QUESTIONING HU’S INVARIANTS

Bad or Good Enough?

Diego Martinoia

Department of Electronics and Information, Politecnico di Milano, P.za Leonardo da Vinci 32, 20133, Milano (MI), Italy

Keywords:

Gesture Recognition, Invariants, Moments, Shape.

Abstract:

Despite Hu’s invariants were proven not to be independent nor complete long time ago, their use in computer

vision applications is still broad, mainly because of their diffusion among common CV libraries and ease of

use by inexperienced users. In this paper I want to investigate whether, given their mathematical ﬂaws, they

are nevertheless good enough to justify such a wide diffusion, also considering that more sophisticated tools

have been developed over the years.

In order to do this, I am going to test the robustness of Hu’s invariants in a comparative way against the more

modern wavelet invariants, in a hand gesture recognition application. Finally, I am going to discuss, basing

my considerations on the experimental data, whether Hu’s invariants are still a viable option for small scale,

amateurish applications, or if the time has come to abandon them for more effective solutions.

1 INTRODUCTION

Image invariants can be deﬁned as characteristic val-

ues of an image, chosen to be invariant to some kind

of distortions, usually a combination of translation,

rotation and scaling. Albeit invariants suffer of var-

ious weaknesses, such as the inability to be used to

match occluded objects, they are still considered one

of the main benchmark references for all other shape

descriptors. Over the course of the years, moment

invariants have become one of the most famous and

common tools in shape recognition. Moment invari-

ants ﬁnd their roots in the theory of algebraic invari-

ants, and they have been used for the ﬁrst time in

pattern recognition from Hu (Hu, 1962), who devel-

oped the very same set of invariants whose robustness

I am going to examine in this paper. After that, many

authors have either improved and generalized Hu’s

work, or applied his results to many different appli-

cation ﬁelds, from satellite images (Wong and Hall,

1978) to hand writing recognition (Flusser and Suk,

1994). Flusser (Flusser, 2000), probably inspired by

the previous work of Reiss (Reiss, 1991), developed

a method to derive a complete set of independent in-

variants for every image, and proved mathematically

that Hu’s invariants are not independent, nor com-

plete. Nevertheless, their use in the ﬁeld of computer

vision is still broad, mainly because of their ease of

use, being implemented in many of the available li-

braries, such as the widely used OpenCV library (Wil-

lowGarage, 2011). On the other hand, we see very

few implementations, in those very same libraries, of

more recent and effective sets of invariants, such as

Hermit-Gaussian’s, Li’s, Zernike’s or wavelet invari-

ants. Therefore, a non-expert user of those libraries,

who might be willing to implement his own ideas, will

probably be using, consciously or not, Hu’s set rather

than other ones. My aim is to test whether this is a vi-

able choice for small scale, amateurish applications,

given the low complexity and the ease of use of Hu’s

set, or if the tradeoff is too extreme, and it is time,

after 50 years since their invention, to start making

pressure on the CV libraries maintainers to have them

deprecate Hu’s set, placing it side by side with the

more recent tools. To address this question, I am go-

ing to compare the performance of Hu’s set with the

one of a subset of the wavelet invariants, in a conﬁg-

uration where the performance of the acquisition de-

vice is comparable to those of amateurish devices, i.e.

low resolution and low frame rate, and the application

is hand gesture recognition.

2 THEORETICAL DESCRIPTION

In this section I am going to give to the reader some

theoretical background about the general procedure

in order to obtain a set of invariant features, immune

to Translation, Rotation and Scaling (TRS) modiﬁca-

tions, as well as describe Hu’s and wavelet sets more

speciﬁcally.

Let f(x,y) by a 2-D binary image object in the (x,y)

311

Martinoia D..

QUESTIONING HU’S INVARIANTS - Bad or Good Enough?.

DOI: 10.5220/0003803103110316

In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2012), pages 311-316

ISBN: 978-989-8565-03-7

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

Cartesian coordinate space, while its corresponding

polar representation is f(ρ, θ). We can achieve TS in-

variance by using two new coordinates, that are de-

ﬁned as:

x −X

y −Y

where (X

) are the coordinates of the center of

mass of the object of interest, and α is the square

root of the ratio between the object’s size and its ex-

pected size. From now on, I am going to refer to f(x,y)

and f(ρ,θ) as the TS normalized version of the origi-

nal image, respectively in Cartesian and polar coordi-

nates.

R normalization is achieved as follows: Let

Z Z

f (ρ,θ)g

(ρ)e

jqθ

ρdρdθ (1)

being g

(ρ) a function of the radial variable ρ, and p

and q are integer parameters. The demonstration of

the rotation invariance of



is achieved as fol-

lows: if the image f(ρ, θ) is rotated of an angle β,

its corresponding moment becomes F

βRot.

= F

jqβ

that as we know doesn’t affect the value of the norm

of the vector. Thus,



is a rotation invariant.

As shown by Shen (Shen, 1999), we can derive

that Hu’s moments, Li’s moments and Zernike mo-

ments are all special cases of (1) and that the extracted

features are global features.

2.1 Hu’s Invariants

For the complete theoretical explanation of Hu’s

work, I address the reader to the original paper by Hu

(Hu, 1962), and the following work by Reiss (Reiss,

1991) and Flusser (Flusser, 2000). The set of the

7 original Hu’s moment invariants can be directly

derived from the geometrical center moments:

= η

+ η

= (η

−η

)

+ 4η

= (η

−3η

)

+ (3η

−η

)

= (η

+ η

)

+ (η

+ η

)

= (η

− 3η

)(η

+ η

)((η

+ η

)

−

3(η

+ η

)

) + (3η

− η

)(η

)(3(η

+ η

)

−(η

+ η

)

= (η

− η

)((η

+ η

)

− (η

+ η

)

) +

4η

(η

+ η

)(η

+ η

)

= (3η

− η

)(η

+ η

)((η

+ η

)

−

3(η

+ η

)

) − (η

− 3η

)(η

)(3(η

+ η

)

−(η

+ η

)

where

(1+

p+q

)

and

+∞

−∞

+∞

−∞

(x −X

)

(y −Y

)

f(x,y)dxdy

is the central moment of the function f(x,y) . One of

the main weaknesses of the set is its explosive com-

plexity, and thus its usage is almost always limited to

these 7 invariants, albeit higher order moments have

been constructed, for example by Li (Li, 1992) and

Wong et Al (Wong et al., 1995), respectively up to or-

der nine and ﬁve. Flusser (Flusser, 2000) proved that

the above mentioned set is not independent: let c

be the pq-order complex moment in polar coordinates

denoted by:

+∞

2π

p+q+1

i(p−q)θ

f(ρ,θ)dρdθ

Under this representation, Hu’s invariants can easily

be rewritten as:

= c

= ℜ(c

= ℑ(c

)

From this representation we can see that:

= c

)

+ φ

and therefore the invariants are not independent.

2.2 Wavelet Invariants

A detailed explanation of the wavelet invariants can

be found in Shen (Shen, 1999), while some of their

applications for image matching can be found in

Zhang et al (Zhang et al., 2009).

Wavelet Transform is a method for obtaining local-

ized analysis, but unlike the traditional short-time

Fourier Transform, Wavelet Transform can provide

both time and frequency localization. While dealing

with the wavelet based approach, we are going to treat

(ρ) from (1) as the wavelet mother function, and

consider the wavelet family:

a,b

(ρ) =

√

ψ(

ρ −b

) (2)

VISAPP 2012 - International Conference on Computer Vision Theory and Applications

312

where a,b ∈ R

are, respectively, the dilatation pa-

rameter and the shifting parameter. In the experiment

I am going to use the cubic B-spline in Gaussian ap-

proximation form (Unser et al., 1992), given by:

ψ(ρ) =

n+1

2π(n + 1)

cos(2π f

(2ρ −1))e

−

(2r−1)

2σ

(n+1)

where n = 3,a = 0.697066, f

= 0.409177,σ

0.561145 . This choice was based on the properties

illustrated in (Ahuja et al., 2005).

The values of the a and b parameters in (2) are usu-

ally discrete, and discretization is achieved by choos-

ing a = a

with m integer and a

6= 1 while b is dis-

cretized by taking the positive and negative multiples

of b

m, with b

chosen so to cover the whole do-

main of ψ((ρ −b)/a) for different values of m or, in

other words:

a = a

,b = nb

with n,m integer. But since the image size ha been

normalized in the domain ρ ≤ 1 , one can set a

and

both to 0.5, and restrict the domain for m and n as:

m = 0,1, 2,3; n = 0,1,..., 2

m+1

;

in this way, we obtain a simpliﬁed form of the

wavelet, deﬁned along a radial axis in any orientation:

m,n

= 2

ψ(2

ρ −0.5n)

Now, if we let this function sweep through all the an-

gular rotation in the moment computation, it will ex-

tract global or local features according to the values

of m and n.

The general form of a wavelet invariant is given by:



wavelet

m,n,q



Z Z

f (ρ,θ)ψ

m,n

(ρ)e

jqθ

ρdρdθ



where ψ

m,n

(ρ) replaces g

(ρ) in (1), m =

0,1, 2,3; n = 0,1,..., 2

m+1

;q = 0,1,2, 3. There-

fore in our case we have 136 possible wavelet

invariants, while Hu’s are only 7. In order to make

the comparison fairer, one has to choose a reduced

set of the wavelet invariants.

2.3 Choosing the Invariants

All of Hu’s invariants were kept, due to their limited

number. Besides, the purpose of this test is to verify

the set’s robustness as it is, including the dependent

values. For the wavelet invariants, the ﬁrst reduction

I have chosen is the restriction of the domain of the

q parameter to the single 0 value. Albeit this might

seem like a drastic choice, we must remember that we

want to go from 136 dimensions to 7, and no matter

what reduction we choose, we are going to lose a lot

of information in the process. At least, this choice al-

lows to avoid working with complex numbers, reduc-

ing the computational load on that part, and allowing

to increase the resolution of the dρ and dθ values of

the discretization of the polar integral, thus reducing

numerical instabilities.

After some exploration of the parameters space, I

found out that the invariants with lower variance over

the training set in the 34 remaining options were (for

each of them, q=0 and therefore I report only the val-

ues of m and n):



wavelet

0,0



wavelet

1,3



wavelet

2,0



wavelet

2,1



wavelet

2,2



wavelet

3,0



wavelet

3,1



which are distributed over all the values of m.

3 EXPERIMENTAL SETUP

The acquisition device I am going to use in this ex-

periment is the Mesa Imaging SwissRanger SR4000

Time of Flight camera (Imaging, 2011). It acquires a

black and white, 176x144 resolution, 16bit depth im-

age, providing reliable information about the distance

of each pixel from the camera when the pixel is lo-

cated in the 30 – 300 cm distance range. The camera

frame-rate used is 20 FPS.

Since the design of an effective hand-detection algo-

rithm is beyond the purpose of this paper, the hand

segmentation is performed using distance informa-

tion, in order to create a scanning layer and obtain

a binary image. Eventual remaining holes and noise

are then subjected to morphological transformations

for ﬁlling and ﬁltering. A similar approach for seg-

mentation has already been used for touch-less inter-

faces, with excellent results (Soutschek et al., 2008).

At the start of the experiment, I generate a number N

of agents, each one storing the K = 7 invariant val-

ues of a speciﬁc gesture out of the M gestures in the

database, independently perturbed by a uniformly dis-

tributed mutation, which magnitude is in the ±V % of

the reference value. Each agent stores the signature of

only one gesture, therefore there are at least bN/Mc

agents evaluating each gesture. I am going to refer to

a set of agents evaluating the same gesture as “com-

mittee”.

The evaluation is performed using the following pro-

cedure: for each frame, for each agent, I am going

to calculate the average percentage variation of the

logarithms of the invariants of the image, compared

to those of the reference, which is the ”distance” be-

tween the stored reference and the current image. If

the result is bigger than a given tolerance T , then the

shape is considered as not matching the shown ges-

QUESTIONING HU'S INVARIANTS - Bad or Good Enough?

313

ture, yes otherwise. This very same decision policy

is used even in the OpenCV libraries implementation

of Hu’s invariants (WillowGarage, 2011). At the end

of this preliminary evaluation, I am going to calcu-

late the number of agents that recognized their gesture

in the current frame, but only when a minimum per-

centage of agreement was reached in their committee,

assign at each gesture a probability accordingly and

randomly extract the output. If no committee reaches

P% recognition, then no gesture is recognized in that

frame.

4 EXPERIMENTAL RESULTS

User dependency is a very big issue in gesture recog-

nition, as proven by the success in identiﬁcation ap-

plications (Schmidt et al., 2010), and the evaluation

of its effects is clearly outside the scope of this pa-

per. Therefore, I am going to use the same users for

both training and testing. Albeit the gesture database

is very limited, consisting only of 5 gesture performed

with both hands, the experiment should be able to give

us some useful material for discussion. I performed 3

different experiments, testing the performances of the

sets over still images, moving images, and still images

with added noise respectively.

Due to the users’ inability to stay perfectly still in

posture, slight changes in the perceived image oc-

curred. In order to avoid setting the value of T too

high to compensate, testing was performed also over

a “Noise” gesture, i.e. a gesture not matching any of

the database ones, but which looked similar to many

of them under different perspective angles, shown ro-

tating on all possible axis.

Rock Paper OK Stop Pointing Noise

I report only the best results, over all the param-

eters combinations tested. In all cases, training was

performed using 15 seconds long recordings of the

gesture, recorded in a static pose, while testing was

done using 10 seconds long recordings (which are

long times, considering the application). Training

consisted in calculating the invariants of each frame

of the training sample, and then averaging the output

only when the value was not further than ±50% of

the median value, in order to ﬁlter out training noise.

Due to the comparative nature of this study, the very

same data-set for training and experiment were used

for both Hu’s and wavelet approaches.

Table 1: Static gestures; Hu’s invariants; N=300; M=5;

V=5%; T=7.5%; P=75%.

Gesture Correct Opposite Total Wrong

R. Rock 96.9 0.0 96.9 0.0

R. Paper 89.3 0.0 89.3 0.0

R. OK 97.2 2.1 99.3 0.0

R. Stop 93.7 0.0 93.7 0.0

R. Pointing 95.7 0.0 95.7 0.0

L. Rock 92.2 0.0 92.2 0.0

L. Paper 97.3 0.0 97.3 0.0

L. OK 99.2 0.0 99.2 0.0

L. Stop 73.7 25.2 99.3 0.0

L. Pointing 97.1 0.0 97.1 0.0

Noise – – – 7.5

Average 93.2 2.8 96.0 0.8

Table 2: Static gestures; Wavelet invariants; N=300; M=5;

V=10%; T=15%; P=75%.

Gesture Correct Opposite Total Wrong

R. Rock 50.4 46.5 96.9 0.8

R. Paper 83.9 0.0 83.9 5.4

R. OK 7.1 0.7 7.8 0.0

R. Stop 44.4 23.8 68.3 6.3

R. Pointing 64.0 3.6 67.6 1.4

L. Rock 53.5 42.9 96.3 1.2

L. Paper 84.6 0.0 84.6 2.0

L. OK 84.1 3.8 87.9 0.0

L. Stop 46.1 52.0 98.0 0.0

L. Pointing 20.1 0.0 20.1 2.2

Noise – – – 21.5

Average 53.8 17.3 71.1 3.7

4.1 Static Experiment

All gestures were presented frontally, with the palm

of the hand towards the camera. Each table repre-

sent the percentage of the ratio between the number of

frames where a gesture was output, and the total num-

ber of frames. Rows represent which gesture sample

was shown to the camera. “Correct” means a correct

hand and gesture recognition, “Opposite” means cor-

rect gesture but wrong hand. These two columns were

summed in the “Total” column, for ease of the reader.

“Wrong” means a completely wrong gesture recog-

nized, while no-output frames are not reported in this

tables. The results are shown in Table 1: Hu’s invari-

ants matching ratio for static gestures is very high,

with complete lack of error when a real gesture is

shown to the system. In the second phase of the ﬁrst

experiment the test was repeated using the wavelet in-

variants approach. The results are shown in Table 2:

as we can see, the performance of the wavelet invari-

ants is generally lower than Hu’s, which is probably

due to both the discretization process for the radial

integrals in such a low-resolution environment, prone

VISAPP 2012 - International Conference on Computer Vision Theory and Applications

314

Table 3: Moving gestures; Hu’s invariants; N=300; M=5;

V=5%; T=7.5%; P=75%.

Gesture Correct Opposite Total Wrong

R. Rock 37.4 1.0 38.4 1.4

R. Paper 45.3 1.1 46.4 0.0

R. OK 51.0 0.0 51.0 3.5

R. Stop 69.1 0.0 69.1 0.0

R. Pointing 83.9 0.0 83.9 0.0

L. Rock 39.4 0.0 39.4 0.0

L. Paper 43.5 13.0 56.5 0.0

L. OK 43.6 12.8 56.4 0.0

L. Stop 71.3 13.1 84.4 0.0

L. Pointing 77.9 3.2 35.2 0.0

Noise – – – 2.5

Average 56.2 4.4 56.1 1.1

Table 4: Moving gestures; Wavelet invariants; N=300;

M=5; V=10%; T=15%; P=75%.

Gesture Correct Opposite Total Wrong

R. Rock 44.6 42.9 87.6 2.8

R. Paper 49.2 1.1 50.3 9.0

R. OK 8.1 6.6 14.7 3.6

R. Stop 17.5 12.4 29.9 25.2

R. Pointing 23.2 6.7 29.9 3.1

L. Rock 45.7 42.8 88.5 3.8

L. Paper 23.6 1.4 25.0 4.6

L. OK 9.6 6.9 16.5 3.7

L. Stop 40.8 33.6 74.3 10.

L. Pointing 12.2 6.5 18.7 1.6

Noise – – – 21.5

Average 25.9 15.7 41.6 8.1

to numerical instabilities. As a matter of fact, in order

to obtain these results I had to use a very high value

of the tolerance T parameter, which led to a big false

positive ratio. To be noted is also the inability of the

wavelet set to distinguish between mirrored images

(in this case left or right hand) properly. Hu’s set, in-

stead, can do this better, being the sign of φ

sensitive

to mirroring.

4.2 Moving Experiment

This time, gestures in the sample were shown trans-

lating all over the camera plane, but without any rota-

tion. Involuntary rotation and shape change happened

due to users’ movement, which is the cause for worse

results. Results are shown in Table 3: as we can see,

they are sensibly worse. Upon close inspection of

the errors for the gestures that had a poorer perfor-

mance (“Pointing”, “OK”), I noticed that this is due

to their similar outer silhouette, which is confused by

the algorithm in most of the non-static situations. The

results for the wavelet invariants tested over moving

gestures are shown in Table 4. Wavelet invariants’

performance is again generally lower than its oppo-

Table 5: 2% Salt&Pepper noise; Hu’s invariants; N=300;

M=5; V=5%; T=12.5%; P=75%.

Gesture Correct Opposite Total Wrong

R. Rock 56.6 6.2 62.8 0.8

R. Paper 26.8 0.0 26.8 0.0

R. OK 61.7 9.2 70.9 0.0

R. Stop 56.4 0.8 57.1 0.0

R. Pointing 63.3 0.0 63.3 0.7

L. Rock 62.9 0.4 63.3 0.4

L. Paper 47.0 0.0 47.0 0.0

L. OK 81.1 0.0 81.1 0.0

L. Stop 47.4 20.4 67.8 0.0

L. Pointing 70.5 0.0 70.5 0.0

Noise – – – 2.5

Average 57.4 3.7 61.1 0.4

nent’s: wavelet invariants are too sensitive to shape

changes to outperform the more ”stubborn” Hu’s set

in this setting. Another thing that needs to be pointed

out is that, just like it happened with Hu’s set, some

gestures (for example “Rock”) are much more stable

than others (“OK”,“Pointing”). The reason for this is

probably in the fact that the user can perform those

gestures in a moving way with less involuntary ro-

tation than the others, which might be an interesting

consideration for a possible human interface applica-

tion.

4.3 Noise Experiment

The last part of the experiment consisted in testing

over the same static set as before, but including a 2%

Salt&Pepper noise, i.e. negating the binary value of

2% of the pixels of the image. It is important to state

that this disturbance was introduced before the mor-

phology operations, and therefore its effect reaches

the invariants extraction phase highly ampliﬁed, as

shown in this sample image:

Original Noisy

Results for this test are shown in Tables 5 and 6,

for Hu’s and wavelet invariants respectively. We can

see that the average performance is deeply affected

in both cases, but Hu’s invariants resist better than the

wavelet ones, albeit they required an increase of the T

parameter, while the wavelet’s did not. The reason for

this is the fact that wavelet invariants are much more

sensitive to the noise-induced shape differences than

Hu’s invariants, and are in fact used, for example, to

distinguish between very similar shapes (Shen, 1999).

QUESTIONING HU'S INVARIANTS - Bad or Good Enough?

315

Table 6: 2% Salt&Pepper noise; Wavelet invariants; N=300;

M=5; V=10%; T=15%; P=75%.

Gesture Correct Opposite Total Wrong

R. Rock 27.1 35.7 62.8 22.5

R. Paper 41.1 0.0 41.1 48.2

R. OK 24.8 5.1 29.8 17.8

R. Stop 31.0 17.5 48.4 25.4

R. Pointing 35.3 5.8 41.0 31.0

L. Rock 30.6 30.2 60.8 20.0

L. Paper 43.0 2.0 45.0 28.2

L. OK 50.0 8.3 58.3 9.9

L. Stop 40.8 33.6 74.3 18.4

L. Pointing 12.2 6.5 18.7 17.3

Noise – – – 22.3

Average 33.6 14.4 48.0 23.6

5 CONCLUSIONS

The goal of this paper was to investigate whether Hu’s

moment invariants, despite their proven mathematical

imprecision, are still a viable option for non-expert

users and small scale applications, thanks to their sim-

plicity. To address this question, I compared their

performance with the more recent set of wavelet in-

variants, in a severe low-resolution environment that

emulates a possible amateurish hardware scenario.

The results, albeit few in quantity, show how Hu’s

invariants perform greatly over still images, deﬁnitely

better than the more advanced wavelet set. I have

then tried to have Hu’s and wavelet sets detect trans-

lating gestures: results were, obviously, much worse,

but still in the 40–60% range, with Hu’s invariants

performing generally better than the wavelet’s also in

this case. Finally the test was conducted over noisy

images. In both cases there is a worsening of the re-

sults of about absolute 30%, but Hu’s invariants are

deﬁnitely better in this case also. The reason for

this is dual: ﬁrstly, the low resolution leads to prob-

lems in the discretization of the radial integral steps

for the wavelet invariants calculation, and secondly

the wavelet set is much more sensitive to the shape

changes, intrinsic in this application, than Hu’s set.

In conclusion, it is my opinion that Hu’s invari-

ants have proven their robustness under difﬁcult con-

ditions, and therefore they are still a great choice, es-

pecially in order to allow amateurs of the ﬁeld to try

and implement their ideas. Nevertheless, this doesn’t

mean that one must think of them as the panacea of

all shape recognition problems: in other applications

other approaches have proven their superiority, and

CV libraries maintainers should indeed start imple-

menting the newer tools in their collections, leaving

the responsibility of picking the right tool for each

problem to the developer.

ACKNOWLEDGEMENTS

The author would like to thank Prof. Koichi Shinoda

at Tokyo Institute of Technology for his suggestions,

and all Shinoda Lab’s members for their cooperation.

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