Texture

I. Texture: A Deep Dive

• The Essence of Texture: The teacher emphasizes that texture isn’t just a random pattern; it’s a repetitive pattern of local variations in image intensity. This repetition is key to understanding and analyzing textures. Importantly, the teacher states, “Texture cannot be defined for a point.” It’s a regional property, requiring a spatial extent to be characterized.

• Why Textures Matter: The lecture highlights the crucial role of texture in computer vision tasks:

  • Object Detection & Recognition: Textures provide vital clues for identifying objects, even when they are partially occluded or have similar shapes.
  • Motion Analysis: Texture helps determine the movement of objects, as the repetitive patterns shift and deform over time.
  • Surveillance and Tracking: Textures are essential for tracking objects and individuals in surveillance scenarios.

• The Iris Example: The teacher uses the example of iris recognition to illustrate the practical application of texture analysis.

  • The iris has unique, intricate texture patterns that can be used for biometric identification.
  • The system extracts salient texture features during registration and stores them in a database.
  • During verification, the system compares the extracted features from a new image with the stored features to authenticate the individual.

• Fingerprint Minutiae: Similarly, fingerprint analysis relies on minutiae, which are specific ridge patterns with local curvature and bending moments. These features are unique to each individual and form the basis of fingerprint-based identification.

II. Limitations of Edge Detectors

• The teacher points out that traditional edge detectors, like differential operators, struggle with textured images.
• This is because the repetitive patterns create many edges within the texture region, making it difficult to isolate meaningful object boundaries.

III. Statistical Underpinnings of Texture

• Repetition Implies Statistics: Because textures involve repeating patterns, statistical measures become powerful tools for analysis.
• Gross Effects vs. Individual Elements: Instead of focusing on individual texels (texture elements), we analyze the overall statistical properties of a region, such as the mean, standard deviation, and variance of pixel intensities.

IV. Analysis vs. Synthesis: Two Sides of the Texture Coin

• Texture Analysis: The process of extracting information from a given texture. The teacher describes three main approaches:
  • Structural: Focuses on the arrangement of texels in the real domain.
  • Statistical: Employs statistical measures to characterize texture in the real domain.
  • Fourier: Analyzes texture in the frequency domain using Fourier transforms.
• Texture Synthesis: The creation of artificial textures based on specific parameters or descriptions.
  • Markov Random Fields: A statistical model used to generate textures by considering the relationships between neighboring pixels.
  • Graph Cut Textures: A technique that uses graph cuts to seamlessly stitch together different texture patches.
  • Image Analogies: A method that learns the relationship between a pair of images (e.g., a texture image and its filtered version) and applies it to synthesize new textures.

V. Delving into Texture Descriptors

• Texels: The Building Blocks: Texels are the fundamental units of texture, the repeating elements that form the pattern.
• Key Descriptors:
  • Size/Granularity: Refers to the dimensions of the texels. Are they small and fine, or large and coarse?
  • Directionality/Orientation: Captures the dominant direction of the texture. Is there a preferred orientation, or is it isotropic (the same in all directions)?
  • Randomness/Regularity: Quantifies the degree of randomness in the texture pattern. Is it a highly regular, predictable pattern, or does it have significant random variations?
• Brodatz Texture Database: The teacher mentions the Brodatz database as a valuable resource for researchers, providing a wide variety of texture images for testing and comparison.

VI. Statistical Approaches: A Closer Look

• Intuitive vs. Statistical: The teacher contrasts intuitive approaches (e.g., trying to understand the underlying cause of a fever) with statistical approaches (e.g., administering a common treatment based on observed effectiveness). Statistical texture analysis is often less intuitive but more computationally efficient and applicable to a wider range of images.
• Classification and Segmentation:
  • Texture Classification: The task of assigning a given texture to a predefined category based on its features.
  • Texture Segmentation: The process of dividing an image into regions with distinct textures. The teacher emphasizes the challenge of defining boundaries between textured regions, especially when they are not well-defined.

VII. Texture Energy: A Measure of Variation

• Energy and Amplitude: The teacher draws an analogy between image energy and the energy of a physical system, where energy is proportional to the square of the amplitude. In images, the amplitude corresponds to pixel intensity.
• Texture Energy Features: These features are derived by applying texture filters to the image, creating filtered images from which texture characteristics are computed.
• Texture Descriptor Image: The filtered image, where each pixel’s value represents a texture descriptor value, providing a more convenient representation for texture analysis.

VIII. Gray Level Co-occurrence Matrix (GLCM): A Powerful Tool

• Capturing Spatial Relationships: The GLCM is designed to quantify the spatial relationships between pixel intensities in a textured image.

• The Displacement Vector (d): This vector, denoted as $d=(dx,dy)$, specifies the spatial offset between the pixel pairs being analyzed.

  • $\delta x$: Represents the horizontal displacement.
  • $\delta y$: Represents the vertical displacement.
  • $d=(1,0)$: Indicates pixel pairs one unit apart horizontally.
  • $d=(0,1)$: Indicates pixel pairs one unit apart vertically.
  • $d=(1,1)$: Represents a 45-degree angle (diagonal neighbors).

• GLCM Construction:

  • $P_d(i,j)$: Represents the number of times pixel pairs with gray levels $i$ and $j$ occur at a distance $d$ apart in the image.
  • $N(i,j)$: Denotes the number of occurrences of pixel values $i$ and $j$ at a distance $d$ apart.

  $P_d(i,j)=\frac{N(i,j)}{{\sum }_{i,j}N(i,j)}$

• Normalization: To make the GLCM independent of image size, we often normalize it, resulting in the Normalized GLCM (NGLCM): $NGLCM(i,j)=\frac{P_d(i,j)}{{\sum }_{i=0}^{L-1}{\sum }_{j=0}^{L-1}{P}_d(i,j)}$

  where $L$ is the number of gray levels.

• Example:

  • Given a 3x3 image with pixel values 0, 1, and 2, and $d=(1,1)$, we count the diagonal co-occurrences.
  • If the pair (0, 1) appears twice along the diagonal, then $P(0,1)=2.$
  • The resulting GLCM will be a 3x3 matrix.
  • If we consider the following image, where d=(1,1)

     2 1 2
     0 1 1
     0 2 2
    

    The GLCM will be:

     0 1 2
    0 0 2 2
    1 2 1 2
    2 0 0 3
    

• Interpretation:

  • The largest value in the GLCM corresponds to the most frequent pixel pair at the defined distance and angle.
  • This provides insights into the repetitive nature of the texture and can be used for texture modeling and synthesis.

IX. Moments: Describing Shape and Distribution

• Moments in Mechanics: The teacher relates image moments to the concept of moments in mechanics, where the moment of a force is the product of the force’s magnitude and its perpendicular distance from a reference point.
• Image Moments: In image processing, moments are used to characterize the shape and distribution of pixel intensities within a region.
• Raw Moment of Order (r, s): $m_{rs}={\sum }_x{\sum }_y{x}^r{y}^{s}f(x,y)$ where:
  • $f(x,y)$ is the pixel intensity at location $(x,y)$.
  • $x$, $y$ are pixel coordinates.
• Central Moment of Order (r, s): ${\mu }_{rs}={\sum }_x{\sum }_y(x-\bar{x}{)}^r(y-\bar{y}{)}^{s}f(x,y)$ where:
  • $\bar{x}=\frac{m_{10}}{m_{00}}$ (x-coordinate of the centroid)
  • $\bar{y}=\frac{m_{01}}{m_{00}}$ (y-coordinate of the centroid)
• Meaning of Moments:
  • ${\mu }_{20}$ and ${\mu }_{02}$ are related to the variance (spread) of intensities along the x and y axes, respectively.
  • Higher-order moments capture more intricate shape information.
• Legendre Moments: These moments use Legendre polynomials $P_m(x)$ instead of $x^r$ and $y^s$ to provide a more robust representation: $L_m={\sum }_x{P}_m(x)f(x)$
• Joint Variation: Measures like correlation and GLCM capture the relationships between pixel intensities, providing further insights into texture structure.

X. Other Essential Texture Analysis Techniques

• Range: A simple measure calculated as the difference between the maximum and minimum intensity values within a neighborhood: $Range=I_{max}-I_{min}$
• Variance: Quantifies the spread of intensity values around the mean within a neighborhood: ${\sigma }^2=\frac{1}{N}{\sum }_{i=1}^N(x_i-\bar{x}{)}^2$ where:
  • $x_i$ are the pixel intensity values in the neighborhood.
  • $\bar{x}$ is the mean intensity value in the neighborhood.
  • $N$ is the number of pixels in the neighborhood.
• Edge-Based Features:
  • Edgeness per unit area: Measures the density of edges within a region, providing information about the texture’s complexity.
  • Edge Magnitude and Orientation: Histograms of edge magnitudes and orientations can reveal valuable textur
  • e characteristics.
• Entropy: A measure of randomness or uncertainty in pixel intensities, computed from the image histogram: $H=-{\sum }_i{p}_i\log p_i$ where $p_i$ is the probability of intensity level $i$.

XI. Macro vs. Micro Textures: A Matter of Scale

• Macro Textures: Characterized by large, distinct primitives, where the shape and arrangement of these primitives are crucial (e.g., tiger stripes).
• Micro Textures: Composed of small, densely packed primitives, where statistical methods are often more effective (e.g., the texture of grass).

XII. Three Pillars of Texture Analysis: Structural, Statistical, and Modeling

• Structural Approach: Defines texture as a collection of primitives arranged in a regular or repetitive manner.
• Statistical Approach: Employs quantitative measures (feature vectors) to describe the arrangement of intensities within a region.
• Modeling Approach: Develops mathematical models (e.g., generative models) to represent and synthesize textures.

XIII. Real-World Applications

• Object Recognition: Textures are vital for identifying objects based on their unique surface patterns.
• Image Segmentation: Separating different regions in an image based on their distinct textures.
• Scene Understanding: Analyzing textures to gain a deeper understanding of the content and context of a scene.
• Medical Imaging: Detecting anomalies or patterns in medical images, such as tissue analysis, by examining texture variations.
• Remote Sensing: Classifying different land cover types based on their characteristic textures.
• Computer Graphics: Generating realistic textures for synthetic images and objects to enhance visual fidelity.