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1 Short answer problems [20 points]

1. Suppose we form a texture description using textons built from a filter bank of multiple anisotropic

derivative of Gaussian filters at two scales and six orientations (as displayed below in Figure 1). Is the

resulting representation sensitive to orientation, or is it invariant to orientation? Explain why.

Figure 1: Filter bank

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2. Consider Figure 2 below. Each small square denotes an edge point extracted from an image. Say

we are going to use k-means to cluster these points’ positions into k=2 groups. That is, we will run

k-means where the feature inputs are the (x,y) coordinates of all the small square points. What is a

likely clustering assignment that would result? Briefly explain your answer.

Figure 2: Edge points

3. When using the Hough Transform, we often discretize the parameter space to collect votes in an

accumulator array. Alternatively, suppose we maintain a continuous vote space. Which grouping

algorithm (among k-means, mean-shift, or graph-cuts) would be appropriate to recover the model

parameter hypotheses from the continuous vote space? Briefly describe and explain.

4. Suppose we have run the connected components algorithm on a binary image, and now have access to

the multiple foreground ‘blobs’ within it. Write pseudocode showing how to group the blobs according

to the similarity of their outer boundary shape, into some specified number of groups. Define clearly

any variables you introduce.

2 Programming [80 points]

1. Color quantization with k-means.[40 points]

For this problem you will write code to quantize a color space by applying k-means clustering to the

pixels in a given input image, and experiment with two different color spaces—RGB and HSV. Write

Matlab(Python) functions as defined below. Save each function in a file called

and submit all of them.

(a) 5 points. Given an RGB image, quantize the 3-dimensional RGB space, and map each pixel in

the input image to its nearest k-means center. That is, replace the RGB value at each pixel with

its nearest cluster’s average RGB value. Use the following form:

[outputImg, meanColors] = quantizeRGB(origImg, k)

where origImg and outputImg are MxNx3 matrices of type uint8, k specifies the number of colors

to quantize to, and meanColors is a k x 3 array of the k centers. (You can use built-in k-means

function in Matlab and Python)

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(b) 5 points. Given an RGB image, convert to HSV, and quantize the 1-dimensional Hue space. Map

each pixel in the input image to its nearest quantized Hue value, while keeping its Saturation and

Value channels the same as the input. Convert the quantized output back to RGB color space.

Use the following form:

[outputImg, meanHues] = quantizeHSV(origImg, k)

where origImg and outputImg are MxNx3 matrices of type uint8, k specifies the number of

clusters, and meanHues is a k x 1 vector of the hue centers. (You can use built-in k-means

function in Matlab and Python)

(c) 5 points. Write a function to compute the SSD error (sum of squared error) between the original

RGB pixel values and the quantized values, with the following form:

function [error] = computeQuantizationError(origImg,quantizedImg)

where origImg and quantizedImg are both RGB images, and error is a scalar giving the total

SSD error across the image.

(d) 5 points. Given an image, compute and display two histograms of its hue values. Let the first

histogram use equally-spaced bins (uniformly dividing up the hue values), and let the second

histogram use bins defined by the k cluster center memberships (i.e., all pixels belonging to hue

cluster i go to the i-th bin, for i=1,…k). Use the following form:

function [histEqual, histClustered] = getHueHists(im, k)

where im is an MxNx3 matrix represeting an RGB image, and histEqual and histClustered are

the two output histograms.

(e) 5 points. Write a script colorQuantizeMain.m(py) that calls all the above functions appropriately using the provided image fish.jpg, and displays the results. Calculate the SSD error for

the image quantized in both RGB and HSV space. Write down the SSD errors in your answer

sheet. Illustrate the quantization with a lower and higher value of k. Be sure to convert an HSV

image back to RGB before displaying with imshow. Label all plots clearly with titles.

(f) 15 points. In your writeup, explain all the results. How do the two forms of histogram differ?

How and why do results vary depending on the color space? The value of k? Across different

runs?

2. Circle detection with the Hough Transform [40 points]

Implement a Hough Transform circle detector that takes an input image and a fixed radius, and returns

the centers of any detected circles of about that size. Include a function with the following form:

[centers] = detectCircles(im, radius, useGradient)

where im is the input image, radius specifies the size of circle we are looking for, and useGradient

is a flag that allows the user to optionally exploit the gradient direction measured at the edge points.

The output centers is an N x 2 matrix in which each row lists the (x,y) position of a detected circles’

center. Save this function in a file called detectCircles.m(py) and submit it.

Then experiment with the basic framework, and in your writeup analyze the following:

(a) 10 points. Explain your implementation in concise steps (English, not code).

(b) 10 points. Demonstrate the function applied to the provided images jupiter.jpg and egg.jpg.

Display the accumulator arrays obtained by setting useGradient to 0 and 1. In each case, display

the images with detected circle(s), labeling the figure with the radius. For Matlab you can use

impixelinfo to estimate the radius of interest manually. For Python, you can directly read the

image pixel information in the plot.

(c) 10 points. For one of the images, display and briefly comment on the Hough space accumulator

array.

(d) 5 points. Experiment with ways to determine how many circles are present by post-processing

the accumulator array.

(e) 5 points. For one of the images, demonstrate the impact of the vote space quantization (bin size).

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3 [OPTIONAL] Extra credit [up to 10 points]

Extend your Hough circle detector implementation to detect circles of any radius. Demonstrate the method

applied to the test images.

Matlab hints:

1. Useful functions: kmeans, rgb2hsv, hsv2rgb, imshow, im2double, reshape, subplot, title,

hist, atan2, hold on; plot, fspecial, conv2, im2double, sin, cos, axis equal; edge

impixelinfo.

2. If the variable im is a 3d matrix containing a color image with numpixels pixels, X = reshape(im,

numpixels, 3); will yield a matrix with the RGB features as its rows

3. im2double will rescaling the data as necessary (0−255 → 0−1). Make sure you rescale the data back.

Python hints:

1. Useful Modules: numpy, scipy, skimage, matplotlib.

2. Useful Functions: scipy.cluster.vq.kmeans2, numpy.gradient, numpy.arctan

skimage.color.rgb2hsv, skimage.color.rgb2gray, skimage.feature.canny.

3. If the variable im is a 3d matrix containing a color image with numpixels pixels, numpy.reshape(im,

(-1, 3)) will yield a matrix with the RGB features as its rows;

This assignment is adapted from the PS2 assignment from Kristen Grauman’s CS 376: Computer Vision at UT Austin.

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