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# Computer Vision Tasks

📗 Unsupervised:
➩ Image segmentation.
📗 Supervised:
➩ Image colorization.
➩ Image reconstruction.
➩ Image synthesis.
➩ Image captioning.
➩ Object detection and tracking.
➩ Medical image analysis.



# Convolution

📗 Pixels are not independent of their neighbors. Ignoring the dependence is inappropriate for most computer vision tasks.
📗 Neighboring pixel intensities can be combined to create one feature that captures the information in the region around the pixel.
📗 Linearly combining pixels in a rectangular region is called convolution: Wikipedia.
➩ The convolution of an \(m \times m\) matrix (representing a training image indexed \(i\)) \(X_{i}\) with a \(\left(2 k + 1\right) \times \left(2 k + 1\right)\) filter \(W = \left[W_{s t}\right]_{s = -k, ..., k, t = -k, ..., k}\) is an \(m \times m\) matrix \(A_{i} = X_{i} \star W\), where the row \(r\) column \(c\) entry is given by \(\left[A_{i}\right]_{r c} = \displaystyle\sum_{s = -k}^{k} \displaystyle\sum_{t = -k}^{k} \left[X_{i}\right]_{r + t, c + s} W_{-s, -t}\).
➩ Technical note: the definition here flips the filter, for example the filter \(\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}\) is flipped to \(\begin{bmatrix} 9 & 8 & 7 \\ 6 & 5 & 4 \\ 3 & 2 & 1 \end{bmatrix}\) before linear combination with the image. Similar linear combination without flipping the filter is called cross-correlation.
➩ The convolution filter is also called "kernel", but different from the kernel matrix for SVMs.
Math Note
📗 Convolution between two 1D vectors is also defined: the convolution of a vector \(x_{i} = \left(x_{i 1}, x_{i 2}, ..., x_{i m}\right)\) with a filter \(w = \left(w_{-k}, w_{-k + 1}, ..., w_{k-1}, w_{k}\right)\) is: \(a_{i} = \left(a_{i 1}, a_{i 2}, ..., a_{i m}\right) = x \star w\), where \(a_{i j} = w_{-k} x_{i \left(j + k\right)} + w_{-k + 1} x_{i \left(j + k - 1\right)} + ... + w_{k} x_{i \left(j - k\right)}\) for \(j = 1, 2, ..., m\).
📗 3D convolution can be defined in a similar way.
Example
filter

TopHat Discussion
📗 [1 points] Upload an image and try one of the following filters: . Image:
. Click on the image to perform convolution with this filter: .

For Gaussian filter: size = , \(\sigma\) = .
TopHat Quiz (Past Exam Question) ID:
📗 [4 points] What is the convolution between the image and the filter using zero padding? Remember to flip the filter first.

📗 Answer (matrix with multiple lines, each line is a comma separated vector): .




# Traditional Computer Vision

📗 Traditional computer vision algorithms use engineered features for images based on image gradients.
➩ Image gradients are changes in pixel intensity due to the change in the location of the pixel: Wikipedia.
➩ Histogram of Gradients (HOG) can be used as a features for images and is often combined with SVM for face detection and recognition tasks: Wikipedia.
Math Note
📗 Image gradients can be computed (approximated) by convolution with the following filters: \(\nabla_{x} I = W_{x} \star I\) and \(\nabla_{y} I = W_{y} \star I\).
➩ (Discrete) derivative filter: \(W_{x} = \begin{bmatrix} -1 & 0 & 1 \\ -1 & 0 & 1 \\ -1 & 0 & 1 \end{bmatrix}\) and \(W_{y} = \begin{bmatrix} -1 & -1 & -1 \\ 0 & 0 & 0 \\ 1 & 1 & 1 \end{bmatrix}\).
➩ Sobel filters: \(W_{x} = \begin{bmatrix} -1 & 0 & 1 \\ -2 & 0 & 2 \\ -1 & 0 & 1 \end{bmatrix}\) and \(W_{y} = \begin{bmatrix} -1 & -2 & -1 \\ 0 & 0 & 0 \\ 1 & 2 & 1 \end{bmatrix}\), which can be viewed as a combination of the Gaussian filter (to blur the image) and the derivative filter: Wikipedia.
📗 Image gradients can be used in edge detection: Link, Wikipedia.
📗 Gradient magnitude at a pixel \((s, t)\) is given by \(G = \sqrt{\nabla_{x}^{2} I\left(s, t\right) + \nabla_{y}^{2} I\left(s, t\right)}\) and gradient direction \(\Theta = arctan\left(\dfrac{\nabla_{y} I\left(s, t\right)}{\nabla_{x} I\left(s, t\right)}\right)\).
Math Note
📗 For HOG features: in every 8 by 8 pixel region of an image, the gradient vectors \(\begin{bmatrix} \nabla_{x} I\left(s, t\right) \\ \nabla_{y} I\left(s, t\right) \end{bmatrix}\) are put into 9 orientation bins, for example, \(\left[0, \dfrac{2}{9} \pi\right], \left[\dfrac{2}{9} \pi, \dfrac{4}{9} \pi\right], ..., \left[\dfrac{16}{9} \pi, 2 \pi\right]\), and the histogram count is used as the HOG features.
📗 The resulting bins are normalized within a block of 2 by 2 regions.
📗 Scale Invariant Feature Transform (SIFT) produces similar feature representation using histogram of oriented gradients: Wikipedia.
➩ It is location invariant.
➩ It is scale invariant: images at different scales are used to compute the features.
➩ It is orientation invariant: dominant orientation in a larger region is calculated and all gradients in the region are rotated by the dominant orientation.
➩ It is illumination and contrast invariant: feature vectors are normalized so that they sum up to 1, and thresholded (values smaller than a threshold, for example 0.2, are made 0).
TopHat Discussion ID:
📗 [1 points] Select the pixels in each of the 9 bins. The histogram is computed by summing up the .



Example Python example: Link
hog




# Haar Features

📗 HOG and SIFT features are too expensive to compute for real-time face detection tasks.
📗 Each image contains large number of locations at different scales, but faces only occur in very few of them.
📗 Each feature and classifier based on the feature should be easy to compute, and boosting can be used to combine simple features and classifiers.
📗 Haar features are differences between sums of pixel intensities in rectangular regions and can be obtained by convolution with filters such as \(\begin{bmatrix} 1 & 1 \\ -1 & -1 \end{bmatrix}\), \(\begin{bmatrix} 1 & -1 \\ 1 & -1 \end{bmatrix}\), ...: Wikipedia.
📗 Integral images (sum of pixel intensities above and to the left of every pixel) can be used to further speed up the computation: Wikipedia.



# Cascade Classifiers

📗 Each weak classifier can be a decision stump (decision tree with only one split) based on a Haar feature.
📗 Finding the threshold by comparing information gain is computationally expensive, so it is usually computed as the mid-point of the averages of the two classes.
📗 A sequence of weak classifiers can be combined into a strong classifier using AdaBoost (Adaptive Boosting).
➩ Start with a weak classifier and equal weights on all training items.
➩ Increase the weights on the items that are classified incorrectly, and train another weak classifier based on the updated weights.
➩ Continue the process to create a sequence of weak classifiers. The combined classifier is called a strong classifier.
📗 A sequence of strong classifiers can be combined into a cascade classifier.
➩ Start with a classifier with close to 100 percent detection rate by a possibly large false-positive rate.
➩ Train the next classifier still with close to 100 percent detection rate but with lower false-positive rate.
➩ Repeat this process to get a sequence of classifiers. The combined classifier called a cascade classifier.
Math Notes
📗 To create each classifier in the cascade classifier:
➩ Start by assigning equal weights for each image: \(w_{i} = \dfrac{1}{n}\) for \(i = 1, 2, ..., n\).
➩ Find optimal feature and threshold given the weights: the cost of mistake on image \(i\) should be weighted by \(w_{i}\), for example, \(C = C_{1} + C_{2} + ... + C_{n}\) where \(C_{i} = w_{i}\) if \(f_{j}\left(x_{i}\right) = y_{i}\) and \(C_{i} = 0\) otherwise.
➩ Choose weight of classifier \(f_{j}\) to be \(\alpha_{j} = \dfrac{1}{2} \log\left(\dfrac{1 - C}{C}\right)\).
➩ Update the weights to \(w_{i} = w_{i} \cdot e^{-y_{i} \alpha_{j} f_{j}\left(x_{i}\right)}\).
➩ Repeat the process until the desired detection rate and false positive rates are reached: the resulting classifier is \(\alpha_{1} f_{1} + \alpha_{2} f_{2} + ...\).
Example
ghost




# Viola Jones

📗 Viola Jones algorithm is a popular real-time face detection algorithm: Wikipedia.
➩ Each classifier operates on a 24 by 24 region of the image at multiple scales (scaling factor of 1.25).
➩ The regions can be overlapping. Nearby detection of faces are combined into a single detection.



📗 Notes and code adapted from the course taught by Professors Jerry Zhu, Yudong Chen, Yingyu Liang, and Charles Dyer.
📗 Content from note blocks marked "optional" and content from Wikipedia and other demo links are helpful for understanding the materials, but will not be explicitly tested on the exams.
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Last Updated: August 22, 2025 at 10:06 AM