Image Processing and Projective Transformations

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Questions and Answers

What is a common drawback of subsampling an image?

Higher resolution than the original image

Increased brightness in the image

Production of aliasing artifacts (correct)

Loss of color saturation

Which interpolation method is typically chosen for better image resampling?

Nearest-neighbor interpolation

Bilinear interpolation

Linear interpolation

Bicubic interpolation (correct)

What process can improve image upsampling when multiple images are available?

Post-processing override

Bilinear interpolation

Super-resolution with multiple images (correct)

Contrast enhancement

What is the primary purpose of blurring before downsampling an image?

To reduce aliasing artifacts Signup and view all the answers

What is created when you repeatedly blur and downsample an image by 2x?

A Gaussian pyramid Signup and view all the answers

What is a characteristic of projective transformations regarding lines?

Lines remain straight but may not be parallel Signup and view all the answers

What problem can occur during the forward warping process?

Pixels may land between two other pixels Signup and view all the answers

Which of the following statements is true regarding the origin in projective transformations?

The origin can map to any point in the image Signup and view all the answers

What is required for inverse warping to be successfully executed?

Calculation of the inverse of the transform Signup and view all the answers

What can result from the process of forward warping?

Holes appearing in the transformed image Signup and view all the answers

How do projective transformations behave with respect to composition?

They are closed under composition Signup and view all the answers

What happens to ratios in projective transformations?

Ratios are not preserved Signup and view all the answers

Which image transformation type allows for inversion and composition?

Projective transformations Signup and view all the answers

What is the purpose of interpolation in the context of image processing?

To resample color values from an interpolated source image Signup and view all the answers

What results from using an insufficient sampling rate in image processing?

Aliasing, producing incorrect signals/images Signup and view all the answers

Which of the following best describes the Nyquist rate?

The minimum sampling rate needed to avoid aliasing Signup and view all the answers

What is one potential outcome of image sub-sampling?

Crumpled appearance or motion blur Signup and view all the answers

Which interpolation filter is likely to produce the smoothest results when resizing an image?

Sinc Signup and view all the answers

What is the wagon-wheel effect an example of?

An optical illusion caused by frame rates Signup and view all the answers

How does image interpolation help prevent jaggies during image processing?

By estimating pixel values between sampled points Signup and view all the answers

Which interpolation method tends to be the simplest and fastest, but may lead to less accurate results?

Nearest neighbor Signup and view all the answers

What is the simplest approach to image interpolation mentioned?

Repeating each row and column 10 times Signup and view all the answers

What are digital images formed from?

Discrete point-sampling of a continuous function Signup and view all the answers

Which interpolation method is considered an 'ideal' reconstruction technique?

Gaussian reconstruction Signup and view all the answers

What is the role of the reconstruction filter in image interpolation?

To convert a discrete image to a continuous function Signup and view all the answers

In the context of image interpolation, what is nearest-neighbor interpolation primarily used for?

Upscaling images through pixel repetition Signup and view all the answers

Which of the following is NOT a method of reconstruction mentioned?

Cubic interpolation Signup and view all the answers

What is the primary goal of image reconstruction in interpolation?

To generate new images at different resolutions and scales Signup and view all the answers

What defines a similarity transformation in the context of image alignment?

Translation, rotation, and uniform scale Signup and view all the answers

How can approximations be made when the original function is unknown?

By applying a principled way, such as filtering Signup and view all the answers

What is the main purpose of image warping in computer vision?

To change the domain of the image based on a new mapping function Signup and view all the answers

What does a parametric (global) warp imply about transformation T?

It is the same for any point p in the image Signup and view all the answers

In the function g(x) = h(f(x)), what does 'f' represent?

The input image before any mapping Signup and view all the answers

What is true about linear transformations in the context of parametric warping?

They can be represented by a 2x2 matrix Signup and view all the answers

What is the outcome of a uniform scaling transformation by a factor of s?

All points move away from the origin equally Signup and view all the answers

What is the key difference between image filtering and image warping?

Image filtering changes the range of values while image warping changes the domain Signup and view all the answers

What type of transformations are considered common linear transformations?

Translation, rotation, and scaling Signup and view all the answers

What is the primary purpose of Gaussian pre-filtering in the downsampling process?

To remove high-frequency components before subsampling Signup and view all the answers

In the context of aliasing, what happens when an image with high-frequency content is downsampled without filtering?

The image may show patterns or artifacts Signup and view all the answers

What visual effect is commonly associated with the wagon-wheel effect in video and animations?

Wheels appearing to rotate backward Signup and view all the answers

What is a Gaussian pyramid, and in which field is it primarily applied?

A multi-resolution representation in computer vision Signup and view all the answers

What should be the result of repeatedly making a checkerboard pattern smaller?

The image turns grey due to averaging Signup and view all the answers

When forming a Gaussian pyramid, what is the process involved?

Blurring and then subsampling the image Signup and view all the answers

What is a consequence of improper subsampling without prior filtering?

Introduction of aliasing artifacts Signup and view all the answers

What would happen to the image quality if an image is upsampled without filtering?

The image would be pixelated and blurred Signup and view all the answers

Study Notes

Course Information

Course title: Introduction to Computer Vision
Course code: CPS834/CPS8307
Instructor: Dr. Omar Falou
University: Toronto Metropolitan University
Semester: Fall 2024

Image Transformations and Warping

Image transformations modify an image's geometric structure.
Image warping changes an image's coordinate domain, not its range.
Transformations include translation, similarity, projective, Euclidean, and affine.
Reading material: Szeliski, Chapter 3.6

Image Alignment

Images may not perfectly align due to differences in perspective or camera position.
Geometric relationships analyze the spatial connections between images.

Image Warping

Image filtering manipulates the intensity range of an image.
Image warping changes the coordinate domain of an image.
Mathematical formula examples: g(x) = h(f(x)) and g(x) = f(h(x))

Parametric (Global) Warping

Examples include translation, rotation, and aspect ratios.
Parametric transformations can be described using a few parameters.
Transformation T is a coordinate-changing machine, where p' = T(p).
Linear transformations (can be represented by a 2x2 matrix) are used, where p' = Tp.

Common Linear Transformations

Uniform scaling (by a factor 's').
Rotation (by an angle θ about the origin).
The inverse of a rotation is its transpose. Mathematical formula example: R= [cos θ -sin θ] [sin θ cos θ]

2x2 Matrices

Some transformations (mirrors across axes or lines) can be represented with 2x2 matrices.
2D translation cannot be represented using a 2x2 matrix.

All 2D Linear Transformations

Linear transformations include scale, rotation, shear, and mirror.
They map the origin to the origin.
They map lines to lines, and parallel lines remain parallel.
They preserve ratios.
They are closed under composition.

Homogeneous Coordinates

Representing points using homogeneous coordinates helps incorporate translation into transformation matrices.
(x, y) becomes (x, y, w).
Conversion from homogeneous coordinates: (x/w, y/w).

Translation

In homogeneous coordinates, translation can be represented using a 3x3 matrix.
Example matrix: T = [1 0 tx] [0 1 ty] [0 0 1 ]

Affine Transformations

Affine transformations are combinations of linear and translation transformations.
The origin does not necessarily map to the origin.
Lines map to lines and parallel lines remain parallel.
Ratios are preserved.
Closed under composition.

Projective Transformations (Homographies)

A projective transformation maps points in 2D space to another plane. It can be represented by a 3x3 matrix.
Example matrix: H = [a b c] [d e f] [g h 1]

Points at Infinity

Projective transformations involve points at infinity

Image Warping with Homographies

These transformations involve mapping points in an image to another location.

Properties of Projective Transformations

Origin does not necessarily map to origin.
Lines map to lines.
Parallel lines do not necessarily remain parallel.
Ratios are not preserved.
Projective transformations are closed under composition.

2D Image Transformations

Various types of 2D transformations (translation, rigid transformations, similarity, affine, projective) exist.
Each transformation has a specific matrix that represents it and the number of degrees of freedom (e.g, 6 degrees of freedom for affine transformations).

Implementing Image Warping

Forward warping involves sending each pixel to its corresponding location in the output image.
Inverse warping involves getting each pixel in the output image by transforming back to the location in the source image.

Forward Warping

Pixels need to be interpolated if mapping falls between pixels.

Inverse Warping

Interpolated values from the source image determine pixel values.

Interpolation

Interpolation is about estimating values between sampled points.
Common filters for interpolation are nearest neighbor, bilinear, bicubic, and sinc filters.

Image Resampling and Interpolation

Includes resampling and interpolating between sample points
Key for upscaling or downscaling images
Nearest neighbor interpolation, bilinear interpolation, and bicubic interpolation are frequent methods.

Image Scaling

Techniques to increase or decrease image dimensions proportionally.
Common method is image sub-sampling where pixels are skipped.

Image Sub-sampling

Throwing away pixels to create images of reduced dimensions (e.g, 1/2 or 1/8 size).
Leads to image blurring or "crushing" of content, known as aliasing.
Gaussian prefiltering can reduce aliasing by blurring the image before sub-sampling

Aliasing

Occurs when sampling rate is too low for image details.
Leads to distortions (like the "wagon wheel" artifact). -Nyquist rate is the minimum sampling rate to avoid aliasing.

Wagon-wheel effect

It's a visual artifact that appears when the scene or object rotates in a video, but the sampling rate doesn't keep up with the rotation.
Leads to the appearance that it's moving in the opposite direction than expected.
The problem can be prevented by higher sample rates.

Temporal Aliasing

The effect that occurs in video when the frame rate isn't high enough to keep up with an object's motion.
Causes the object to appear to be moving slower or backwards.

Gaussian Pre-filtering (Antialiasing)

Blurring the image using a Gaussian filter before downsampling.
Reduces aliasing effects and preserves image detail more accurately.
Enables generation of proper downsampled images while resolving possible artifacts.

Gaussian pyramids

A set of images in which each successive level has lower resolution.
Created by blurring and sub-sampling images repeatedly.
Useful for image processing tasks that involve multiple resolution levels.

Upsampling

Increasing image dimensions
Typically involves interpolating values for new pixels.
Nearest neighbor interpolation and other methods like Bicubic interpolation can be used.

Image Interpolation

Estimating values for new pixels between sampled points in upsampling.
Important for generating images at higher resolutions.
Reconstruction filters are useful for estimating values at any resolution.

Raster-to-vector Graphics

Technique to convert pixel-based graphics to vector graphics (lines and curves, not pixels). Image tracing and vector magic are examples of this transformation.
Useful for maintaining image quality when scaling and manipulating images.
Good for image compression and scalability when scaling the images.

Depixelating Pixel Art

Methods to improve the quality of pixel-based art.
Aims at increasing the detail within pixelated art.

Modern Methods

Modern techniques for image upsampling, including algorithms like RAISR.
Advanced techniques for pixelated image resolution enhancement and better quality image generation.

Super-resolution with Multiple Images

Utilizing multiple images of a scene with slight shifts to enhance resolution.
Effectively used in some cellphone cameras.

Google Pixel 3 Super Resolution Zoom

Cell phone technique for enhancing resolution while keeping the quality of the image.
Improves the quality of images at higher resolutions while addressing problems associated with hand tremors and other artifacts

Edge Detection

Identifying and highlighting boundaries or transitions in image intensities.
Extracting vital image characteristics and details.

Origin of Edges

Factors behind edges, including surface normal discontinuities, depth discontinuities, surface color discontinuities, and illumination discontinuities.

Images as Functions...

Representation of images as functions showing how abrupt changes in intensity correspond to edges in images.
Useful for analyzing and interpreting images.

Characterizing Edges

Edges are characterized as rapid changes in image intensity functions.

Image Derivatives

Using differential calculus to extract edge information from images.
Finite difference approximations are methods to estimate derivatives from image data.

Image Gradient

The gradient shows the direction and magnitude of the maximum rate of increase in image intensity.
The magnitude corresponds to the edge strength and its direction represents the edge orientation.

Effects of Noise

Noisy images have random variations in image values causing artifacts and affecting the accuracy in edge detection. -Methods for removing noise are needed before edge detection or characterization procedures.

Solution: Smoothing First

Applying a smoothing filter to an image before edge detection reduces noise effects.

Associative Property of Convolution

Convolution can be used to smooth images, estimate derivatives or edges and it's associative.

The 1D Gaussian and its Derivatives

Gaussian distributions and their derivatives are used for image smoothing and edge detection.
Gaussian distribution can be used to smooth the image values and enhance edge characteristics.

2D Edge Detection Filters

2D Gaussian or Laplacian filters are used to detect edges at different resolutions, and to highlight important image features.

The Sobel Operator

An approximation of the derivative of Gaussian used to find edges in images by using partial derivatives on each dimension.
It's often used to locate and/or delineate edges.

Canny Edge Detector

Detecting edges in images to reveal boundaries or significant changes in image characteristics.
A commonly employed method in computer vision.
The parameters include smoothing, gradient magnitude, and thresholding rules to refine edge locations.

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Description

Test your knowledge on various aspects of image processing, focusing on subsampling, interpolation methods, and projective transformations. This quiz covers fundamental concepts like blurring, warping, and the effects of transformations on images. Challenge yourself with questions that probe your understanding of these advanced techniques in image manipulation.