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Introduction

In this homework, you will implement an interest point (keypoint) detector, a feature

descriptor and an image stitching tool based on feature matching and homography.

Interest point (keypoint) detectors find particularly salient points in an image. We

can then extract a feature descriptor that helps describe the region around each of the

interest points. SIFT, SURF and BRIEF are all examples of commonly used feature

descriptors. Once we have extracted the interest points, we can use feature descriptors to match them (find correspondences) between images to do interesting things like

panorama stitching or scene reconstruction.

We will implement the BRIEF feature descriptor in this homework. It has a compact

representation, is quick to compute, has a discriminative yet easily computed distance

metric, and is relatively simple to implement. This allows for real-time computation, as

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you have seen in class. Most importantly, as you will see, it is also just as powerful as

more complex descriptors like SIFT for many cases.

After matching the features that we extract, we will explore the homography between

images based on the locations of the matched features. Specifically, we will look at the

planar homographies. Why is this useful? In many robotics applications, robots must

often deal with tabletops, ground, and walls among other flat planar surfaces. When two

cameras observe a plane, there exists a relationship between the captured images. This

relationship is defined by a 3 × 3 transformation matrix, called a planar homography.

A planar homography allows us to compute how a planar scene would look from a

second camera location, given only the first camera image. In fact, we can compute

how images of the planes will look from any camera at any location without knowing

any internal camera parameters and without actually taking the pictures, all using the

planar homography matrix.

1 Keypoint Detector (Talha)

We will implement an interest point detector similar to SIFT. A good reference for its

implementation can be found in [3]. Keypoints are found by using the Difference of

Gaussian (DoG) detector. This detector finds points that are extrema in both scale and

space of a DoG pyramid. This is described in [1], an important paper in computer vision.

Here, we will implement a simplified version of the DoG detector described in Section 3

of [3].

HINT: All the functions to implement here are located in keypointDetect.py.

NOTE: The parameters to use for the following sections are:

σ0 = 1, k =

√

2, levels = [−1, 0, 1, 2, 3, 4], θc = 0.03 and θr = 12.

1.1 Gaussian Pyramid

In order to create a DoG pyramid, we will first need to create a Gaussian pyramid.

Gaussian pyramids are constructed by progressively applying a low pass Gaussian filter

to the input image. This function is already provided to your in keypointDetect.py.

GaussianPyramid = createGaussianPyramid(im, sigma0, k, levels)

The function takes as input an image which is going to be converted to grayscale

with intensity between 0 and 1 (hint: cv2.cvtColor(…)), the scale of the zeroth level

of the pyramid sigma0, the pyramid factor k, and a vector levels specifying the levels

of the pyramid to construct.

At level l in the pyramid, the image is smoothed by a Gaussian filter with σl = σ0k

l

.

The output GaussianPyramid is a R × C × L matrix, where R × C is the size of the

input image im and L is the number of levels. An example of a Gaussian pyramid

can be seen in Figure 1. You can visualize this pyramid with the provided function

displayPyramid(pyramid).

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Figure 1: Example Gaussian pyramid for model chickenbroth.jpg

Figure 2: Example DoG pyramid for model chickenbroth.jpg

1.2 The DoG Pyramid (5 pts)

The DoG pyramid is obtained by subtracting successive levels of the Gaussian pyramid.

Specifically, we want to find:

Dl(x, y, σl) = (G(x, y, σl−1) − G(x, y, σl)) ∗ I(x, y) (1)

where G(x, y, σl) is the Gaussian filter used at level l and ∗ is the convolution operator.

Due to the distributive property of convolution, this simplifies to

Dl(x, y, σl) = G(x, y, σl−1) ∗ I(x, y) − G(x, y, σl) ∗ I(x, y) (2)

= GPl − GPl−1 (3)

where GPl denotes level l in the Gaussian pyramid.

Q 1.2: Write the following function to construct a Difference of Gaussian pyramid:

DoGPyramid, DoGLevels = createDoGPyramid(GaussianPyramid, levels)

The function should return DoGPyramid an R × C × (L − 1) matrix of the DoG

pyramid created by differencing the GaussianPyramid input. Note that you will have

one level less than the Gaussian Pyramid. DoGLevels is an (L−1) vector specifying the

corresponding levels of the DoG Pyramid (should be the last L−1 elements of levels).

An example of the DoG pyramid can be seen in Figure 2.

1.3 Edge Suppression (10 pts)

The Difference of Gaussian function responds strongly on corners and edges in addition

to blob-like objects. However, edges are not desirable for feature extraction as they are

not as distinctive and do not provide a substantially stable localization for keypoints.

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Figure 3.a: Without edge

suppression

Figure 3.b: With edge suppression

Figure 3: Interest Point (keypoint) Detection without and with edge suppression for

model chickenbroth.jpg

Here, we will implement the edge removal method described in Section 4.1 of [3], which

is based on the principal curvature ratio in a local neighborhood of a point. The paper

presents the observation that edge points will have a “large principal curvature across

the edge but a small one in the perpendicular direction.”

Q 1.3: Implement the following function:

PrincipalCurvature = computePrincipalCurvature(DoGPyramid)

The function takes in DoGPyramid generated in the previous section and returns PrincipalCurvature,

a matrix of the same size where each point contains the curvature ratio R for the corresponding point in the DoG pyramid:

R =

Tr(H)

2

Det(H)

=

(λmin + λmax)

2

λminλmax

(4)

where H is the Hessian of the Difference of Gaussian function (i.e. one level of the DoG

pyramid) computed by using pixel differences as mentioned in Section 4.1 of [3]. (hint:

use Sobel filter cv2.Sobel(…)).

H =

Dxx Dxy

Dyx Dyy

(5)

This is similar in spirit to but different than the Harris corner detection matrix you saw

in class. Both methods examine the eigenvalues λ of a matrix, but the method in [3]

performs a test without requiring the direct computation of the eigenvalues. Note that

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you need to compute each term of the Hessian before being able to take the trace and

determinant.

We can see that R reaches its minimum when the two eigenvalues λmin and λmax

are equal, meaning that the curvature is the same in the two principal directions. Edge

points, in general, will have a principal curvature significantly larger in one direction

than the other. To remove edge points, we simply check against a threshold R > θr.

Fig. 3 shows the DoG detector with and without edge suppression.

1.4 Detecting Extrema (10 pts)

To detect corner-like, scale-invariant interest points, the DoG detector chooses points

that are local extrema in both scale and space. Here, we will consider a point’s eight

neighbors in space and its two neighbors in scale (one in the scale above and one in the

scale below).

Q 1.4: Write the function :

locsDoG = getLocalExtrema(DoGPyramid, DoGLevels, PrincipalCurvature,

th contrast, th r)

This function takes as input DoGPyramid and DoGLevels from Section 1.2 and PrincipalCurvature

from Section 1.3. It also takes two threshold values, th contrast and th r. The threshold θc should remove any point that is a local extremum but does not have a Difference

of Gaussian (DoG) response magnitude above this threshold (i.e. |D(x, y, σ)| > θc). The

threshold θr should remove any edge-like points that have too large a principal curvature

ratio specified by PrincipalCurvature.

The function should return locsDoG, an N × 3 matrix where the DoG pyramid

achieves a local extrema in both scale and space, and also satisfies the two thresholds.

The first and second column of locsDoG should be the (x, y) values of the local extremum

and the third column should contain the corresponding level of the DoG pyramid where

it was detected. (Try to eliminate loops in the function so that it runs efficiently.)

NOTE: In all implementations, we assume the x coordinate corresponds to

columns and y coordinate corresponds to rows. For example, the coordinate

(10, 20) corresponds to the (row 21, column 11) in the image.

1.5 Putting it together (5 pts)

Q 1.5: Write the following function to combine the above parts into a DoG detector:

locsDoG, GaussianPyramid = DoGdetector(im, sigma0, k, levels,

th contrast, th r)

The function should take in a gray scale image, im, scaled between 0 and 1, and

the parameters sigma0, k, levels, th contrast, and th r. It should use each of the

above functions and return the keypoints in locsDoG and the Gaussian pyramid in

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GaussianPyramid. Figure 3 shows the keypoints detected for an example image. Note

that we are dealing with real images here, so your keypoint detector may find points

with high scores that you do not perceive to be corners.

Include the image with the detected keypoints in your report (similiar to the one

shown in Fig. 3 (b) ). You can use any of the provided images.

2 BRIEF Descriptor (Rishi)

Now that we have interest points that tell us where to find the most informative feature

points in the image, we can compute descriptors that can be used to match to other

views of the same point in different images. The BRIEF descriptor encodes information

from a 9 × 9 patch p centered around the interest point at the characteristic scale of the

interest point. See the lecture notes to refresh your memory.

HINT: All the functions to implement here are located in BRIEF.py.

2.1 Creating a Set of BRIEF Tests (5 pts)

The descriptor itself is a vector that is n-bits long, where each bit is the result of the

following simple test:

τ (P; x, y) := (

1, if P[x] < P[y].

0, otherwise.

(6)

Set n to 256 bits. There is no need to encode the test results as actual bits. It is fine to

encode them as a 256 element vector.

There are many choices for the 256 test pairs {x, y} (remember x and y are 2D

vectors relating to discrete 2D coordinates within the 2D image patch matrix P) used

to compute τ (P; x, y) (each of the n bits). The authors describe and test some of them

in [2]. Read Section 3.2 of that paper and implement one of these solutions. You should

generate a static set of test pairs and save that data to a file. You will use these pairs

for all subsequent computations of the BRIEF descriptor.

Q 2.1: Write the function to create the x and y pairs that we will use for comparison

to compute τ :

compareX, compareY = makeTestPattern(patchWidth, nbits)

patchWidth is the width of the image patch (usually 9) and nbits is the number of

tests n in the BRIEF descriptor. compareX and compareY are linear indices into the

patchWidth × patchWidth image patch and are each nbits × 1 vectors. Run this routine for the given parameters patchWidth = 9 and n = 256 and save the results in

testPattern.npy. Include this file in your submission.

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2.2 Compute the BRIEF Descriptor (10 pts)

Now we can compute the BRIEF descriptor for the detected keypoints.

Q 2.2: Write the function:

locs,desc = computeBrief(im, GaussianPyramid, locsDoG, k, levels,

compareX, compareY)

Where im is a grayscale image with values from 0 to 1, locsDoG are the keypoint

locations returned by the DoG detector from Section 1.5, levels are the Gaussian scale

levels that were given in Section 1, and compareX and compareY are the test patterns

computed in Section 2.1 and were saved into testPattern.npy.

The function returns locs, an m×3 vector, where the first two columns are the image

coordinates of keypoints and the third column is the pyramid level of the keypoints, and

desc is an m × n bits matrix of stacked BRIEF descriptors. m is the number of valid

descriptors in the image and will vary. You may have to be careful about the input DoG

detector locations since they may be at the edge of an image where we cannot extract

a full patch of width patchWidth. Thus, the number of output locs may be less than

the input locsDoG. Note: Its possible that you may not require all the arguments to

this function to compute the desired output. They have just been provided to permit

the use of any of some different approaches to solve this problem.

2.3 Putting it all Together (5 pts)

Q 2.3: Write a function :

locs, desc = briefLite(im)

which accepts a grayscale image im with values between 0 and 1 and returns locs, an

m × 3 vector, where the first two columns are the image coordinates of keypoints and

the third column is the pyramid level of the keypoints, and desc, an m × n bits matrix

of stacked BRIEF descriptors. m is the number of valid descriptors in the image and

will vary. n is the number of bits for the BRIEF descriptor.

This function should perform all the necessary steps to extract the descriptors from

the image, including

• Compute DoG pyramid.

• Get keypoint locations.

• Compute a set of valid BRIEF descriptors.

2.4 Check Point: Descriptor Matching (5 pts)

A descriptor’s strength is in its ability to match to other descriptors generated by the

same world point despite change of view, lighting, etc. The distance metric used to

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compute the similarity between two descriptors is critical. For BRIEF, this distance

metric is the Hamming distance. The Hamming distance is simply the number of bits

in two descriptors that differ. (Note that the position of the bits matters.)

To perform the descriptor matching mentioned above, we have provided the function in

BRIEF.py):

matches = briefMatch(desc1, desc2, ratio)

Which accepts an m1 × n bits stack of BRIEF descriptors from a first image and a

m2×n bits stack of BRIEF descriptors from a second image and returns a p×2 matrix of

matches, where the first column are indices into desc1 and the second column are indices

into desc2. Note that m1, m2, and p may be different sizes and p ≤ min (m1, m2).

Q 2.4: Write a test script or utilize the code provided in the main function of BRIEF.py

to load two of the chickenbroth images and compute feature matches. Use the provided

plotMatches and briefMatch functions to visualize the result.

plotMatches(im1, im2, matches, locs1, locs2)

where im1 and im2 are two colored images stored as uint8, matches is the list of

matches returned by briefMatch and locs1 and locs2 are the locations of keypoints

from briefLite.

Save the resulting figure and submit it in your PDF report. Also, present results with

the two incline images and with the computer vision textbook cover page (template

is in file pf scan scaled.jpg) against the other pf * images. Briefly discuss any cases

that perform worse or better. See Figure 4 for an example result.

Suggestion for debugging: A good test of your code is to check that you can match

an image to itself.

2.5 BRIEF and rotations (10 pts)

You may have noticed worse performance under rotations. Let’s investigate this!

Q 2.5: Take the model chickenbroth.jpg test image and match it to itself while rotating the second image (hint: cv2.getRotationMatrix2D(…), cv2.warpAffine(…))

in increments of 10 degrees. Count the number of correct matches at each rotation and

construct a bar graph showing rotation angle vs the number of correct matches. Include

this in your PDF and explain why you think the descriptor behaves this way. Create

a separate script briefRotTest.py that performs this task.

3 Planar Homographies: Theory (20 pts) (Harsh)

Suppose we have two cameras looking at a common plane in 3D space. Any 3D point

w on this plane generates a projected 2D point located at u˜ = [u1, v1, 1]T on the first

camera and x˜ = [x2, y2, 1]T on the second camera. Since w is confined to a plane, we

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Figure 4: Example of BRIEF matches for model chickenbroth.jpg and

chickenbroth 01.jpg.

expect that there is a relationship between u˜ and x˜. In particular, there exists a common

3 × 3 matrix H, so that for any w, the following condition holds:

λx˜ = Hu˜ (7)

where λ is an arbitrary scalar weighting. We call this relationship a planar homography.

Recall that the ˜ operator implies a vector is employing homogenous coordinates such

that x˜ = [x, 1]T

. It turns out this homography relationship is also true for cameras that

are related by pure rotation without the planar constraint.

Q 3.1 We have a set of N 2D homogeneous coordinates {x˜1, . . . , x˜N } taken at one camera

view, and {u˜1, . . . , u˜N } taken at another. Suppose we know there exists an unknown

homography H between the two views such that,

λnx˜n = Hu˜n, for n = 1 : N (8)

where again λn is an arbitrary scalar weighting.

(a) Given the N correspondences across the two views and using Equation 8, derive a

set of 2N independent linear equations in the form:

Ah = 0 (9)

where h is a vector of the elements of H and A is a matrix composed of elements

derived from the point coordinates. Write out an expression for A.

Hint: Start by writing out Equation 8 in terms of the elements of H and the homogeneous coordinates for u˜n and x˜n.

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(b) How many elements are there in h?

(c) How many point pairs (correspondences) are required to solve this system?

Hint: How many degrees of freedom are in H? How much information does each

point correspondence give?

(d) Show how to estimate the elements in h to find a solution to minimize this homogeneous linear least squares system. Step us through this procedure.

Hint: Use the Rayleigh quotient theorem (homogeneous least squares).

4 Planar Homographies: Implementation (10 pts) (Talha)

Note: Implement the method in planarH.py.

Now that we have derived how to find H mathematically in Q 3.1, we will implement

in this section.

Q 4.1 (10pts) Implement the function

H2to1 = computeH(X1,X2)

Inputs: X1 and X2 should be 2 × N matrices of corresponding (x, y)

T

coordinates between

two images.

Outputs: H2to1 should be a 3 × 3 matrix encoding the homography that best matches

the linear equation derived above for Equation 8 (in the least squares sense). Hint:

Remember that a homography is only determined up to scale. The numpy.linalg function

eigh() or svd() will be useful. Note that this function can be written without an explicit

for-loop over the data points.

5 RANSAC (15 pts) (Harsh)

Note: Implement the method in planarH.py.

The least squares method you implemented for computing homographies is not robust to

outliers. If all the correspondences are good matches, this is not a problem. But even a

single false correspondence can completely throw off the homography estimation. When

correspondences are determined automatically (using BRIEF feature matching for instance), some mismatches in a set of point correspondences are almost certain. RANSAC

(Random Sample Consensus can be used to fit models robustly in the presence of outliers.

Q 5.1 (15pts): Write a function that uses RANSAC to compute homographies automatically between two images:

bestH = ransacH(matches, locs1, locs2, nIter, tol)

The inputs and output of this function should be as follows:

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Figure 5.a:

incline L.jpg (img1)

Figure 5.b:

incline R.jpg (img2)

Figure 5.c: img2 warped to

img1’s frame

Figure 5: Example output for Q 6.1: Original images img1 and img2 (left and center)

and img2 warped to fit img1 (right). Notice that the warped image clips out of the

image. We will fix this in Q 6.2

• Inputs: locs1 and locs2 are matrices specifying point locations in each of the

images and matches is a matrix specifying matches between these two sets of

point locations. These matrices are formatted identically to the output of the

provided briefMatch function.

• Algorithm Input Parameters: nIter is the number of iterations to run RANSAC

for, tol is the tolerance value for considering a point to be an inlier. Define your

function so that these two parameters have reasonable default values.

• Outputs: bestH should be the homography model with the most inliers found

during RANSAC.

6 Stitching it together: Panoramas (40 pts) (Shumian)

NOTE: All the functions to implement here are in panoramas.py.

We can also use homographies to create a panorama image from multiple views of the

same scene. This is possible for example when there is no camera translation between the

views (e.g., only rotation about the camera center). First, you will generate panoramas

using matched point correspondences between images using the BRIEF matching in

Section 2.4. We will assume that there is no error in your matched point

correspondences between images (Although there might be some errors, and

even small errors can have drastic impacts). In the next section you will extend

the technique to deal with the noisy keypoint matches.

You will need to use the perspective warping function from OpenCV: warp im =

cv2.warpPerspective(im, H, out size), which warps image im using the homography transform H. The pixels in warp_im are sampled at coordinates in the rectangle (0, 0)

to (out_size[0]-1, out_size[1]-1). The coordinates of the pixels in the source image

are taken to be (0, 0) to (im.shape[1]-1, im.shape[0]-1) and transformed according

to H. To understand this function, you may review Homework 1.

Q 6.1 (15pts) In this problem you will implement and use the function:

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panoImg = imageStitching(img1, img2, H2to1)

on two images from the Dusquesne incline. This function accepts two images and the

output from the homography estimation function. This function:

(a) Warps img2 into img1’s reference frame using the aforementioned perspective warping function

(b) Blends img1 and warped img2 and outputs the panorama image.

For this problem, use the provided images incline L as img1 and incline R as img2.

The point correspondences pts are generated by your BRIEF descriptor matching.

Apply your ransacH() to these correspondences to compute H2to1, which is the

homography from incline R onto incline L. Then apply this homography to incline R

using warpH().

Note: Since the warped image will be translated to the right, you will need a larger

target image.

Visualize the warped image and save this figure as results/6 1.jpg using OpenCV’s

cv2.imwrite() function and save only H2to1 as results/q6 1.npy using Numpy np.save()

function.

Q 6.2 (15pts) Notice how the output from Q 6.1 is clipped at the edges? We will fix

this now. Implement a function

[panoImg] = imageStitching noClip(img1, img2, H2to1)

that takes in the same input types and produces the same outputs as in Q 6.1.

To prevent clipping at the edges, we instead need to warp both image 1 and image 2

into a common third reference frame in which we can display both images without any

clipping. Specifically, we want to find a matrix M that only does scaling and translation

such that:

warp im1 = cv2.warpPerspective(im1, M, out size)

warp im2 = cv2.warpPerspective(im2, np.matmul(M,H2to1), out size)

This produces warped images in a common reference frame where all points in im1 and

im2 are visible. To do this, we will only take as input either the width or height of

out size and compute the other one based on the given images such that the warped

images are not squeezed or elongated in the panorama image. For now, assume we only

take as input the width of the image (i.e., out size[0]) and should therefore compute

the correct height(i.e., out size[1]).

Hint: The computation will be done in terms of H2to1 and the extreme points

(corners) of the two images. Make sure M includes only scale (find the aspect ratio of

the full-sized panorama image) and translation.

Again, pass incline L as img1 and incline R as img2. Save the resulting panorama

in results/q6 2 pan.jpg.

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Q 6.3 (10pts): You now have all the tools you need to automatically generate panoramas. Write a function that accepts two images as input, computes keypoints and descriptors for both the images, finds putative feature correspondences by matching keypoint

descriptors, estimates a homography using RANSAC and then warps one of the images

with the homography so that they are aligned and then overlays them.

im3 = generatePanorama(im1, im2)

Run your code on the image pair data/incline L.jpg, data/incline R.jpg. However

during debugging, try on scaled down versions of the images to keep running time low.

Save the resulting panorama on the full sized images as results/q6 3.jpg. (see Figure 6

for example output). Include the figure in your writeup.

Figure 6: Final panorama view. With homography estimated using RANSAC.

7 Submission and Deliverables

The assignment should be submitted to Canvas as a zip file named .zip and

to Gradescope as a pdf file named hw3.pdf. You can use check files.py to

ensure you have all required files ready before submission. The zip file should contain:

• A folder code containing all the .py files you were asked to write.

• A PDF hw3.pdf.

• A folder results containing all the .npy files you were asked to generate.

The PDF hw3.pdf should contain the results, explanations and images

asked for in the assignment along with to the answers to the questions on homographies.

Submit all the code needed to make your panorama generator run. Make sure all the

.py files that need to run are accessible from the code folder without any editing of the

path variable. You may leave the data folder in your submission, but it is not needed.

Note: Missing to follow the structure will incur huge penalty in scores!

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Appendix: Image Blending

Note: This section is not for credit and is for informational purposes only.

For overlapping pixels, it is common to blend the values of both images. You can

simply average the values but that will leave a seam at the edges of the overlapping

images. Alternatively, you can obtain a blending value for each image that fades one

image into the other. To do this, first create a mask like this for each image you wish

to blend:

mask = np.zeros((im.shape[0], im.shape[1]))

mask[0,:] = 1

mask[-1,:] = 1

mask[:,0] = 1

mask[:,-1] = 1

mask = scipy.ndimage.morphology.distance_transform_edt(1-mask)

mask = mask/mask.max(0)

The function distance_transform_edt computes the distance transform of the binarized input image, so this mask will be zero at the borders and 1 at the center of the

image. You can warp this mask just as you warped your images. How would you use

the mask weights to compute a linear combination of the pixels in the overlap region?

Your function should behave well where one or both of the blending constants are zero.

References

[1] P. Burt and E. Adelson. The Laplacian Pyramid as a Compact Image Code. IEEE

Transactions on Communications, 31(4):532–540, April 1983.

[2] Michael Calonder, Vincent Lepetit, Christoph Strecha, and Pascal Fua. BRIEF :

Binary Robust Independent Elementary Features.

[3] David G. Lowe. Distinctive Image Features from Scale-Invariant Keypoints. International Journal of Computer Vision, 60(2):91–110, November 2004.

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