Scheda di revisione: Fundamentals of Image Processing and Transformations

📋 Course Outline

1. Mathematical Notation
2. Image Foundations
3. Discrete Images
4. Coordinate System
5. Image Visualization
6. Discretization Methods
7. Image Interpolation
8. Image Transformations
9. Linear Transformations
10. Geometric Transformations
11. Affine Transformations
12. Image Characteristics

📖 1. Mathematical Notation

🔑 Key Concepts & Definitions

• Vectors and Matrices: Small boldface letters (e.g., v) denote vectors; capital boldface letters (e.g., A) denote matrices. Vectors can be transposed (vᵗ) to switch between row and column forms.

• Complex Conjugation: The symbol * indicates complex conjugation, performed component-wise for vectors and matrices.

• Hadamard Product and Division: Element-wise multiplication ($\circ$) and division ($/$) between vectors or matrices, applied element-by-element.

• Image Function: Images are modeled as functions f(x) with domain Ω (spatial coordinates) and range in color space, possibly continuous or discrete, multi-dimensional (e.g., 2D, 3D+t).

• Coordinate Transformations: Mapping between physical and normalized coordinates using scaling and offset parameters, enabling switching between image and real-world sizes.

• Discrete vs. Continuous Images: Discrete images are represented on index sets (pixels/voxels), while continuous images are functions over real-valued domains.

• Pixel and Intensity: Pixels are the smallest units in discrete images, characterized by position, value, and physical size; intensities are the color or gray level values within pixels.

• Array Representation: Multidimensional arrays store pixel data in contiguous memory, with linear (1D) indexing and Cartesian (multi-dimensional) indexing strategies.

• Transformations and Kernels: Linear transformations are represented via kernels (matrices or functions), with properties like separability (can be decomposed into simpler 1D operations) and symmetry.

• Shift Invariance: A property where the transformation's response does not depend on the position, allowing representation as a convolution with a point-spread function (PSF).

📝 Essential Points

• Mathematical notation standardizes image processing operations, with vectors/matrices and their conjugates, transposes, and element-wise products being fundamental.

• Images are functions over a domain, either continuous or discrete, with coordinate transforms enabling conversion between physical and normalized domains.

• Discrete images can be represented as matrices or arrays, with indexing strategies (linear vs. Cartesian) affecting computational efficiency, especially in neighborhood operations.

• Image transformations, especially linear ones, are often shift-invariant and can be expressed as convolutions with kernels, which may be separable for computational efficiency.

• Proper discretization (sampling and quantization) is crucial to avoid artifacts like aliasing, and the choice of kernel and boundary conditions impacts the quality of interpolation and transformation.

💡 Key Takeaway

Mathematical notation provides a precise framework for representing and manipulating images and their transformations, enabling efficient computation and analysis in image processing workflows.

📖 2. Image Foundations

🔑 Key Concepts & Definitions

• Discrete Image: An image represented as a finite set of pixel values arranged in a grid, either as a matrix or array, suitable for digital processing.
• Coordinate System: A framework defining pixel positions, typically with the origin at the upper-left corner, axes pointing right (horizontal) and down (vertical), using one-based indexing.
• Pixel: The smallest unit of a digital image, characterized by its position, intensity value, and physical or dimensionless size.
• Sampling: The process of converting a continuous image into a discrete form by measuring the image at specific points.
• Quantization: The process of mapping continuous intensity values into a finite set of levels, often using a fixed number of bits.
• Spatial Resolution: The ability to distinguish fine details in an image, determined by pixel size and physical properties of the imaging system.

📝 Essential Points

• Images can be modeled mathematically as functions with domain (spatial coordinates) and range (intensity or color values). They may be continuous or discrete.
• Discrete images are stored as matrices or arrays, with indexing strategies (Cartesian or linear) affecting neighbor operations and processing efficiency.
• Coordinate systems typically use one-based indexing with the origin at the upper-left pixel; the axes extend rightward and downward.
• Visualization methods include matrix value display, color images, surface plots, and profile plots, each suited for different analysis needs.
• Discretization involves sampling (domain) and quantization (range). Proper sampling prevents artifacts like aliasing; quantization limits the number of intensity levels.
• Image resolution impacts the level of detail; higher resolution captures finer features but increases memory and processing requirements.
• Noise degrades image quality and can be modeled as an additive random component; contrast measures the difference between light and dark regions.
• Memory footprint depends on image size and bit depth; efficient storage is crucial for large datasets.

💡 Key Takeaway

Understanding the mathematical and physical basis of images—including their discretization, coordinate systems, and resolution—is fundamental for effective image processing and analysis.

📖 3. Discrete Images

🔑 Key Concepts & Definitions

• Discrete Image: An image represented on a grid of distinct, separate points (pixels or voxels), with both domain (coordinates) and range (intensity values) discretized.

• Pixel (Picture Element): The smallest unit of a digital image, characterized by its position (coordinate), intensity value, and physical or dimensionless size.

• Matrix Representation: A 2D array where each element corresponds to a pixel's intensity, facilitating neighbor and local operations.

• Array Representation: Multidimensional data stored linearly in memory, with indexing strategies (Cartesian vs. linear) to access elements efficiently.

• Coordinate System: Defines pixel positions with a right-handed system, usually with the origin at the upper-left corner, axes directed downwards (vertical) and rightwards (horizontal).

• Sampling & Quantization:

  • Sampling: Discretizing the continuous domain by selecting specific points.
  • Quantization: Discretizing the range by assigning continuous intensity values to a finite set of levels, often represented with a fixed number of bits.

📝 Essential Points

• Discrete images are mathematically modeled as functions over a finite index set, often represented as matrices or multidimensional arrays.
• The matrix and array representations are interchangeable; matrices are more suitable for neighbor-based operations, while arrays facilitate efficient memory access.
• Indexing can be Cartesian (multi-dimensional) or linear (single index), with the choice depending on the application.
• The coordinate system typically uses one-based indexing with the origin at the upper-left pixel, axes pointing down (rows) and right (columns).
• Sampling involves selecting discrete points from a continuous signal, which can lead to artifacts like aliasing if not performed properly.
• Quantization reduces the precision of intensity values, with the number of bits determining the number of intensity levels; fewer bits cause visible quantization effects.

💡 Key Takeaway

Discrete images are fundamental in digital image processing, represented as matrices or arrays with specific indexing and coordinate conventions, where sampling and quantization are crucial steps that influence image quality and fidelity.

📖 4. Coordinate System

🔑 Key Concepts & Definitions

• Coordinate System: A mathematical framework that assigns a set of numbers (coordinates) to each point in an image, enabling precise localization and manipulation.
• Origin: The reference point in the coordinate system, typically located at the upper-left corner of an image (pixel (1,1)).
• Axes: The horizontal (x-axis) and vertical (y-axis) directions; in images, the x-axis points right, and the y-axis points downward.
• Right-Handed System: A coordinate system where the axes follow the right-hand rule; in images, this is often rotated 90° from standard Cartesian.
• Pixel Coordinates: Integer-based indices (i,j) used to locate pixels, with (1,1) at the top-left corner.
• Physical Coordinates: Real-world measurements (meters, millimeters) associated with pixel positions, obtained via scaling factors.
• Coordinate Transformations: Mathematical operations that convert between pixel indices and physical coordinates, often involving scaling and offsets.
• Image Center: The point in the image that divides it into symmetrical halves, often at (width/2, height/2) in pixel coordinates.
• Cartesian Indexing: Standard grid-based indexing where rows and columns are numbered starting from 1.
• Linear Indexing: A single-dimensional index that traverses the image array in a linear fashion, often used for efficient memory access.

📝 Essential Points

• The origin is typically at the upper-left corner, with axes directed rightward (x) and downward (y).
• Coordinates are one-based in many programming languages (e.g., MATLAB, Julia), meaning counting starts at 1.
• The aspect ratio of an image is determined by the resolution parameters; it influences the shape (landscape or portrait).
• Transformations between pixel and physical coordinates are crucial for accurate image analysis and processing.
• The image center can be defined either as the midpoint of pixel indices or in terms of physical coordinates, depending on application needs.
• Linear indexing simplifies memory access, while Cartesian indexing is more intuitive for neighborhood operations.

💡 Key Takeaway

The coordinate system in image processing provides a structured way to locate, manipulate, and analyze image data, bridging pixel indices with real-world measurements through transformations that are fundamental for accurate image analysis.

📖 5. Image Visualization

🔑 Key Concepts & Definitions

• Visualization: The process of representing image data visually to interpret or analyze it effectively, using various display methods such as matrix view, image view, surface plots, and profile plots.

• Matrix View: Displaying the raw pixel or intensity values of an image as a numerical matrix, useful for precise quantitative analysis.

• Image View: Rendering the image with color mapping to represent intensity or other features visually, often using grayscale or color maps for human perception.

• Surface Plot: A 3D visualization where pixel intensities are represented as height values on a surface, providing insight into the topography of the image data.

• Profile Plot: A 1D graph showing the variation of intensity along a specific line or region within the image, useful for analyzing spatial changes.

• Color Mapping: The process of assigning colors to intensity values to enhance visual interpretation, often involving colormaps like grayscale, jet, or custom schemes.

📝 Essential Points

• Multiple visualization techniques serve different purposes: matrix view for data accuracy, image view for human perception, surface plots for topographical analysis, and profile plots for detailed regional analysis.

• Proper color mapping is crucial for meaningful image display, especially when interpreting intensity variations.

• Visualization methods can be combined; for example, overlaying a profile plot on an image or using surface plots to understand 3D structures.

• Visualization aids in identifying artifacts, noise, resolution issues, and contrast problems in the image data.

• The choice of visualization depends on the analysis goal: quantitative inspection (matrix view), qualitative assessment (image view), or topographical understanding (surface plot).

💡 Key Takeaway

Effective image visualization employs multiple methods tailored to the analysis task, enabling both detailed quantitative evaluation and intuitive qualitative understanding of image data.

📖 6. Discretization Methods

🔑 Key Concepts & Definitions

• Discretization: The process of converting a continuous signal or image into a finite set of discrete values, enabling digital processing.

• Sampling: Discretizing the domain of a continuous function by measuring or recording values at specific, evenly spaced points. It transforms a continuous signal into a discrete one.

• Quantization: Discretizing the range (amplitude) of a signal by mapping continuous values to a finite set of levels, typically represented by a fixed number of bits.

• Sampling Theorem: A fundamental principle stating that a continuous signal can be perfectly reconstructed from its samples if it is band-limited and sampled at a rate greater than twice its highest frequency (Nyquist rate).

• Aliasing: An artifact that occurs when a signal is undersampled, causing different signals to become indistinguishable, leading to distortion or moiré patterns.

• Quantization Error: The difference between the actual continuous value and its quantized level, often introducing noise or distortion in the digital signal.

📝 Essential Points

• Discretization involves two key steps: sampling (domain discretization) and quantization (range discretization).
• Proper sampling rate is critical to avoid aliasing; undersampling causes artifacts like moiré effects.
• Quantization reduces the infinite set of possible amplitude values to a finite set, which can introduce quantization noise.
• The choice of bit depth in quantization affects the fidelity and dynamic range of the digital image.
• In image processing, sampling is often performed via sensors, and quantization is implemented through bit depth (e.g., 8-bit, 16-bit).
• Aliasing artifacts can be mitigated using anti-aliasing filters before sampling.
• The Nyquist criterion guides the minimum sampling rate needed for accurate reconstruction.

💡 Key Takeaway

Discretization transforms continuous images into digital data through sampling and quantization, requiring careful consideration of sampling rates and bit depth to prevent artifacts and preserve image quality.

📖 7. Image Interpolation

🔑 Key Concepts & Definitions

• Interpolation: The process of estimating pixel values at non-grid (off-grid) positions based on known pixel values, effectively creating a continuous representation of a discrete image.

• Spline Interpolation: A method that uses low-degree polynomials (splines) to smoothly interpolate data points, avoiding oscillations common in high-degree polynomial interpolation.

• Nearest-Neighbor Interpolation: The simplest method where the value of the closest pixel is assigned to the off-grid point, resulting in a blocky appearance.

• Linear Interpolation: Uses the two nearest pixels to estimate the value at an off-grid position by a weighted average, producing smoother results than nearest-neighbor.

• Cubic Interpolation: Employs neighboring pixels to fit a cubic polynomial, offering a smoother transition and differentiability at pixel boundaries.

• Boundary Conditions: Strategies to handle image edges during interpolation, such as flat (constant), linear, reflect, or periodic extrapolation.

📝 Essential Points

• Interpolation enables operations like image rotation, scaling, and warping by evaluating pixel values at arbitrary, non-integer coordinates.

• The choice of interpolation affects image quality: nearest-neighbor is fast but produces artifacts; cubic provides smoothness but is computationally more intensive.

• Spline interpolation, especially cubic splines, is preferred for high-quality image transformations due to its smoothness and stability.

• Boundary conditions are crucial to prevent artifacts at image edges; common strategies include reflection and periodic extension.

• Interpolation can be viewed as an approximation or an exact reconstruction depending on the trustworthiness of the data and the presence of noise.

💡 Key Takeaway

Image interpolation transforms discrete pixel data into a continuous function, enabling smooth and accurate image manipulations; the method chosen balances computational efficiency and visual quality.

📖 8. Image Transformations

🔑 Key Concepts & Definitions

• Image Transformation: A process that maps an input image to a new image, often altering geometry, intensity, or other features. Mathematically, it is a function $T: \mathbb{R}^n \to \mathbb{R}^m$.

• Linear Transformation: An operation where the output image is obtained by applying a linear operator $T$ that satisfies additivity and homogeneity: $T(aI + bJ) = aT(I) + bT(J)$. Represented as matrix multiplication in discrete images.

• Shift Invariance: A property where the transformation's response does not depend on the position in the image. Formally, $T(I(x - x_0)) = (T I)(x - x_0)$. Often associated with convolution operations.

• Convolution: A fundamental shift-invariant linear transformation where the output is the integral or sum of the input image weighted by a kernel $h$: $(I * h)(x) = \int I(t)h(x - t) dt$. Used for filtering, blurring, sharpening.

• Kernel (Filter): A small matrix or function used in convolution to modify the image, such as smoothing or edge detection. Can be separable (product of two 1D kernels) or non-separable.

• Inverse Transformation: A transformation $T^{-1}$ that reverses the effect of $T$, satisfying $T^{-1}(T(I)) = I$. Not all transformations are invertible.

📝 Essential Points

• Linear transformations can be efficiently implemented using matrix operations, especially when kernels are separable, reducing computational complexity from $O(n^2)$ to $O(n)$.

• Shift invariance is crucial in many image processing tasks, enabling the use of convolution kernels (e.g., Gaussian blur, edge detectors).

• Convolution kernels can be designed to perform various operations: smoothing (low-pass filters), sharpening (high-pass filters), or edge detection.

• Separable kernels allow 2D transformations to be performed as two successive 1D transformations, significantly improving efficiency.

• Transformation kernels may be symmetric or asymmetric, affecting the nature of the filtering effect.

• Inverse transformations are essential in image reconstruction and registration but may not always exist, especially for non-invertible operations.

💡 Key Takeaway

Image transformations, particularly linear and shift-invariant ones like convolution, form the backbone of many image processing techniques, enabling efficient filtering, geometric modifications, and reconstruction tasks. Understanding kernel properties and invertibility is essential for designing effective image algorithms.

📖 9. Linear Transformations

🔑 Key Concepts & Definitions

• Linear Transformation: A function $T: V \to W$ between vector spaces that preserves vector addition and scalar multiplication, i.e., for all vectors $\mathbf{u}, \mathbf{v}$ and scalar $c$: $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}), \quad T(c\mathbf{u}) = cT(\mathbf{u})$
• Matrix Representation: Any linear transformation $T$ can be represented as a matrix $A$ such that for a vector $\mathbf{x}$: $T(\mathbf{x}) = A\mathbf{x}$
• Separable Kernel: A transformation kernel $K(x,y)$ that can be expressed as a product of two functions: $K(x,y) = f(x)g(y)$ allowing the transformation to be performed as two successive 1D operations.
• Shift Invariance: A property where the transformation's output shifts correspondingly when the input is shifted; mathematically, for shift $\mathbf{h}$: $T(\mathbf{x} + \mathbf{h}) = T(\mathbf{x}) + \text{constant}$ implying the transformation can be expressed as a convolution with an impulse response.
• Convolution: A mathematical operation combining two functions $f$ and $g$ to produce a third function: $(f * g)(x) = \int f(t)g(x - t) dt$ fundamental in shift-invariant linear transformations.

📝 Essential Points

• Linear transformations are fundamental in image processing, enabling operations like filtering, scaling, and rotation.
• They can be efficiently implemented via matrix multiplication, especially when the kernel is separable.
• Shift invariance simplifies many image processing tasks, allowing transformations to be expressed as convolutions with a kernel or impulse response.
• In discrete images, linear transformations are represented as matrix-vector products, facilitating computational implementation.
• Understanding the properties of kernels (separable, symmetric) impacts the computational complexity and implementation strategies.

💡 Key Takeaway

Linear transformations in image processing are mathematical functions that preserve vector operations and can be efficiently implemented using matrices and convolutions, especially when shift invariance and kernel separability are exploited.

📖 10. Geometric Transformations

🔑 Key Concepts & Definitions

• Geometric Transformation: A process that modifies the spatial arrangement of an image, such as translation, rotation, scaling, or shearing, to produce a new image with altered geometry.

• Affine Transformation: A type of linear transformation combined with translation, preserving points, straight lines, and planes. Mathematically represented as $\mathbf{x'} = A\mathbf{x} + \mathbf{b}$, where $A$ is a matrix and $\mathbf{b}$ is a translation vector.

• Linear Transformation: A transformation where the output is a linear function of the input, represented by a matrix multiplication $\mathbf{x'} = A\mathbf{x}$. It includes operations like rotation, scaling, and shearing but excludes translation.

• Transformation Matrix: A matrix that encodes the parameters of a geometric transformation, such as rotation or scaling, which when multiplied with coordinate vectors, produces transformed coordinates.

• Shift (Translation): Moving every point in the image by a fixed amount in the x and y directions, represented as $\mathbf{x'} = \mathbf{x} + \mathbf{t}$.

• Homogeneous Coordinates: An extension of Cartesian coordinates used to represent affine transformations uniformly, enabling the combination of linear transformations and translations in a single matrix operation.

📝 Essential Points

• Geometric transformations are fundamental for image registration, correction, and manipulation tasks.

• Affine transformations include translation, rotation, scaling, and shearing; they preserve parallelism and ratios of distances along lines.

• The transformation matrix for affine transformations in 2D typically is a 3×3 matrix in homogeneous coordinates, combining linear and translational components.

• Transformation operations can be combined by matrix multiplication, allowing complex transformations to be performed sequentially.

• Interpolation is often necessary after transformations to estimate pixel values at non-integer coordinates, especially in rotation and scaling.

• Shift invariance means the transformation's effect is consistent regardless of the image's position, crucial for convolution-based methods.

💡 Key Takeaway

Geometric transformations manipulate the spatial structure of images through matrix operations, enabling tasks like alignment and resizing, with affine transformations being the most commonly used due to their simplicity and preservation of key geometric properties.

📖 11. Affine Transformations

🔑 Key Concepts & Definitions

• Affine Transformation: A geometric operation combining linear transformations (rotation, scaling, shear) with translation, preserving points, straight lines, and planes. Mathematically, it can be expressed as $\mathbf{x'} = A \mathbf{x} + \mathbf{b}$, where $A$ is a linear transformation matrix and $\mathbf{b}$ is a translation vector.

• Linear Part: The matrix $A$ in the affine transformation, responsible for scaling, rotation, shear, and reflection. It preserves collinearity and ratios of distances along parallel lines.

• Translation: The shift of all points by a fixed vector $\mathbf{b}$, moving the entire image without altering its shape or size.

• Homogeneous Coordinates: An extension of Cartesian coordinates used to represent affine transformations as matrix multiplications, enabling combined linear and translation operations in a single matrix.

• Affine Matrix: A $(n+1) \times (n+1)$ matrix in homogeneous coordinates that encapsulates both linear transformation and translation, typically written as: $\begin{bmatrix} A & \mathbf{b} \\ \mathbf{0}^T & 1 \end{bmatrix}$

• Preservation of Parallelism: Affine transformations maintain the parallelism of lines but not necessarily angles or lengths, distinguishing them from rigid transformations.

📝 Essential Points

• Affine transformations include operations like translation, scaling, rotation, shear, and combinations thereof, and are fundamental in image processing for geometric manipulations.

• They can be efficiently represented and computed using matrix operations in homogeneous coordinates, facilitating concatenation of multiple transformations.

• The inverse of an affine transformation exists if the linear part $A$ is invertible, allowing for reverse mapping.

• Affine transformations do not necessarily preserve angles or distances but do preserve straight lines and ratios of distances along parallel lines.

• Common applications include image registration, object scaling, rotation, skewing, and perspective correction.

• When applying affine transformations to images, interpolation methods (e.g., bilinear, bicubic) are used to evaluate pixel values at non-grid positions.

💡 Key Takeaway

Affine transformations are versatile tools in image processing that modify images through combined linear and translational operations, enabling flexible geometric adjustments while maintaining fundamental properties like straight lines and parallelism.

📖 12. Image Characteristics

🔑 Key Concepts & Definitions

• Spatial Resolution: The ability of an image to distinguish small details, determined by pixel size (pixel resolution) and physical resolution (real-world detail discernible).
  Physical resolution refers to the minimum distinguishable distance between features, often measured in millimeters or cycles per unit length.

• Contrast: The difference in intensity between the darkest and brightest parts of an image, often quantified using Michelson contrast:

  $C = \frac{I_{max} - I_{min}}{I_{max} + I_{min}}$

  where $I_{max}$ and $I_{min}$ are the maximum and minimum intensities.

• Noise: Random variations in pixel intensity that degrade image quality, modeled as an additive random component:

  $I_{noisy} = I_{true} + N$

  where $N$ is noise with certain statistical properties (mean, standard deviation).

• Intensity Resolution: The smallest distinguishable difference in pixel intensity, often related to the number of bits used for storage (bit depth).
  Higher bit depth allows finer gradations of intensity, but actual resolution can be limited by noise.

• Memory Footprint: The amount of storage required for an image, calculated as:

  $\text{Memory} = \text{Width} \times \text{Height} \times \text{Bits per pixel}$

  indicating how image size impacts storage needs.

📝 Essential Points

• Resolution Types: Pixel resolution (discrete sampling) versus physical resolution (real-world detail). Proper sampling prevents artifacts like aliasing (e.g., moiré patterns).
• Contrast and Noise: Higher contrast improves feature visibility; noise can obscure details and reduce effective contrast.
• Quantization: Discretization of intensity values into a finite set of levels, limited by bit depth. Quantization effects become visible at low bit depths (e.g., ≤4 bits).
• Image Characteristics Independence: Resolution, contrast, and noise are largely independent; optimizing one does not automatically improve others.
• Memory Considerations: Larger images or higher bit depths exponentially increase storage requirements, impacting processing and transmission.

💡 Key Takeaway

Image quality and interpretability depend on a balance of resolution, contrast, noise, and storage capacity; understanding these characteristics enables optimal image acquisition, processing, and analysis.

📊 Synthesis Tables

Aspect	Continuous Images	Discrete Images
Representation	Functions over real domain (Ω)	Arrays/matrices over index set
Sampling	Not applicable	Sampling converts continuous to discrete
Resolution	Infinite (theoretically)	Finite, determined by pixel grid
Processing operations	Mathematical functions, integrals	Matrix/array operations, convolutions
Storage	Not stored directly	Stored as pixel values in memory

Transformation Type	Linear Transformations	Geometric Transformations
Definition	Operations represented by matrices (A)	Spatial manipulations (rotation, scaling)
Shift invariance	Yes (via convolution)	Not necessarily shift-invariant
Kernel usage	Yes (for convolution)	Often, but includes non-linear methods
Examples	Rotation, scaling, shearing	Rotation, translation, affine, projective

⚠️ Common Pitfalls & Confusions

1. Confusing vectors (v) with matrices (A); vectors are 1D, matrices are 2D.
2. Misinterpreting complex conjugation (*); forgetting it applies component-wise.
3. Assuming shift invariance applies to all transformations; only applies to linear, convolution-based ones.
4. Mixing coordinate systems: physical vs. normalized; leads to incorrect transformations.
5. Overlooking sampling artifacts like aliasing when discretizing continuous images.
6. Assuming all images are naturally in RGB; many images are grayscale or multi-channel.
7. Confusing pixel size with image resolution; higher resolution means smaller pixels, not necessarily better quality.
8. Ignoring boundary effects when applying kernels or transformations; can cause artifacts.
9. Misapplying interpolation methods; choosing inappropriate methods for the task.
10. Overlooking the importance of proper quantization levels; too few bits cause banding.

✅ Exam Checklist

• Understand the notation for vectors, matrices, and their transposes.
• Know the difference between continuous and discrete images.
• Be able to describe the coordinate system conventions used in images.
• Explain the processes of sampling and quantization and their effects.
• Recognize how images are stored as arrays/matrices and the implications for processing.
• Describe the properties of linear transformations, including shift invariance and kernel representations.
• Differentiate between geometric, affine, and linear transformations.
• Understand the concept of shift invariance and its relation to convolution.
• Identify common discretization artifacts such as aliasing and quantization effects.
• Know various image visualization techniques and their purposes.
• Be familiar with the concepts of pixel size, spatial resolution, and their impact.
• Recognize the importance of boundary conditions in image processing operations.
• Master the vocabulary and basic operations related to image characteristics.