Windows into Elementary Mathematics:Alternate public images of mathematics and mathematicians

Research on students’ (and teachers’) images of mathematics and mathematicians reveals a number of stereotypical images, most of which are negative. In this paper we present an overview of some these images and stereotypes and consider the questions: (1) how might the image of mathematics and mathematicians be a problem in mathematics education, and (2) what can be done to remedy the situation? Also, we consider an outreach project called Windows into Elementary Mathematics. In this project mathematicians are interviewed about their perspectives on elementary mathematics topics and their interviews are videotaped and are posted online, along with supporting images and interactive content. In this context we consider the questions: (3) what is the Windows project about, and (4) how might it offer an alternate (and perhaps better) image of mathematics and mathematicians? Lastly, we share an example where activities from the project were used in a math-for-teachers course.

Automatic classification of prostate stromal tissue in histological images using Haralick descriptors and Local Binary Patterns

In this paper we presente a classification system that uses a combination of texture features from stromal regions: Haralick features and Local Binary Patterns (LBP) in wavelet domain. The system has five steps for classification of the tissues. First, the stromal regions were detected and extracted using segmentation techniques based on thresholding and RGB colour space. Second, the Wavelet decomposition was applied in the extracted regions to obtain the Wavelet coefficients. Third, the Haralick and LBP features were extracted from the coefficients. Fourth, relevant features were selected using the ANOVA statistical method. The classication (fifth step) was performed with Radial Basis Function (RBF) networks. The system was tested in 105 prostate images, which were divided into three groups of 35 images: normal, hyperplastic and cancerous. The system performance was evaluated using the area under the ROC curve and resulted in 0.98 for normal versus cancer, 0.95 for hyperplasia versus cancer and 0.96 for normal versus hyperplasia. Our results suggest that texture features can be used as discriminators for stromal tissues prostate images. Furthermore, the system was effective to classify prostate images, specially the hyperplastic class which is the most difficult type in diagnosis and prognosis.

Fractal dimension of the mandibular trabecular bone measured on digital and digitized images.

Aims: This study compared fractal dimension (FD) values on mandibular trabecular bone in digital and digitized images at different spatial and contrast resolutions. Materials and Methods: 12 radiographs of dried human mandibles were obtained using custom-fabricated hybrid image receptors composed of a periapical radiographic film and a photostimulable phosphor plate (PSP). The film/ PSP sets were disassembled, and the PSPs produced images with 600 dots per inch (dpi) and 16 bits. These images were exported as tagged image file format (TIFF), 16 and 8 bits, and 600, 300 and 150 dpi. The films were processed and digitized 3 times on a flatbed scanner, producing TIFF images with 600, 300 and 150 dpi, and 8 bits. On each image, a circular region of interest was selected on the trabecular alveolar bone, away from root apices and FD was calculated by tile counting method. Two-way ANOVA and Tukey’s test were conducted to compare the mean values of FD, according to image type and spatial resolution (α = 5%). Results: Spatial resolution was directly and inversely proportional to FD mean values and standard deviation, respectively. Spatial resolution of 150 dpi yielded significant lower mean values of FD than the resolutions of 600 and 300 dpi ( P < 0.05). A nonsignificant variability was observed for the image types ( P > 0.05). The interaction between type of image and level of spatial resolution was not signi fi cant (P > 0.05). Conclusion: Under the tested, conditions, FD values of the mandibular trabecular bone assessed either by digital or digitized images did not change. Furthermore, these values were in fluenced by lower spatial resolution but not by contrast resolution.

Automatic method to classify images based on multiscale fractal descriptors and paraconsistent logic

In this study is presented an automatic method to classify images from fractal descriptors as decision rules, such as multiscale fractal dimension and lacunarity. The proposed methodology was divided in three steps: quantification of the regions of interest with fractal dimension and lacunarity, techniques under a multiscale approach; definition of reference patterns, which are the limits of each studied group; and, classification of each group, considering the combination of the reference patterns with signals maximization (an approach commonly considered in paraconsistent logic). The proposed method was used to classify histological prostatic images, aiming the diagnostic of prostate cancer. The accuracy levels were important, overcoming those obtained with Support Vector Machine (SVM) and Bestfirst Decicion Tree (BFTree) classifiers. The proposed approach allows recognize and classify patterns, offering the advantage of giving comprehensive results to the specialists.

Comparison of linear measurements and analyses taken from plaster models and three-dimensional images

Digital models are an alternative for carrying out analyses and devising treatment plans in orthodontics. The objective of this study was to evaluate the accuracy and the reproducibility of measurements of tooth sizes, interdental distances and analyses of occlusion using plaster models and their digital images. Thirty pairs of plaster models were chosen at random, and the digital images of each plaster model were obtained using a laser scanner (3Shape R-700, 3Shape A/S). With the plaster models, the measurements were taken using a caliper (Mitutoyo Digimatic(®), Mitutoyo (UK) Ltd) and the MicroScribe (MS) 3DX (Immersion, San Jose, Calif). For the digital images, the measurement tools used were those from the O3d software (Widialabs, Brazil). The data obtained were compared statistically using the Dahlberg formula, analysis of variance and the Tukey test (p < 0.05). The majority of the measurements, obtained using the caliper and O3d were identical, and both were significantly different from those obtained using the MS. Intra-examiner agreement was lowest when using the MS. The results demonstrated that the accuracy and reproducibility of the tooth measurements and analyses from the plaster models using the caliper and from the digital models using O3d software were identical.

Estimation of chlorophyll-a concentration and the trophic state of the Barra Bonita hydroelectric reservoir using oli/landsat-8 images

Reservoirs are artificial environments built by humans, and the impacts of these environments are not completely known. Retention time and high nutrient availability in the water increases the eutrophic level. Eutrophication is directly correlated to primary productivity by phytoplankton. These organisms have an important role in the environment. However, high concentrations of determined species can lead to public health problems. Species of cyanobacteria produce toxins that in determined concentrations can cause serious diseases in the liver and nervous system, which could lead to death. Phytoplankton has photoactive pigments that can be used to identify these toxins. Thus, remote sensing data is a viable alternative for mapping these pigments, and consequently, the trophic. Chlorophyll-a (Chl-a) is present in all phytoplankton species. Therefore, the aim of this work was to evaluate the performance of images of the sensor Operational Land Imager (OLI) onboard the Landsat-8 satellite in determining Chl-a concentrations and estimating the trophic level in a tropical reservoir. Empirical models were fitted using data from two field surveys conducted in May and October 2014 (Austral Autumn and Austral Spring, respectively). Models were applied in a temporal series of OLI images from May 2013 to October 2014. The estimated Chl-a concentration was used to classify the trophic level from a trophic state index that adopted the concentration of this pigment-like parameter. The models of Chl-a concentration showed reasonable results, but their performance was likely impaired by the atmospheric correction. Consequently, the trophic level classification also did not obtain better results.