A NOVEL WAVELET MEASUREMENT SCHEME

BASED ON OVERSAMPLING

Albert Gilg, Utz Wever and Yayun Zhou

Siemens AG, CT T DE TC3/GTF MSO, Otto-Hahn-Ring 6, 81739 Munich, Germany

Keywords:

Wavelet, Oversampling, Sensor, Measurement.

Abstract:

In this paper, a novel wavelet image measurement scheme is developed inspired by the Haar wavelet oversam-

pling. It is equivalent to the dyadic Haar wavelet decomposition, but has a simpler hardware implementation

architecture. It contains three basis patterns and one ﬁxed selection template, which enables parallel computa-

tions. The measurement scheme is veriﬁed by simulation results and a hardware implementation is proposed.

This measurement scheme records the difference of neighboring pixels, which is independent of illumination

conditions.

1 INTRODUCTION

Starting with Haar’s work (Haar, 1910) at early 20th

century, wavelet becomes a more and more popu-

lar tool in signal processing. Its ability to localize

both time and frequency and provide multi-resolution

representation of image enables its wide applications

in many ﬁelds of signal and image analysis, such

as speech recognition, image compression, image

segmentation, image denoising/enhancing, and etc.

Most of the researches focus on the software-based

wavelet transform (Antonini et al., 1992) (Lewis and

Knowles, 1992) (Porwik and Lisowska, 2004) (Ravi-

raj and Sanavullah, 2007), though the transform re-

quires extensive computational resources for the real-

time implementation. Later a number of techniques

for realizing the wavelet transform in hardware sys-

tems are developed. The use of Digital Signal Pro-

cessors (DSPs) provides a quick and ﬂexible way to

compute the wavelet transform (Haapala et al., 2000).

However, it requires signiﬁcant area and power re-

sources. Besides, an analog-to-digital converter to

quantize the analog input is required for such digital

processors.

In recent years, some researchers try to integrate

the wavelet transform in image sensors, where the

transform is implemented in the analog domain di-

rectly on the focal plane (Luo and Harris, 2002) (Mos-

queron et al., 2006) (Shoushun et al., 2006). Analog

circuits perform area-efﬁcient and low-power com-

putation directly on the focal plane, eliminating the

need for an external processor (Olyaei and Genov,

2007). The wavelet embedded image sensor com-

bines image acquisition, signal processing and quan-

tization in a compact architecture, yielding high com-

putational throughput. Their performance is often be-

yond that of modern digital processors, allowing to

perform complex image processing operations in real

time. Those wavelet sensors are mostly developed

based on the Haar wavelet transform, because Haar

wavelet transform requires only shift and addition op-

erations, which are suitable for the hardware imple-

mentation.

The ﬁxed circuit design of the standard wavelet

transform has limited scalability due to the prior de-

termined level of the wavelet decomposition. High

decomposition levels are usually too complicated to

be realized in the digital circuit design. In this pa-

per, we propose a novel wavelet image measurement

scheme developed based on the Haar wavelet over-

sampling. It is equivalent to the dyadic Haar wavelet

decomposition, but has a simpler structure for the

hardware implementation. This measurement scheme

records the difference of neighboring pixels, which is

independent of illumination conditions. It truly cap-

tures the ratio between the various features of an ob-

ject. Besides, the difference is generally much smaller

than the absolute pixel value, hence less bits are

required after quantization, reducing the throughout

signiﬁcantly. Furthermore, the parallelism of imaging

architecture guarantees the real-time processing prop-

erty.

The organization of this paper is as follows. Com-

bined with the conventional 1D Haar wavelet trans-

Gilg A., Wever U. and Zhou Y..

A NOVEL WAVELET MEASUREMENT SCHEME BASED ON OVERSAMPLING.

DOI: 10.5220/0003274500270032

In Proceedings of the International Conference on Imaging Theory and Applications and International Conference on Information Visualization Theory

and Applications (IMAGAPP-2011), pages 27-32

ISBN: 978-989-8425-46-1

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

formation, the concept of oversampling is introduced

in Section 2. In 2D space, the dyadic Haar wavelet

transform is introduced. Based on this transform, we

propose a novel image measurement scheme, whose

results can be transformed to dyadic Haar wavelet

coefﬁcients easily. This equivalence is veriﬁed both

by the mathematical proof and simulation results. In

Section 4, the simulation results are presented and a

schematic hardware implementation is proposed. Fi-

nally, a conclusion is drawn based on the discussion

above.

2 WAVELET MEASUREMENT IN

1D SPACE

Wavelet transform converts a 1D-signal into a se-

ries of wavelet coefﬁcients using basis functions that

are bounded in frequency as well as in space do-

mains. There exists a variety of wavelet transforms,

which range from the oldest Haar wavelet, the gen-

eral Daubechies wavelets to the more complicated

biorthogonal wavelets. Among all the wavelet trans-

forms, the discrete wavelet transform using Haar

wavelet functions is one of the most promising tech-

nique in image coding and sensor design due to its

simplicity and small computation costs.

However, the classical Haar wavelet transform has

several limitations, for instance, the lack of transla-

tional shift invariance. If the input data is updated

with new samples, the majority of the coefﬁcients

changes and needs to be recalculated. Aiming at

the incremental update, Z. Struzik proposes an ex-

tended formulation of the Haar wavelet decomposi-

tion (Struzik, 2001). It oversamples the decompo-

sition by calculating coefﬁcients on a shift invari-

ant grid. Oversampling on the position axis can be

done to the highest resolution required or any required

lower resolution. Fig. 1 shows an oversampled grid

for an 8-point signal, in which the highest available

resolution is used. In the ﬁgure, MA denotes the

the moving average ﬁltering operation and DMA de-

notes the convolution of the derivative operator and

the moving average ﬁlter. The oversampling scheme

provides a representation with shift invariant coefﬁ-

cients and incremental update on new samples. It also

provides a possibility to extend the Haar representa-

tion over higher order (Struzik, 2001).

Inspired by this oversampling scheme, we develop

a new measurement scheme. It eliminates the redun-

dancy in the oversampled representation, while still

records all the information needed for restoring the

Haar wavelet coefﬁcients as well as the oversampled

representation. Fig. 2 shows the measurement scheme

Figure 1: Oversampled scheme of Haar wavelet transform.

with a 1D sensor array. This measurement scheme is

suitable for hardware implementation and records the

difference of adjacent input signals. Obviously, the

Haar wavelet coefﬁcients can be obtained by a simple

recursive calculation. In this sense, this measurement

is equivalent to the Haar wavelet coefﬁcients.

Figure 2: Measurement scheme with 1D sensor array.

3 WAVELET MEASUREMENT IN

2D SPACE

The standard 2D Haar wavelet decomposition is ob-

tained by computing a 1D Haar transform on each

row, followed by a 1D Haar transform on each column

(or conversely). Fig. 3(a) shows the basis functions

of the standard 2D Haar wavelet transform. Pixel

values are added where the white color appears and

subtracted where the black color appears. From the

ﬁgure, we conclude that the standard Haar transform

generates different spectral coefﬁcients on different

decomposition levels. A naive extension of the pre-

vious 1D measurement is not working in 2D case. We

have to turn to some alternative wavelet decomposi-

tion method.

3.1 Dyadic Wavelet Transform

It is well known that the dyadic wavelet transform is

a more efﬁcient representation for the entropy cod-

IMAGAPP 2011 - International Conference on Imaging Theory and Applications

(a) (b)

Figure 3: Haar wavelet transform in 2D space (a): Basis

functions of the standard Haar wavelet transform; (b): Basis

functions of the dyadic Haar wavelet transform.

ding in image compression. It is a slightly modiﬁed

2D Haar wavelet transform, whose basis functions are

shown in Fig. 3(b). The alternation between rows

and columns are applied within each decomposition

steps, leading to a multi-scale version of three inde-

pendent basis patterns. The wavelet basis functions

can be interpreted as three independent forms in dif-

ferent scales:



+1 −1



(1a)



+1 +1

−1 −1



(1b)



+1 −1

−1 +1



(1c)

This structure avoids the sequential operation along

rows and columns in the standard Haar wavelet trans-

form, providing a possibility to implement a parallel

measurement.

3.2 2D Oversampling-based

Measurement

Inheriting the spirit of the oversampling scheme, we

propose a novel measurement scheme in 2D space.

It contains three different basis patterns. Those pat-

terns are independent of each other, which leads

to a parallel architecture in hardware implementa-

tion. As we know, the oversampling scheme has

severe redundancy. In this measurement, only the

dyadic wavelet coefﬁcient related measurements are

reserved. The selection principle is a matter of art,

which induces a delicate symmetric template. Com-

bining the three patterns with a selection template, the

2D oversampling-based measurement is determined.

Fig. 4 shows the measurement scheme for a 4×4 sen-

sor array.

The chessboard-like patterns represent the differ-

ent manners interpreting the sample value from each

Figure 4: Measurement scheme for 2D sensor array.

sensor. White color means counting the sample value

as positive, while the black color means counting it as

a negative sample value. For 4 × 4 sensor array, the

selection pattern has two levels. The green blocks are

the ﬁrst level and the red blocks are the second level.

Each block contains four pixels, whose signed sum-

mations constitute the measurements. The acquired

measurements can be reassembled in a similar form

as dyadic wavelet coefﬁcients. Obviously, the ﬁrst

level measurements are consistent with the ﬁrst level

dyadic wavelet coefﬁcients. The second level dyadic

wavelet coefﬁcients can be derived by a linear combi-

nation of the derived measurements. The relationship

is shown in Fig. 5.

Similarly, in the 8 × 8 and 16 × 16 cases, the

dyadic wavelet coefﬁcients in higher levels can be

computed recursively. The three different patterns re-

main the same for larger sensor arrays. They are con-

structed by repeating the basis functions deﬁned in

Eq. 1 respectively. The selection template distribution

obeys strict rules and is highly symmetric. This ar-

chitecture beneﬁts the hardware implementation. The

independence of three patterns provides a parallel ar-

chitecture shown in Fig. 6. Compared with the tra-

ditional dyadic wavelet transform, this measurement

has a simpler structure, which avoids complicated

switch operations in hardware design. The pattern

number and the rule for selection template construc-

tion remain unchanged when the size of sensor array

increases, which simpliﬁes the circuit design process.

All the computation can be carried out simultane-

ously, leading to less process time. Besides, only the

difference of adjacent pixels is recorded, which cap-

tures the true features of an object and eliminates the

affect of the illumination condition. Since the vari-

ance of the difference value is generally smaller than

the variance of absolute value, the throughout can be

reduced after the quantization step.

A NOVEL WAVELET MEASUREMENT SCHEME BASED ON OVERSAMPLING

(a)

(b)

(c)

Figure 5: Second level dyadic wavelet functions.

4 SIMULATION AND

HARDWARE DESIGN

This measurement scheme is equivalent to the dyadic

Haar wavelet decomposition. In this section, we sim-

ulate the measuring process and compare the results

with the coefﬁcients of the dyadic Haar wavelet trans-

form. The measurements can be converted to dyadic

Haar wavelet coefﬁcients in any level, which guaran-

tees the ﬂexibility of postprocess. Furthermore, we

propose a design guideline for a CMOS sensor de-

sign.

Figure 6: Parallel architecture.

4.1 Simulation Results

The measurement process is simulated with a given

matrix, and a test image in order to verify its accuracy.

We start with a given 4 × 4 matrix:

A =







64 2 3 61

9 55 54 12

17 47 49 20

40 26 27 37







(2)

The oversampling-based measurement is given as:

M =







32.6875 0.2500 4.0000 −4.0000

−3.2500 0.7500 −4.0000 4.7500

0.5000 −0.5000 27.0000 −25.0000

−0.5000 1.2500 −11.0000 9.7500







(3)

In order to verify the equivalent of the measure-

ments and the dyadic Haar wavelet decomposition,

we convert the measurements based on the relation-

ship given in Fig. 5. The derived matrix is the same

as the results applying the dyadic wavelet transform

directly.

W =







32.6875 −0.1875 4.0000 −4.0000

−0.1875 0.1875 −4.0000 4.7500

0.5000 −0.5000 27.0000 −25.0000

−0.5000 1.2500 −11.0000 9.7500







(4)

Fig. 7 shows a simulation example with an image

input. The value of the novel measurement scheme

is compatible to the dyadic Haar wavelet transform

with the highest level. Through recursive computa-

tions, the measurement matrix can be converted to the

dyadic Haar wavelet coefﬁcients. The equivalence is

veriﬁed. Then an inverse dyadic Haar wavelet trans-

form is applied to the coefﬁcients, the reconstructed

image is shown in Fig. 7(d).

IMAGAPP 2011 - International Conference on Imaging Theory and Applications

(a) (b)

Figure 7: 2D measurement simulation (a): Test image;

(b): Oversampling-based measurement; (c): Converted to

dyadic wavelet coefﬁcients; (d): Reconstructed image.

4.2 Hardware Design

This measurement scheme can be applied in dif-

ferent sensor systems together with certain postpro-

cessing blocks, such as noise removal, image cod-

ing/compression and etc. Its ﬁxed template simpliﬁes

the complexity of hardware implementation.

Here, we propose a schematic of hardware imple-

mentation in a CMOS sensor array in Fig. 8. It con-

tains four main parts: pattern control unit, sensor ar-

ray, measurement template and AD converter.

The pattern control unit generates the control sig-

nal with respect to three patterns, such as sample/hold

signal and sign control signal. The sign control signal

is generated by logic circuits for each basis pattern.

A modiﬁed S/H circuit is described in Fig. 9.

It is set up based on the conventional APS design

(Chi et al., 2009) with a sign control function, which

counts the accumulated photons as a positive or nega-

tive output voltage. Clocks S/H are non-overlapping.

If SC (stands for sign control) is low during the sam-

ple phase S and goes high during the hold phase H, the

amount of charge transferred is C(V

int

i j

−V

re f

), where

int

i j

is the voltage collected in the sample phase. V

re f

is deﬁned as the pixel voltage when there is no light.

Conversely, if SC is high during the sample phase S

and goes low during the hold phase H, the amount of

charge transferred is C(V

re f

−V

int

i j

The main computation unit is the measurement

template unit. It contains basic measurement el-

ements arranged as the selection template deﬁned

above. A basic measurement element is shown in

Fig. 10, which computes the weighted sum of adja-

cent pixel in an image. The capacitor is set to 4C in

order to get a normalized measurement. The output

of the ampliﬁer is given as:

i j

∑

i, j

int

i, j

−V

re f

) · SC

i j

. (5)

where SC

i j

is either +1 or −1. In the end, the mea-

sured differences are quantized through a AD con-

verter. This hardware design is only an example of

this measurement realization, in which the measure-

ment process is done consecutively. It contains a

switch circuits transferring one basis pattern to an-

other. The measurement scheme can also be carried

out simultaneously, which means implementing the

measurement template concerning each pattern layer-

by-layer. It avoids the switch operation between the

three patterns, hence it is faster, but the manufacture

cost for the chip is also higher. With different applica-

tions, certain circuits should be changed accordingly.

Figure 8: Hardware architecture.

Figure 9: S/H circuit with sign control.

5 CONCLUSIONS

In this paper, we propose an oversampling-based

wavelet measurement scheme. It records only the dif-

ference of adjacent pixels, capturing the true charac-

ter of an object. It eliminates the redundancy which

A NOVEL WAVELET MEASUREMENT SCHEME BASED ON OVERSAMPLING

Figure 10: Basic measurement element.

occurs commonly in the oversampling system and re-

stores only the measurements equivalent to the dyadic

wavelet decomposition. The measurement scheme

can be carried out by combining three independent

chess-board like patterns with one ﬁxed selection

template. The hardware architecture is much simpler

than the traditional wavelet transform. The indepen-

dent pattern structure provides a possibility of paral-

lel computations. Combined with certain postprocess

units, it can be used in different applications, espe-

cially the occasions requiring real-time image acqui-

sition and processing.

REFERENCES

Antonini, M., Barlaud, M., and Daubechies, I. (1992). Im-

age coding using wavelet transform. IEEE transac-

tions on image processing, pages 205–220.

Chi, Y. M., Abbas, A., Chakrabartty, S., and Cauwenberghs,

G. (2009). An active pixel CMOS separable transform

image sensor. In Proc. IEEE Int. Symp. Circuits and

Systems, Taibei, Taiwan.

Haapala, K., Kolinummi, P., Hamalainen, T., and Saarinen;,

J. (2000). Parallel DSP implementation of wavelet

transform in image compression. In The Proc. IEEE

International Symposium on Circuits and Systems,

pages 89 – 92, Geneva , Switzerland.

Haar, A. (1910). Zur theorie der orthogonalen funktionen-

systeme. Math. Annal., pages 331–371.

Lewis, A. S. and Knowles, G. (1992). Image compression

using the 2-D wavelet transform. IEEE transactions

on image processing, pages 244–250.

Luo, Q. and Harris, J. (2002). A novel integration of on-

sensor wavelet compression for a CMOS imager. Cir-

cuits and System, 3:III 325–III 328.

Mosqueron, R., Dubois, J., and Paindavoine, M. (2006).

Embedded image processing/compression for high-

speed CMOS sensor. In European Signal Processing

Conference, Florence.

Olyaei, A. and Genov, R. (2007). CMOS focal-plane

spatially-oversampling computational image sensor.

Circuits and System I, pages 26–34.

Porwik, P. and Lisowska, A. (2004). The Haar-wavelet

transform in digital image processing: its status and

achievements. Machine graphics & vision, 13:79–98.

Raviraj, P. and Sanavullah, M. Y. (2007). The modiﬁed 2D-

Haar wavelet transformation in image compression.

Middle-East Journal of Scientiﬁc Research, 2(2):73–

78.

Shoushun, C., Bermak, A., Yang, W., and Martinez, D.

(2006). A CMOS image sensor with combined

adaptive-quantization and QTD-based on-chip com-

pression processor. In Custom Integrated Circuits

Conference, pages 329–332, San Jose, CA.

Struzik, Z. R. (2001). Oversampling the Haar wavelet trans-

form. Technical report, Information Systems in CWI.

IMAGAPP 2011 - International Conference on Imaging Theory and Applications