructions according to the used principal components. (a) Entropy of reconstructions.

In this section, we will consider the coordinates of the first vectors that form the principal components. If we consider that the vectors have been obtained as 23 x23 dimension blocks, vectors will have 64 coordinates. Figure 13 shows the coordinates of the first six principal components with respect to the canonical base.

As can be seen from Figure 13, all coordinates seem to have some component with period 8. This suggests that there may be some relationship with the shape of the blocks chosen and that most vectors are close to being periodic with period 8, because when we consider each of the

Figure 12. Entropy of the errors of the reconstructions converted into images according to the used principal components. (a) Entropy of differences (b) Entropy of some differences.

Figure 13. Coordinates of the first six principal components with respect to the canonical base. (a) First component. (b) Second component. (c) Third component. (d) Fourth component. (e) Fifth component. (f) Sixth component.

Figure 14. Coordinates of the first three principal components when vectors are constructed from blocks of 22x22 and 24x24. (a) First component 22x22 (b) Second component 22x22 (c) Third component 22x22 (d) First component 24x24 (e)

Application of Principal Component Analysis to Image Compression

http://dx.doi.org/10.5772/intechopen.75007

131

Second component 24x24 (f) Third component 24x24.

Application of Principal Component Analysis to Image Compression http://dx.doi.org/10.5772/intechopen.75007 131

Figure 14. Coordinates of the first three principal components when vectors are constructed from blocks of 22x22 and 24x24. (a) First component 22x22 (b) Second component 22x22 (c) Third component 22x22 (d) First component 24x24 (e) Second component 24x24 (f) Third component 24x24.

Figure 13. Coordinates of the first six principal components with respect to the canonical base. (a) First component.

(b) Second component. (c) Third component. (d) Fourth component. (e) Fifth component. (f) Sixth component.

130 Statistics - Growing Data Sets and Growing Demand for Statistics

4096 vectors of 64 components, the first 8 pixels are adjacent to the next 16 pixels, and these are adjacent to the next 8 pixels, and so on, up to 8 times.

Since the first principal components collect a large part of the characteristics of the vectors, it is plausible that they also reflect the periodicity of the vectors. Recall that principal components are linear combinations of vectors and that if all of them had all their periodic coordinates with the same period, then all components would be periodic as well.

In Figure 14, the coordinates of the first three principal components are shown when the vectors are constructed from blocks of 22x22(see Figure 14 (a-c)) and from blocks of 24x24 (see Figure 14 (d-f)). As can be seen, the periodicity in the first components is again appreciated.

Figure 15. Compression with 2 and 8 original and periodic principal components. (a) Compression with two components (b) Compression with two componentsper. (c) Compression with eight components. (d) Compression with eight componentsper.

Figure 16. Differences between the image and the reconstruction according to the number of chosen components.

Application of Principal Component Analysis to Image Compression

http://dx.doi.org/10.5772/intechopen.75007

133

(a) 1-norm (b) 2-norm (c) ∞-norm.

4096 vectors of 64 components, the first 8 pixels are adjacent to the next 16 pixels, and these are

Since the first principal components collect a large part of the characteristics of the vectors, it is plausible that they also reflect the periodicity of the vectors. Recall that principal components are linear combinations of vectors and that if all of them had all their periodic coordinates with

In Figure 14, the coordinates of the first three principal components are shown when the vectors are constructed from blocks of 22x22(see Figure 14 (a-c)) and from blocks of 24x24 (see Figure 14 (d-f)). As can be seen, the periodicity in the first components is again appreciated.

Figure 15. Compression with 2 and 8 original and periodic principal components. (a) Compression with two components (b) Compression with two componentsper. (c) Compression with eight components. (d) Compression with eight

componentsper.

adjacent to the next 8 pixels, and so on, up to 8 times.

132 Statistics - Growing Data Sets and Growing Demand for Statistics

the same period, then all components would be periodic as well.

Figure 16. Differences between the image and the reconstruction according to the number of chosen components. (a) 1-norm (b) 2-norm (c) ∞-norm.


*Limitations*

Some of the limits of the research are related to self–reported data for understanding the concepts of mindfulness and self-regulation, the convenience sample typically biases [55,56]. Other limitations are due to the cross-sectional research design used; therefore, we cannot express any causal inferences from the obtained results.

## **9. Conclusions**

The data obtained is relevant both for those who practice handball and their coaches. Often, young aged Romanian sportsmen have the tendency of being competitive, using an external referential, either the opinion of the coach, or the motor qualities of other fellow players. They leave for a second level the internal referential, obtaining a state of well-being and the development of capacities by relating to their own self. Thus, they tend to experience a high level of anxiety and mistrust in their own forces, which visibly diminishes sport performances. They are aware to a too small extent that internal limits are greater than external limits. The competition with one's own self, accepting the level of performance as a departing point in the sport evolution, diminishing internal limitations, and the belief in one's own abilities to succeed, together with a good relation with one's self, giving conscious, constant attention, not only to the exterior environment but also to inner feelings, experienced emotions, and their e ffects on the body, are indicators of sport performance.

Attaining a state of flux has short term and long-term e ffects in preparing sportsmen, the optimum dosing of resources allowing openness towards new possibilities. Our results are consistent with some studies [1,57–59], even if the studies did not have handball players as respondents. One of the possible practical applications of this research would be to improve the training programme by introducing components of mindfulness and self-regulation strategies. We also recommend that, at the end of each training session, the coach use ways to facilitate the development of these two essential components in obtaining sports performance.

Future researches could focus on longitudinal studies that investigate the influence of self-regulation and practising mindfulness on sport performance. Another research direction could be exploring the methods of formation, by teaching these important competencies in fostering performance.

**Author Contributions:** Conceptualization: D.P. and L.-G.T.; Formal analysis: V.M., T.-M.I. and L.-G.T.; Investigation: V.M.; Methodology: V.M. and T.-M.I.; Resources: D.P., T.-M.I. and L.-G.T.; Software: D.P.; Supervision: T.-M.I.; Validation: L.-G.T.; Writing – original draft: D.P. and V.M.; Writing – review & editing: D.P., T.-M.I. and L.-G.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** Acknowledgements to NGO Handbal Veraflor management board and Irimia Marin Viorel, president of NGO Sporting Ghimbav.

**Conflicts of Interest:** No potential conflict of interest was reported by the authors.