CSCI 5561: Assignment 1


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Histogram of Oriented Gradients (HOG)


Figure 1: Histogram of oriented gradients. HOG feature is extracted and visualized for
(a) the entire image and (b) zoom-in image. The orientation and magnitude of the red
lines represents the gradient components in a local cell.

In this assignment, you will implement a variant of HOG (Histogram of Oriented Gradients) in Python proposed by Dalal and Trigg [1] (2015 Longuet-Higgins Prize Winner).

It had been long standing top representation (until deep learning) for the object detection task with a deformable part model by combining with a SVM classifier [2]. Given
an input image, your algorithm will compute the HOG feature and visualize as shown in
Figure 1 (the line directions are perpendicular to the gradient to show edge alignment).

The orientation and magnitude of the red lines represents the gradient components in
a local cell.
def extract_hog(im):

return hog
Input: input gray-scale image with uint8 format.

Output: HOG descriptor.
Description: You will compute the HOG descriptor of input image im. The pseudocode can be found below:
Algorithm 1 HOG

1: Convert the gray-scale image to float format and normalize to range [0, 1].
2: Get differential images using get_differential_filter and filter_image
3: Compute the gradients using get_gradient

4: Build the histogram of oriented gradients for all cells using build_histogram
5: Build the descriptor of all blocks with normalization using get_block_descriptor
6: Return a long vector (hog) by concatenating all block descriptors.

2.1 Image filtering
(a) Input image (b) Differential along x
(c) Differential along y
Figure 2: (a) Input image dimension. (b-c) Differential image along x and y directions.
def get_differential_filter():

return filter_x, filter_y
Input: none.

Output: filter_x and filter_y are 3×3 filters that differentiate along x and y directions, respectively.
Description: You will compute the gradient by differentiating the image along x and
y directions. This code will output the differential filters.
def filter_image(im, filter):

return im_filtered
Input: im is the gray scale m × n image (Figure 2(a)) converted to float format and
filter is a filter (k × k matrix)

Output: im_filtered is m × n filtered image. You may need to pad zeros on the
boundary on the input image to get the same size filtered image.
Description: Given an image and filter, you will compute the filtered image. Given the
two functions above, you can generate differential images by visualizing the magnitude
of the filter response as shown in Figure 2(b) and 2(c).

2.2 Gradient Computation
(a) Magnitude

(b) Angle (c) Gradient (d) Zoomed eye (e) Zoomed neck
Figure 3: Visualization of (a) magnitude and (b) orientation of image gradients. (c-e)
Visualization of gradients at every 3rd pixel (the magnitudes are re-scaled for illustrative
def get_gradient(im_dx, im_dy):

return grad_mag, grad_angle

Input: im_dx and im_dy are the x and y differential images (size: m × n).
Output: grad_mag and grad_angle are the magnitude and orientation of the gradient
images (size: m × n). Note that the range of the angle should be [0, π), i.e., unsigned
angle (θ == θ + π).

Description: Given the differential images, you will compute the magnitude and angle
of the gradient. Using the gradients, you can visualize and have some sense with the
image, i.e., the magnitude of the gradient is proportional to the contrast (edge) of the local patch and the orientation is perpendicular to the edge direction as shown in Figure 3.

2.3 Orientation Binning
Ignore this shaded area
Store gradient mag
4,3 
(a) ori histo
165 15 45 75 105 135 165

(b) Histogram per cell
Figure 4: (a) Histogram of oriented gradients can be built by (b) binning the gradients
to corresponding bin.
def build_histogram(grad_mag, grad_angle, cell_size):

return ori_histo

Input: grad_mag and grad_angle are the magnitude and orientation of the gradient
images (size: m × n); cell_size is the size of each cell, which is a positive integer.

Output: ori_histo is a 3D tensor with size M × N × 6 where M and N are the
number of cells along y and x axes, respectively, i.e., M = ⌊m/cell_size⌋ and N =
⌊n/cell_size⌋ where ⌊·⌋ is the round-off operation as shown in Figure 4(a).

Description: Given the magnitude and orientation of the gradients per pixel, you can
build the histogram of oriented gradients for each cell.
ori histo(i, j, k) = X
grad mag(u, v) if grad angle(u, v) ∈ θk (1)
where Ci,j is a set of x and y coordinates within the (i, j) cell, and θk is the angle
range of each bin, e.g., θ1 = [165◦
, 180◦
) ∪ [0◦
, 15◦
), θ2 = [15◦
, 45◦
), θ3 = [45◦
, 75◦
θ4 = [75◦
, 105◦
), θ5 = [105◦
, 135◦
), and θ6 = [135◦
, 165◦
). Therefore, ori_histo(i,j,:)
returns the histogram of the oriented gradients at (i, j) cell as shown in Figure 4(b).

Using the ori_histo, you can visualize HOG per cell where the magnitude of the line
proportional to the histogram as shown in Figure 1. Typical cell_size is 8.

2.4 Block Normalization
Blo2cxk2 block
Concatenation of HOG
and normalization
(a) Block descriptor

(b) Block overlap with stride 1
Figure 5: HOG is normalized to account illumination and contrast to form a descriptor
for a block. (a) HOG within (1,1) block is concatenated and normalized to form a long
vector of size 24. (b) This applies to the rest block with overlap and stride 1 to form
the normalized HOG.
def get_block_descriptor(ori_histo, block_size):

return ori_histo_normalized

Input: ori_histo is the histogram of oriented gradients without normalization.
block_size is the size of each block (e.g., the number of cells in each row/column),
which is a positive integer.

Output: ori_histo_normalized is the normalized histogram (size: (M−(block_size−
1)) × (N − (block_size − 1)) × (6 × block_size2

Description: To account for changes in illumination and contrast, the gradient strengths
must be locally normalized, which requires grouping the cells together into larger, spatially connected blocks (adjacent cells).

Given the histogram of oriented gradients, you
apply L2 normalization as follow:

1. Build a descriptor of the first block by concatenating the HOG within the block.

You can use block_size=2, i.e., 2 × 2 block will contain 2 × 2 × 6 entries that
will be concatenated to form one long vector as shown in Figure 5(a).

2. Normalize the descriptor as follow:

i = p
i + e
where hi
is the i
th element of the histogram and hˆ
is the normalized histogram.
e is the normalization constant to prevent division by zero (e.g., e = 0.001).

3. Assign the normalized histogram to ori_histo_normalized(1,1) (white dot location in Figure 5(a)).

4. Move to the next block ori_histo_normalized(1,2) with the stride 1 and iterate
1-3 steps above.

The resulting ori_histo_normalized will have the size of (M − 1) × (N − 1) × 24.

2.5 Application: Face Detection
(a) Template image (b) Target image

(c) Response map

(d) Thresholding (e) Non-maximum suppression
Figure 6: You will use (a) a single template image to detect faces in (b) the target
image using HOG descriptors. (c) HOG descriptors from the template and target image
patches can be compared by using the measure of normalized cross-correlation (NCC).

(d) Thresholding on NCC score will produce many overlapping bounding boxes. (e)
Correct bounding boxes for faces can be obtained by using non-maximum suppression.
Using the HOG descriptor, you will design a face detection algorithm. You can download the template and target images from the Canvas Assignment #1 page.
def face_recognition(I_target, I_template):

return bounding_boxes
Input: I_target is the image that contains multiple faces. I_template is the template
face image that will be matched to the image to detect faces.

Output: bounding_boxes is n×3 array that describes the n detected bounding boxes.
Each row of the array is [xi
, yi
, si
] where (xi
, yi) is the left-top corner coordinate of the
th bounding box, and si
is the normalized cross-correlation (NCC) score between the bounding box patch and the template:
s =
a · b
where a and b are two normalized descriptors, i.e., zero mean:
ai = ai − ea (4)
where ai
is the i
th element of a, and ai
is the i
th element of the HOG descriptor. ea is
the mean of the HOG descriptor.

Description: You will use thresholding and non-maximum suppression with IoU
50% to localize the faces. You may use
visualize_face_detection(I_target, bounding_boxes, bb_size)
to visualize your detection.

[1] N. Dalal and B. Triggs. Histograms of oriented gradients for human detection. In
CVPR, 2005.
[2] P. F. Felzenszwalb, R. B. Girshick, D. McAllester, and D. Ramanan. Object detection with discriminatively trained part based models. TPAMI, 2010.