JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 30, 1719-1731 (2014)

Fast Calculation of Histogram of Oriented Gradient

Feature by Removing Redundancy in Overlapping Block*

SOOJIN KIM AND KYEONGSOON CHO1

Department of Electronics Engineering
Hankuk University of Foreign Studies
Gyeonggi-do, 449-791 Korea
E-mail: {ksjsky9888; kscho}@hufs.ac.kr

In order to improve pedestrian detection accuracy, histogram of oriented gradient

(HOG) feature is widely used in many applications. Although HOG feature can provide
high detection accuracy, fast detection time is hardly achieved due to its computational
complexity. Therefore, this paper describes a novel algorithm for fast calculation of HOG
feature. In the proposed algorithm, HOG feature is calculated based on cells instead of
overlapping blocks to avoid redundancy. Furthermore, by identifying key rules and shar-
ing common operations in trilinear interpolation, the number of required operations in
HOG feature calculation is reduced up to 60.5% while detection accuracy is not degraded
at all. Therefore, the proposed method is applicable to many applications such as intelli-
gent vehicles, robots, and surveillance systems in which both high detection rate and fast
detection time are strongly required.

Keywords: pedestrian detection, histogram of oriented gradient, trilinear interpolation,

high detection rate, fast detection time

1. INTRODUCTION

A robust and discriminative feature is strongly required for pedestrian detection be-
cause of a wide range of pedestrian’s appearance, illumination, and cluttered background.
Since histogram of oriented gradient (HOG) [1] feature has achieved great success on
pedestrian detection, it is widely used in many applications such as intelligent vehicles,
robots, and surveillance systems. In these applications, detection speed is essential as
well as detection accuracy [2]. However, its detection speed is relatively slow due to the
high computational complexity of HOG feature calculation. Especially trilinear interpo-
lation, one of the most effective techniques to improve detection accuracy in HOG fea-
ture calculation, degrades detection speed significantly. In order to accelerate detection
speed, therefore, we present a novel method to significantly reduce the number of re-
quired operations in HOG feature calculation. By replacing block-based operation with
cell-based one and by identifying key rules in trilinear interpolation, each cell in over-
lapping blocks is considered only once and redundant operations are totally removed.
Furthermore, the number of required operations is reduced up to 60.5% by sharing the
common operations in trilinear interpolation.
The rest of this paper is organized as follows. Section 2 introduces the related works
to pedestrian detection using HOG feature. Section 3 briefly overviews the algorithm of

Received November 16, 2013; revised January 9, 2014; accepted January 20, 21014.
Communicated by Chu-Song Chen.
1
Corresponding author.
*
This research was supported by Basic Science Research Program through the National Research Foundation
of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2063422).
1719
1720 SOOJIN KIM AND KYEONGSOON CHO

HOG feature calculation, and Section 4 describes the proposed algorithm to accelerate
the speed of HOG feature calculation. The experimental results are given in Section 5.
Finally, Section 6 concludes this paper.

2. RELATED WORK

In recent years, many researches have been conducted for pedestrian detection.
Symmetry, corners, texture, shadow, color, edges, and gradients can be used to represent
the characteristics of the pedestrian. Among the various types of feature, HOG feature
that is based on gradient information is widely used since it is considered to be the most
discriminative feature for pedestrian detection. The performances of 16 state-of-the-art
pedestrian detectors across six data sets are evaluated in [2]. They reported that nearly all
modern detectors employ some form of gradient histograms. In [3], a diverse set of pe-
destrian detectors is evaluated with identical test criteria and data sets. According to their
experiments, an approach using HOG feature clearly outperforms others. In order to im-
prove detection accuracy, several features are often combined with HOG feature. A
combination of several features with HOG feature outperforms any individual feature in
[4]. By combining HOG feature with local binary pattern (LBP) [5], the detection rate is
increased by 7% in [6]. However, the increase in detection accuracy has been paid for
with increased computational costs [7]. In many applications, including automotive
safety, surveillance, and robotics, fast detection time is as important as high detection
rate.
Many researches have been proposed to improve detection speed for pedestrian de-
tection. Several methods to simplify the computation of HOG feature extraction are pro-
posed in [8-10]. In order to reduce the amount of calculations, the interpolation tech-
nique is discarded in [8], a simplified linear interpolation technique is applied in [9], and
an approximated trilinear interpolation is applied in [10]. Consequently, detection rates
are significantly degraded. Even though trilinear interpolation is one of the most effec-
tive techniques to improve detection accuracy, it is not easy to apply trilinear interpola-
tion technique in its original form to provide real-time detection due to its high computa-
tional complexity. Nonetheless, trilinear interpolation technique cannot be discarded,
simplified or approximated in HOG feature calculation since high detection rate is also
important in many applications. Instead the method to accelerate the computation of tri-
linear interpolation is strongly required.

3. ALGORITHM OF HOG FEATURE CALCULATION

3.1 Brief Review on Overall Algorithm

HOG feature is presented by N. Dalal and B. Triggs to improve accuracy of human

detection. It is based on evaluating well-normalized local histograms of image gradient
orientations in a dense grid [1]. Given pre-defined parameters in Table 1, HOG feature is
calculated as follows. As shown in Fig. 1, the first step of HOG feature calculation is to
calculate image gradient for each pixel in SX×SY detection window. Gradient for x-direc-
tion is calculated by Eq. (1) and y-direction is calculated by Eq. (2). In these equations,
 FAST HOG FEATURE CALCULATION BY REMOVING REDUNDANCY 1721

f(x, y) represents a pixel value for (x, y) position in an image I. Then the gradients are
used to calculate magnitude M and orientation θ by Eqs. (3) and (4). The third step is to
accumulate weighted votes for gradient magnitude into N orientation bins over p×p spa-
tial cells. When inter-bin distance (θdist) is 20° over 0~180°, N is determined as 9. In or-
der to avoid aliasing, block-based interpolation is applied to interpolate weighted votes
bilinearly between the neighboring bins in both orientation and position. The next step is
to normalize contrast within each block, and two representative normalization schemes
are presented in Eqs. (5) and (6). In these equations, Bk represents 36-dimensional vector
for a block, c represents each element in the vector, and ε is a small constant used to
avoid division by zero. The dimension of each block is determined by the number of
orientation bins in the block. Finally, the last step is to collect histogram vectors for all
overlapping blocks over detection window. When Sx=48, Sy=96, p=6, c=2, L=6, and N=9,
as shown in Fig. 1, the dimension of the final HOG feature for each detection window is
3,780 since the total number of 36-dimensional overlapping blocks in a window is 105.

Table 1. Parameters for HOG feature calculation.

window size (pixels) SX  SY cell size (pixels) pp
# of cells in block cc block size (pixels) pc  pc
block stride (pixels) (L, L) # of bins per cell N
# of cells in window CX × CY (CX = SX /p, CY = SY/p)
BX × BY
# of blocks in window
(BX = (SX – pc + L)/L, BY = (SY – pc + L)/L)

Fig. 1. HOG feature calculation.

1722 SOOJIN KIM AND KYEONGSOON CHO

3.2 Interpolation Technique in HOG Feature Calculation

As mentioned in Section 3.1, an interpolation technique is applied at the third step

of HOG feature calculation in order to avoid aliasing and to improve detection accuracy.
Fig. 2 shows a simple example of linear and trilinear interpolations. When N=9 over
0~180° and a gradient orientation for a pixel in Cell0 is 32°, the two nearest neighboring
bins are determined as 1 and 2 as shown in the figure. Since only orientation of a pixel is
considered in linear interpolation, the weighted votes are calculated by multiplying mag-
nitude (M), Gaussian coefficient (G), and weight for orientation (Wθ). Then, the weighted
votes are distributed into only two bins (bin 1 and bin 2 for Cell0 only). On the other
hand, orientation and the position of a pixel in block are all considered in trilinear inter-
polation. Therefore, the weighted votes are calculated by multiplying magnitude (M),
Gaussian coefficient (G), weight for orientation (Wθ), and weights for pixel position (Wx
and Wy), and distributed into eight surrounding bins (bin 1 and bin 2 for Cell0~Cell3) as
shown in Fig. 2.

Fig. 2. Example of linear and trilinear interpolations.

Even though trilinear interpolation improves detection rate considerably, fast detec-
tion time is hardly achieved due to its high computational complexity. Since HOG fea-
ture is calculated by considering all overlapping blocks in detection window, the amount
of computations is further increased. Table 2 shows the number of required multiplica-
tions per detection window including interpolation and L2norm normalization. As
shown in the table, the number of required multiplications for trilinear interpolation is
much larger than linear interpolation. In order to accelerate detection speed, therefore,
most researches usually discard trilinear interpolation and often apply linear interpolation
instead.

Table 2. Number of required operations in interpolations.

interpolation # of multiplications per detection window
linear {(4×p2×c2)+(c2×N)+1}×Bx×By
trilinear [{32×p ×(c1) }+{32×p2×(c1)}+(8×p2)+(c2×N)+1]×Bx×By
2 2
 FAST HOG FEATURE CALCULATION BY REMOVING REDUNDANCY 1723

4. PROPOSED ALGORITHM

Trilinear interpolation is applied in HOG feature calculation to avoid aliasing, but it

increases computational complexity significantly due to the overlapping-block-based
operation. However, there have been no attempts to remove the redundant operations in
trilinear interpolation. Moreover, trilinear interpolation technique is not applied in many
researches to achieve high detection speed, resulting in degraded detection accuracy. In
this paper, therefore, we address the problem of HOG feature calculation and propose a
novel algorithm to totally remove the redundant operations to accelerate detection speed
without any loss of detection accuracy.

Fig. 3. Number of required cells in HOG feature calculation.

As described in the previous section, redundant operations are inherently involved

in the original algorithm due to the overlapping-block-based operation. When SX=48,
SY=96, c=2, and L=p=6, for example, 40 cells are overlapped twice and 84 cells are
overlapped four times as shown in Fig. 3. Although the total number of cells in detection
window is 128, the number of required cells for HOG feature calculation is 420 due to
the overlapping blocks. In this paper, therefore, we propose a novel algorithm of HOG
feature calculation to totally remove the redundant operations. Since the overlapping
operations for each cell are considered in advance, each cell is required only once in the
proposed algorithm. From now on, we will describe the proposed algorithm in case that
c=2 and L=p=6.

Fig. 4. Pre-defined regions in block.

1724 SOOJIN KIM AND KYEONGSOON CHO

Trilinear interpolation requires the largest computational efforts in HOG feature

calculation. Since the amount of operations in trilinear interpolation depends on the pixel
position in a block, we define region xR and yR in a block and divide the block into four
regions (SB1~SB4) as shown in Fig. 4. An example of trilinear interpolation for the pixel
position is shown in Fig. 5. When a pixel is in xR∩yR as indicated (a) in Fig. 5, weighted
votes are distributed into eight surrounding bins (two nearest bins to θ for each of four
cells). In case that a pixel is in xR∪yR–xR∩yR as indicated (b), weighted votes are distrib-
uted into four bins (two nearest bins to θ for each two nearest cells that the pixel is posi-
tioned). Otherwise, weighted votes are distributed into only two bins (two nearest bins to
θ for the cell that the pixel is positioned). The last case is indicated as (c) in Fig. 5.

Fig. 5. Trilinear interpolation for pixel position.

Fig. 6. Cells with pre-defined regions and types.

Since interval of each overlapping block is one cell in each direction, most cells in
detection window have up to four types, depending on the position in overlapping block.
Therefore, we divide a cell into four regions (SC1~SC4) and define four cell types (type #
1~type #4) as shown in Fig. 6. For example, since cell_a in Fig. 6 is overlapped by four
blocks (block_a~block_d), cell_a is type #4 for block_a, type #3 for block_b, type #2 for
block_c, and type #1 for block_d. As shown in the figure, only Gaussian coefficient
among the operands in trilinear interpolation differs for each type (G1~G4 represent
Gaussian coefficients in region SB1~SB4). Therefore, we modified the equation of trilin-
ear interpolation to share the common operations for each cell type as shown in Table 3.
In this table, A~D represent weights for pixel position (Wx×Wy). Table 4 shows the
 FAST HOG FEATURE CALCULATION BY REMOVING REDUNDANCY 1725

equations for A~D. Note that G1~G4 and A~D are pre-computed in the proposed algo-
rithm since they are already determined by p and c. The modified equations of trilinear
interpolation are applied in Fig. 7 where the proposed algorithm of HOG feature calcula-
tion is presented.

Table 3. Proposed trilinear interpolation for each cell type.

cell region trilinear interpolation cell region trilinear interpolation
SC1 [Nk+α]←M×A×G1×β [N(k4)+α]←M×C×G2×β
SC1
[Nk+α]←M×A×G1×β [N(k3)+α]←M×A×G2×β
SC2
[N(k+1)+α]←M×C×G1×β SC2 [N(k3)+α]←M×A×G2×β
[Nk+α]←M×A×G1×β [N(k4)+α]←M×C×G2×β
type SC3 type
[N(k+2)+α]←M×B×G1×β [N(k3)+α]←M×A×G2×β
#1 #2 SC3
[Nk+α]←M×A×G1×β [N(k2)+α]←M×D×G2×β
[N(k+1)+α]←M×C×G1×β [N(k1)+α]←M×B×G2×β
SC4
[N(k+2)+α]←M×B×G1×β [N(k3)+α]←M×A×G2×β
SC4
[N(k+3)+α]←M×D×G1×β [N(k1)+α]←M×B×G2×β
[N(k28)+α]←M×B×G3×β [N(k32)+α]←M×D×G4×β
SC1
[N(k26)+α]←M×A×G3×β [N(k31)+α]←M×B×G4×β
SC1
[N(k28)+α]←M×B×G3×β [N(k30)+α]←M×C×G4×β
[N(k27)+α]←M×D×G3×β [N(k29)+α]←M×A×G4×β
type SC2 type
[N(k26)+α]←M×A×G3×β [N(k31)+α]←M×B×G4×β
#3 #4 SC2
[N(k25)+α]←M×C×G3×β [N(k29)+α]←M×A×G4×β
SC3 [N(k26)+α]←M×A×G3×β [N(k30)+α]←M×C×G4×β
SC3
[N(k26)+α]←M×A×G3×β [N(k29)+α]←M×A×G4×β
SC4
[N(k25)+α]←M×C×G3×β SC4 [N(k29)+α]←M×A×G4×β

Table 4. Pre-computed weights for pixel position.

(x, y) position for a pixel in cell weights
x=(x+0.5)/p0.5 y=(y+0.5)/p0.5 A← XH × YH
x=floor(x’) y=floor(y) B← XH × (1YH)
x1=xx, x2=1 x1 y1=yy, y2=1 y1 C← (1XH) × YH
XH=max(x1, x2) YH=max(y1, y2) D← (1XH) × (1YH)

In order to totally remove redundant operations, HOG feature is calculated by

cell-based operation in the proposed algorithm. Fig. 7 shows the proposed algorithm for
fast HOG feature calculation. As shown in Fig. 7, each cell is considered only once by
identifying key rules and sharing common operations in trilinear interpolation. Unlike
the original algorithm, block normalization is conducted after the interpolation for all
cells in detection window is finished in our algorithm. In Fig. 7, i represents an index of
cell row in detection window, j represents an index of cell number, k represents an index
of group of N bins for cell, n1 and n2 represent the two nearest bins to θ, and na and nb
represent the corresponding weights. In order to further reduce the redundant operations
in the proposed trilinear interpolation in Table 3, we shared common operations and
found the best order as shown in Fig. 7. By multiplying the operands in trilinear interpo-
lation with the specific order, the number of required multiplications in trilinear interpo-
1726 SOOJIN KIM AND KYEONGSOON CHO

lation is reduced up to 60.5% (from 272,160 to 107,496). When the proposed algorithm
is applied with c=3, the number of required operations is further reduced up to 68%
(from 604,800 to 193,896).

Fig. 7. Proposed algorithm for fast HOG feature calculation.

Fig. 8 shows the examples of HOG feature calculation using the proposed algorithm.
When the current pixel is at (0, 0) in the 0th cell, only type #1 is considered since the 0th
cell belongs to only one block as shown in the figure. Since the pixel is in SC1 of type #1,
the operation of trilinear interpolation is required for two bins as shown in Eq. (7). When
the current pixel is at (4, 0) in the 79th cell, type #2 and type #4 are considered due to the
two overlapping blocks as shown in Fig. 8. In this case, the operation of trilinear inter-
polation is required for six bins since the pixel is in SC2 of type #2 and type #4. As
shown in Eqs. (8) and (9), a total of 18 multiplications are required for two cell types.
 FAST HOG FEATURE CALCULATION BY REMOVING REDUNDANCY 1727

Fig. 8. Example of HOG feature calculation using the proposed method.

However, it is reduced to 12 by sharing the common operations in the specific order as

presented in step 5 of Fig. 7. By applying the proposed method in HOG feature calcula-
tion, the total number of required multiplications for each detection window is reduced
up to 60.5%.
We identified key rules and presented the revised equations of trilinear interpolation.
By using the proposed algorithm, detection speed can be significantly accelerated with-
out any loss of detection accuracy since the redundant operations in trilinear interpola-
tion are totally removed and HOG feature is calculated by cell-based operation. There-
fore, the proposed algorithm is considerably useful in many applications in which both
high detection rate and fast detection speed are strongly required.

5. EXPERIEMTNAL RESULTS

We have conducted experiments in order to demonstrate that trilinear interpolation

is superior to linear interpolation in terms of detection rate. In order to evaluate detection
rate, we tested the pedestrian detectors on Daimler [11], INRIA [1], and MIT [12] pedes-
trian datasets using linear support vector machine (SVM) [13]. For Daimler datasets,
5,000 positive and 5,000 negative samples are used to train the detectors, and 10,000
positive and 12,870 negative samples are randomly selected to test them. For INRIA and
MIT datasets, 2,878 positive and 12,180 negative samples are used to train the detectors,
and 1,594 positive and 10,560 negative samples are randomly selected for testing. Detec-
tion rates on Daimler and INRIA+MIT datasets are shown in Fig. 9. As shown in the
figure, the detection rate of the detector using trilinear interpolation is 78% at 10-4 false
positive per window (FPPW) on Daimler dataset and 84% on INRIA+MIT dataset. In
comparison, the detection rate of the detector using linear interpolation is 65% on Daim-
ler dataset and 82.5% on INRIA+MIT dataset. Therefore, trilinear interpolation tech-
nique should be applied in HOG feature calculation for high detection accuracy.
1728 SOOJIN KIM AND KYEONGSOON CHO

Fig. 9. Detection rates on Daimler and INIRA+MIT pedestrian datasets.

In the previous section, we proposed a novel algorithm to remove redundant opera-

tions totally in trilinear interpolation. In order to further reduce the redundant operations
in the proposed trilinear interpolation, we shared common operations and found the best
order. Table 5 shows the comparison results of the number of required multiplications in
trilinear interpolation per detection window. As shown in the table, the number of re-
quired multiplications is reduced for each order of six kinds. By multiplying the oper-
ands in trilinear interpolation with the specific order (order #2), the number of required
multiplications in trilinear interpolation is reduced up to 60.5% (from 272,160 to
107,496) in the proposed method.

Table 5. Number of required multiplications in trilinear interpolation.

# of multiplications comparison
conventional method 272,160 
1) Q←M×β
order #1 2) R←Q×(A~D) 110,736 59.3% ↓
3) S←R×(G1~G4)
1) Q←M×β
order #2 2) R←Q×(G1~G4) 107,496 60.5% ↓
3) S←R×(A~D)
1) Q←M×(G1~G4)
order #3 2) R←Q×(A~D) 117,180 56.9% ↓
3) S←R× β
1) Q←M×(G1~G4)
order #4 2) R←Q× β 113,400 58.3% ↓
3) S←R×(A~D)
1) Q←M×(A~D)
order #5 2) R←Q× β 117,540 56.8% ↓
3) S←R×(G1~G4)
1) Q←M×(A~D)
order #6 2) R←Q×(G1~G4) 118,800 56.4% ↓
3) S←R× β
 FAST HOG FEATURE CALCULATION BY REMOVING REDUNDANCY 1729

In order to evaluate the actual processing time, we implemented our algorithm using
C/C++ programming language and OpenCV2.0 library [14]. Table 6 shows the compar-
ison results of CPU processing time per detection window. Five test images from Daim-
ler dataset and an Intel Core i7-2600@3.40GHz with 16GB RAM are used in the evalua-
tion. By comparing CPU processing time, we found that our algorithm reduces the pro-
cessing time up to 58.6% for these test images.

Table 6. CPU processing time per detection window.

CPU processing time (ms)
conventional method proposed method comparison
test image 1 8.61 3.58 58.4% ↓
test image 2 8.51 3.56 58.2% ↓
test image 3 8.64 3.58 58.6% ↓
test image 4 8.37 3.54 57.7% ↓
test image 5 8.48 3.62 57.3% ↓

6. CONCLUSIONS
Since HOG feature is calculated by collecting histograms for all overlapping blocks
in detection window, most cells in the overlapping blocks are considered redundantly in
trilinear interpolation. Since trilinear interpolation requires the largest computational
efforts in HOG feature calculation, it significantly increases the entire processing time of
pedestrian detection. In this paper, therefore, we proposed a novel algorithm of fast HOG
feature calculation to remove the redundant operations totally in trilinear interpolation.
By identifying key rules and sharing common operations in trilinear interpolation, high
detection rate is still achieved but the number of required multiplications is reduced up to
60.5%. Since trilinear interpolation is applied without any loss of accuracy and the num-
ber of required operations in HOG feature calculation is significantly reduced, the pro-
posed algorithm can be applied to accelerate detection speed in many applications in
which both high detection speed and accuracy are crucial.

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14. Intel Open CV Library, http://www.opencv.org.

Soojin Kim was born in 1983 at Seoul, Korea. She received

her B.S. and M.S. degrees in Electronics Engineering from Han-
kuk University of Foreign Studies, Korea, in 2007 and 2009, re-
spectively. She received her Ph.D. degree from the Department of
Electronics Engineering at Hankuk University of Foreign Studies,
Korea, in 2014. From 2010 to 2013, she was a Researcher at the
SoC Platform Research Center at Korea Electronics Technology
Institute, Korea. Her research interests are the SoC architecture
and design for multimedia and communications, pattern recogni-
tion and their application to vision systems.

Kyeongsoon Cho was born in 1959 at Seoul, Korea. He re-

ceived his B.S. and M.S. degrees in Electronics Engineering from
Seoul National University, Korea, in 1982 and 1984, respectively.
He received his Ph.D. degree from the Department of Electrical
and Computer Engineering at Carnegie Mellon University, U.S.A,
in 1988. From 1988 to 1994, he was a Senior Researcher at the
 FAST HOG FEATURE CALCULATION BY REMOVING REDUNDANCY 1731

Semiconductor ASIC Division of the Samsung Electronics Company. He was responsi-

ble for the research and development of the ASIC cell library and design automation.
Since 1994, he has been a Professor at the Department of Electronics Engineering at
Hankuk University of Foreign Studies. From 1999 to 2003, he was a Senior Director at
Enhanced Chip Technology. From 2003 to 2004, he was a head of the CoAsia Korea
Research and Development Center and a technical advisor of Dongu HiTek from 2005 to
2012. From 2005 to 2011, he was a Vice Director of the Collaborative Project for Excel-
lence in System IC Technology sponsored by the Ministry of Knowledge Economy, Ko-
rea. Since 2012, he has been a Technical Advisor of DawinTech. His current research
activities include the SoC architecture and design for multimedia and communications,
SoC design and verification methodology.