Description
1. Redundancy in the Color Space
Here you will explore how one can account for information that is perceptually redundant,
specifically, visually redundant color information.
The YCBCR colour space is derived from the RGB colour space and has the following three
components: Y, Luminance or Luma component obtained from RGB after gamma correction;
CB=B-Y, how far the blue component is from Luma; and CR=R-Y, how far the red component is
from Luma. This colour space separates the luminance and chrominance components into
different channels, and is mostly used in compression for TV transmission.
(a) Load ‘peppers.png’ (already included in default MATLAB path) into MATLAB with imread.
Convert the image into YCBCR and spatially downsample the luma channel Y by a factor of
4 using a ‘bilinear’ interpolator. Upsample the downsampled luma channel to its original
size, then compute and report the mean squared error with the original luma channel. Next,
reconstruct the image by combining the resampled luma with the original CB, and CR
information and converting it back to RGB. Compare this reconstruction with the original
‘peppers.png’ and comment on your observations.
(b) This time, rather than resampling the luma, downsample both chroma channels CB, and
CR. Measure and report the mean squared error between each resampled channel and its
original values separately. Finally, reconstruct the image by combining the two resampled
chroma channels with the original luma information and converting it back to RGB.
Compare this reconstruction with the original ‘peppers.png’ and comment on your
observations.
(c) Which of the two iterations resulted in a better looking image? Why? How is this relevant in
the context of image compression?
Assignment 3 – EECE 570: Fundamentals of Visual Computing
2. Reconstruction from Projections & CT
Figures (A-C) represent X-ray beams passing through a slice of brain tissue. The numbers at
the tail and head of each arrow indicate the intensities of the X-ray beam at the source and
detector, respectively. The grid lines are represented by 1 unit. The relationship between the
signal strength at the transmitter 𝐼0 and signal strength at the receiver 𝐼 is defined by the
following equation:
𝐼 = 𝐼0exp (− ∫ 𝜇(𝑥, 𝑦)𝑑𝑠)
(a) What is the relationship between the input and the output signal strength and the radon
transform? Consider the equation given above for understanding this relationship.
(b) Figure (D) shows how a line in x-y space is represented using ρ and θ. Your goal is to
report the Radon transform values at the 4 indicated points (a-d) in Figure (E), e.g. point d
is at (ρ, θ) = (3√2, 3𝜋⁄4)
Now, you will explore different interpolators and filters that can be employed to enhance the 2D
reconstruction from linear projections. You will use the Signal-to-Noise Ratio and Mean Squared
Error similarity metrics to evaluate the effectiveness of each parameter. The iradon function in
MATLAB automatically includes the following filters and interpolators which you will be
investigating.
interpolators = {‘nearest’;’linear’;’spline’;’pchip’;’cubic’;’v5cubic’};
filters = {‘none’;’Ram-Lak’;’Shepp-Logan’;’Cosine’;’Hamming’;’Hann’};
Load the Q2_student.m file and perform the following:
(c) Try all pairwise-combinations of the parameters listed above. Which pair of parameters
produces the best results in terms of a low MSE and a high SNR? (Hint: use loops to
Assignment 3 EECE 570: Fundamentals of Visual Computing
automate the process!) Does this correspond with what you visually observe to be the
best?
(d) Change the number of angles from 200 to 50 with steps of 50 degrees. Are the parameters
chosen for part (c) still the best candidates? Why or why not?
3. JPEG Compression
JPEG compression is a commonly used method of compression for digital images. The degree
of compression can be adjusted, allowing a selectable trade-off between storage size and
image quality.
The JPEG encoding process in color images comprises the following 5 steps:
1. The representation of the colors in the image is converted from RGB to YCBCR.
2. The resolution of the chroma data is reduced, usually by a factor of 2.
3. The image is split into blocks of 8×8 pixels, and for each block, each of the Y, CB, and CR
data undergoes the Discrete Cosine Transform (DCT). A DCT is similar to a Fourier
transform in the sense that it produces a kind of frequency spectrum.
4. The amplitudes of the frequency components are quantized.
5. The resulting data for all 8×8 blocks is further compressed with a lossless algorithm, a
variant of Huffman encoding.
In question 1 (“Redundancy in the Color Space”), you observed the effects of subsampling the
color space (steps 1 and 2 above). In this problem you will focus on steps 3 and 4. For the sake
of simplicity, you will perform JPEG compression on a scalar gray scale image from the default
MATLAB path: cameraman.tif.
(a) Use MATLAB’s dctmtx function to generate an 8×8 DCT transformation matrix. Plot this
matrix using imshow and briefly explain what the matrix does.
(b) Prior to performing DCT on the image cameraman.tif, the intensity values must be shifted
from a positive range of [0,255] to one centered around zero [-128,127] by subtracting 128
from every intensity value. Explain why.
(c) MATLAB’s documentation for dctmtx states: “In JPEG compression, the DCT of each 8-by8 block is computed. To perform this computation, use dctmtx to determine D, and then
calculate each DCT using D*A*D’ (where, A is each 8-by-8 block).” To perform DCT on
every 8×8 block of the image, use MATLAB’s blockproc function. Plot the resulting
transformed image and explain the results.
(d) To quantize the transformed image, the following matrix is used:
Assignment 3 – EECE 570: Fundamentals of Visual Computing
Q =
1 1 1 2 2 2 4 4
1 1 2 2 2 4 4 4
1 2 2 2 4 4 4 8
2 2 2 4 4 4 8 8
2 2 4 4 4 8 8 8
2 4 4 4 8 8 8 16
4 4 4 8 8 8 16 16
4 4 8 8 8 16 16 16
And the quantized DCT coefficients are computed as:
B = round(G./(q_level*Q)),
where, G is the un-quantized DCT coefficients, Q is the quantization matrix, q_level is the
level of quantization we want to achieve, and B is the resulting quantized (normalized)
coefficients. To achieve a higher level of quantization, G is divided by integer multiples of Q.
You will be asked to perform this quantization in the next question. For this part, just briefly
explain why this quantization matrix has the given values.
4. JPEG Decoding
You will now explore step 4 of JPEG compression. Before performing data compression, you
will study the effects of quantization of DCT coefficients.
(a) Perform the quantization as described in Q3(d). As you divide each block by the
quantization matrix and round it, you are increasing the sparsity of how the image is
represented effectively reducing many values of G to zero. To observe this, plot the
histogram of G using the hist function and compare it to the histogram of B with
q_level=10. Briefly comment on the histograms.
(b) Now that you quantized and compressed the image, try to reconstruct it. The first step is to
shift back the coefficients. To reconstruct, multiply B by Q the same number of times that
you divided it in the quantization step to obtain a new G:
G = B.*(q_level*Q)
Now perform the inverse block-wise DCT. The forward DCT transform was defined as
G=D*A*D’, the inverse DCT transform is defined as A=D’*G*D. Plot the reconstructed
image as shown below and explain the differences compared to the original image.
Hint: Remember to add back 128 and round the image values to [0 255].
Assignment 3 –
(c) Perform JPEG compression at different levels of quantization and explain your
observations. What happens at very high q_level values? Zoom in on the camera region of
the image and explain the artefacts that you observe as a result of this quantization.
5. Lossless (Huffman) Encoding
In this problem, you will focus on step 5 of the JPEG encoding process. The JPEG compression
standard uses two forms of lossless compression, run-length and Huffman, to further compress
the quantized DCT coefficients. Explore the effects of Huffman coding on the cameraman
image:
(a) Briefly explain information entropy and its relationship to compression.
(b) Compute the entropy of the image and calculate the maximum compression that can be
expected.
(c) Check whether the maximum compression calculated in 5(b) is obtained for your data, or if
not, how close it gets to that.
6. Predictive Coding
Predictive coding is a lossless compression technique that allows one to further compress an
image. A simple way to employ predictive coding on an image is by storing the difference
between adjacent rows in an image rather than the pixel values such that:
𝑒(𝑥, 𝑦) = 𝑓(𝑥, 𝑦) − 𝑓̂(𝑥, 𝑦)
𝑓̂(𝑥, 𝑦) = round[𝛼𝑓(𝑥, 𝑦 − 1)], 𝛼 = 1.
(a) Encode the cameraman.tif image using this predictive scheme and plot the resulting
(predicted) image. What does this image look like?
(b) To appreciate why this simple method can produce good compression, plot the histogram
of the cameraman image before and after the predictive encoding step and comment on the
differences between the two. Calculate the entropy of the two histograms to quantify the
difference between potential compression ratios that may be achieved.
Assignment 3 –
(c) Reconstruct the predictive encoded image and calculate the error between the original and
reconstructed versions. Is this method truly lossless, why or why not?
End of assignment 3
