## Tomography Problems at Monochromatic Sounding

**Valery P.**** ****Yushchenko**^{*}

Ministry of Education and Science of the Russian Federation, Novosibirsk State Technical University, Novosibirsk, Russia

^{*}**Corresponding author: **Valery P. Yushchenko, Ministry
of Education and Science of the Russian Federation,
Novosibirsk State Technical University, 630073, Novosibirsk, K.Marx Avenue, 20, Russia. Tel:+73832243421; Email: jwp@aport2000.ru

**Received Date:** 13 August, 2018; **Accepted Date:** 24 August, 2018; **Published Date:** 03 September, 2018

**Citation:** Yushchenko VP (2018) Tomography
Problems at Monochromatic Sounding. Arch Laser
Photonics: ALP-102. DOI: 10.29011/ALP-102.100002

**1. ****Summary**

In this article it is discussed methods of creation of
the image of objects on the Doppler signal reflected from it by means of the
synthesized antenna aperture. The known property of the synthesized aperture to
be focused on the set range is the cornerstone of a method. Difference of the
offered methods from the known methods that the focused spot scans interiors of
object and is developed the image of object in section. Such chance is given by
repeated correlation of the object signal received as a result of a Doppler
location with basic trajectory signals for the dot objects received as a result
of calculation. Cartograms of the model objects reconstructed on a Doppler
signal are given. Problems of practical realization are discussed.

**1.
****Introduction**

Tomography is the most informative measuring process. The tomography assumes
reconstruction of the image of object in section with display of internal
structure of object. For a tomography it is necessary to choose the probing
signals which get in inside the studied object. It can be the X-rays,
radioactive radiations, microwave ovens radiation provoking a nuclear magnetic
resonance in a gradient magnetic field, a radio emission and, at last,
ultrasonic radiation.

All types of a tomography are
useful. They supplement, but do not replace each other. Likely the most
qualitative are nuclear magnetic resonances tomographs, however they are not
deprived of shortcomings. The patient when diagnosing has to lie not movably
within 30 minutes. For the patient with feeling sick it is not comfortable
procedure.

Let's stop on ultrasonic
radiation. It gets in inside object and at the small capacities is not capable
to do harm to the patient. It is possible to make a critical remark that in an
ultrasonic tomography it is difficult to think up something new. Besides
ultrasonography tomographs are rather perfect. However, there are claims to
results of ultrasonography. The received image is vague and its clearness is
not sufficient. Figure 1.

Dispersive distortions are the reason for that.
Fabrics of a live organism are the dispersive environment for an ultrasonic
wave. The essence of these distortions consists that the short probing impulse
has a wide range. In the dispersive environment components of a range extend
with an unequal speed. It leads to the fact that the short impulse which passed
through the dispersive environment is washed away (becomes wider and gets out
of a shape). It
does not allow to construct the accurate image.

**2.
****Problem Definition**

It is offered to refuse sounding by short impulses and
to pass to the monochromatic probing signals. The range of such signal contains
one frequency therefore there will be no dispersive distortions.

However, it is not simple to apply monochromatic
signals in a location and in a tomography. Monochromatic signals have no radial
permission. And it is necessary for reconstruction of the image. For a solution
of the problem of radial permission we will use property of the synthesized
aperture to be focused on the set range. The synthesized aperture is similar to
a lens and has focal length.

To use synthesis of an aperture it is necessary to
organize the movement of a locator concerning object at small distance on a
rectilinear or circular trajectory with the known speed and to register a
trajectory Doppler signal.

For a Tomography, it is supposed to use a one-position
ultrasonic Doppler locator at which the radiating and accepting antennas are
combined, that is approximately are in one point of space, and have a weak
focus in the equatorial plane and a sharp orientation 6^{º} in the
meridional plane [1].

**3.
****Description of Laboratory
Installation**

The function chart of a Doppler locator is submitted
in Figure 2.

In Figure 2, the following designations are accepted:
the 1-radiating piezo element 2-accepting a piezo element, 90^{º} -a phase-shifting chain on 90^{º}, CM-The Mixer of
Signals (remultiplier), the Cut. Mustache.RA- The Resonant Amplifier, DA-The
Amplifier Of Doppler Frequencies, ADC-The Analog-Digital Converter.

Installation allows to realize both rectilinear, and
circular aperture synthesis. For realization of rectilinear aperture synthesis,
the tank with water
is fixed in motionless situation, Tomography objects move by piezo
elements 1 and 2 on a rectilinear trajectory of Figure 2-a. So, for example,
movement of a thin delay is equivalent to movement of dot object on a
rectilinear trajectory as the vertical size of a delay is limited to
directional pattern width in the meridional plane of Figure 2-b.

Figure 3 shows an ultrasonic experimental setup for
tomography of objects in water. Tomographized objects, a hand or wire are
placed in the water tank. Two ultrasonic antennas are built into the white
square on the side wall of the tank from the inside: a transmitting and
receiving antenna. On the carton box there is an experimental locator circuit.
A trajectory Doppler signal is visible on the monitor screen. The signal is
complex. The real part of the signal is located at the top of the monitor, and
the imaginary part of the signal is at the bottom.

Length of an ultrasonic wave λ=1.34 mm. Relative speed of the movement of object of* Vot* of =30 cm/page.
Movement of object was carried out with a constant speed on a rectilinear
trajectory with *a* = 3,4 cm. At circular synthesis of an aperture the
speed of rotation of the tank made one turn in three seconds.

Thus, the presented installation gives the chance to
register digital discrete counting of trajectory Doppler signals. After
registration of a trajectory signal there is a problem of reconstruction of the
image of object of a trajectory Doppler signal.

The idea of a method of reconstruction consists that
it is necessary to operate focal length of the synthesized aperture and this
focus, due to the movement of a locator, it is possible to scan the space
surrounding a locator, including object interiors.

**4.
****Reconstruction's
Algorithms. Management of Focal Length **

The focal length can be operated by change of parameters of a basic
signal. Before starts reconstruction of the image of object it is necessary to
create in the settlement way basic trajectory signals for dot objects. The look
and parameters of a trajectory signal depend on a trajectory of synthesis of an
aperture. The trajectory is chosen rectilinear or circular. Let's begin
consideration with rectilinear synthesis of an aperture.

**5.1.
****Rectilinear Synthesis of an Aperture**

The being of a method of reconstruction of the image
of object by means of rectilinear synthesis of an aperture can be described
mathematically as follows [2,3]. Let's be guided by the geometrical scheme of
data collection in Figure 4.

Complex amplitude of the signal reflected from *i-oh* point of object with coordinates (*xi,* *yi*),
*v* accepted by a locator at the
movement by a point on a rectilinear trajectory with a constant speed and a
miss *of yi*, it is possible to write
down (Figure 4) as

(1)

where a(x_{i,}y_{i}) the complex amplitude of dispersion *i-oh* a point, *Ca* length
of the synthesized aperture, *t*-the
current time, *v-*the constant speed of
movement of the receiver with a radiator of rather Tomography object, -the wavelength of the probing signal.

Complex amplitude of the signal reflected from all
object consisting of the* I* points it
is possible to present in the form the sums of reflections from all points of
object.

(2)

where the complex amplitude of the total signal reflected from
the all I points object.

To reconstruct the image of object on one registered
trajectory Doppler signal, it is necessary to create basic trajectory signals
from basic dot objects in advance.

And reference points need to be "placed" on
different removal from a rectilinear trajectory on a perpendicular to it with
some step, blocking all range of ranges in which the Tomography object settles
down.

The complex amplitude of a basic signal corresponding
to reflection from basic dot object with coordinates (0,), it is possible to write down as

(3)

where- a step on a miss where the miss is understood as the
shortest distance on a normal from a trajectory to a Tomography point, - number of a step on a miss.

Then the set of one-dimensional mutual correlation
functions between basic signals of* f0 (t)*
and an object signal of *f(t)* can be
presented in the form of function of two variables.

(4)

where - time of synthesizing of an aperture.

The ratio allows to pass from temporary shift to spatial shift and to
construct on expressions (4) two-dimensional function of dispersion of object
in coordinates of a miss and range on a trajectory.

It is necessary to pay attention that expression (4) is not two-dimensional mutual correlation function. It is a set of one-dimensional correlation functions for different ranges of a reference point.

However, it is presented by two-dimensional function (4). This function can be displayed on the plane and to construct a surface which will be similar to two-dimensional correlation function. If reset this surface the planes of different height and to combine the received sections, having painted them in different tone, then we will receive the tomogram of object.

Results of model
reconstruction of the image of dot object are presented in Figure 5. The same
results are yielded by reconstruction of section of a moving delay, Figure 2-a
because the delay truncated down by the directional pattern of a piezo element
(Figure 2-b), it is equivalent to dot object.

It is accepted to call a
surface from correlation functions of Figure 5-b in literature transfer
function of a point [4]. It characterizes possibilities of the tomograph, in
particular resolution of this method of reconstruction.

If reset a correlation
surfaces the horizontal plane at the level of 0,7 from the maximum value, then
we receive a circle diameter λ/5 which characterizes resolution of a method.
Such resolution is known in literature as Rayleigh's limit [4].

Ideally for a tomography
it would be desirable to have transfer function of a point in the form of
Dirac's impulse. Such chance is given by multi-angles synthesis of an aperture.
Let's explain essence of multi-angles synthesis of an aperture on the example
of two-courses synthesis of an aperture. Let's register a trajectory Doppler
signal, having provided the movement of a locator on a rectilinear trajectory.

After correlation
processing of the registered trajectory signal basic trajectory signals from
dot objects we will receive a correlation surface, that is transfer function of
a point.

Then we will force a
locator to move on other trajectory, a perpendicular initial trajectory. After
correlation processing of again received trajectory signal we will have other
correlation surface turned on 900 in relation to an initial surface.

Let's impose correlation
matrixes of the turned and not turned surface and we will multiply the
coincided elements of matrixes. Let's receive the resulting correlation surface
which will correspond to two-courses aperture synthesis. It is presented in
Figure 6. Thus, Figure 6 shows the result of superimposing two correlation
surfaces, one of which is rotated by 90 relative to the original correlation
surface.

When the values of the
correlation functions were superimposed, they multiplied. As a result, two
aperture synthesis of the aperture along two mutually perpendicular
trajectories was obtained.

Multi-angles synthesis
of an aperture is supposed to be used for elimination of an interference. To
show how it is possible to eliminate an interference it is possible on the
example of recovery of the image of multipoint object in the form of an obtuse
angle of Figure 7 on one trajectory signal of Figure 9(the trajectory signal is
shown on a Figure 8), on two signals from two orthogonal trajectories for
Figure 10, an and four trajectory signals of Figure 10-b. At recovery of the
image of object on four trajectory signals we will turn coordinate basis from
orthogonal trajectories for the second couple from four trajectory signals on
45^{°}. Figure 11.

Figure 9 shows the dense
interferential picture hiding sixteen dot object. In Figure 10, the result of
restoration of sixteen dot objects is presented in the form of an obtuse angle
for orthogonal trajectories. Points of this object will defend from each other
at distance 0,651

In Figure 10, b the
image of sixteen dot objects in the form of an obtuse angle restored on four
trajectory signals is presented. At the same time the basis of the second
couple of orthogonal trajectories is turned concerning basis of the first
couple of orthogonal trajectories on 45º (see Figure 11).

That at matrixes the
bigger number of elements coincided, it is necessary to reduce distance between
elements of one of matrixes in time. For
this purpose at registration and processing the step on a miss and a step along
a trajectory also should be reduced in
time. On a site of full overlapping of matrixes coincidence of elements
reaches 50%. At the same time, it is necessary to multiply only the coincided elements
of matrixes and to reject not coincident. The object on the turned matrix will
hold the same position, as at not turned matrix.

Comparing the results
presented in Figure 9 and Figure 10 it is possible to see that multi-angles
synthesis of an aperture allows to weaken an interference.

**5.2. Synthesis of an Aperture on a Circular
Trajectory **

From this geometric
diagram it is possible to determine the current distance R (t) from the moving
locator to the i-th point of the object. It is supposed that the locator with
the isotropic chart moves on a circular trajectory, irradiating the reflecting
point with a monochromatic signal, and registers a trajectory Doppler signal of
Figure 12 [5,6,7].

The distance from a
Tomography point to a locator changes under the law

(5)

Where, - the radius of a circular trajectory on
which the locator moves and is carried out aperture synthesis.

- radial removal of a Tomography point from
the center of synthesizing of an aperture.

- the current corner between radiuses of R0
and at circular synthesizing of an
aperture.

The trajectory signal of
s(t) pays off on a formula:

for a case and without
radial attenuation

(6)

for a case taking into account radial attenuation

(7)

where ,,λ-wavelength

The Tomography object is
representable multipoint model. In case of a large number of points the signal
of s(t) reflected from them according to the principle of superposition can be
calculated on a formula

(8)

Where n - total number of Tomography points of object, i - serial number of a Tomography point of object, si(t) - the trajectory Doppler signal reflected from i-oh point of object and calculated on a formula (6) or (7).

For spatial selection of
points of object, it is necessary to scan area of reconstruction of Figure 13
the reconstructed focal spot of the synthesized aperture for what the accepted
Doppler signal of a look (8) is exposed to repeated correlation with the basic
trajectory signals calculated for all reference points in a circular aperture
of Figure 13.

In the case of Figure
13, the reference reflection points with known coordinates are uniformly
distributed in the square reconstruction area, allowing to calculate the
reference trajectory signals according to formulas (6) or (7).

If to take Fourier's
transformation from a trajectory signal for a reference point, then the
received range will define actually frequency characteristic of the
reconstructed filter which is adjusted on k-yu a reference point.

That is, it is necessary
to calculate mutually - the correlation Rk (t) functions of a look.

(9)

where , (10)

(11)

In expressions (9),
(10), (11) the following designations are accepted:

- a range of the scanned signal (8) - the frequency characteristic of the filter which is adjusted on k - yu a space point in a circle on which aperture synthesis is carried out,

sk(t) - the basic signal
calculated on a formula (6) or (7) for one reference point located in k - oh
areas of space. All square areas in the sum form a square, inscribed in a
circle which defines area of reconstruction of Figure 13.

- strip of frequencies of integration

T - time of synthesizing
of an aperture, that is time of flight of object locator around.

Thus, process of
creation of the image was reduced to repeated correlation (9) of a trajectory
signal of a view (8) with the basic signals of a look (6) or (7) calculated at
various ranges , and development of the central counting mutually - correlation
functions in the Cartesian coordinates recoded in tone symbols. For this
purpose, it is necessary to express
as where xk and yk the Cartesian
coordinates of each k-oh point of a two-dimensional matrix of the image of
Figure 13.

5. Results of Reconstruction of Objects by
Means of Circular Synthesis of the Aperture

Trajectory signals from dot objects at the movement of a locator are around presented in Figure 14. On a Figure 16 the top view on the correlation surfaces presented on a Figure 15 is shown.

**6. Problem of Circular Synthesis of the
Aperture**

The type of transfer
function from above in essence is the tomogram of object. It is necessary to
pay attention to a pedestal of transfer function (Figure 15). The pedestal is
the basis of the transfer function, that is, the lower part of the correlation
surface. It looks like a cone. The less pedestal, the result of a Tomography is
better. At the level of a pedestal the interference is observed. Let's consider
a pedestal in more detail than Figure 17.

Figure 17 consists of
rings whose height grows as the diameter of the ring decreases. A cone is
formed from the rings, passing into the delta function.

We see rings which it is
possible to call "Airey's Rings". The reason of their emergence
easily speaks. Circular synthesis of an aperture should be finished, having
made one turn. We are forced to stop a moving locator and to tear off a
trajectory signal rice 14. Break of a trajectory signal is followed by
manifestation of effect of Gibbs, that is not the steady state is followed by
refluctuations. These refluctuations eventually are made out in Airey's rings.

This problem is
designated in D. Mensah [4] Predlozhennoye's work by it the solution disposal
of rings consists in applying multi frequency sounding, that is to leave from a
coherent tomography.

Until now circular
synthesis of an aperture without radial attenuation of a trajectory signal of
Figure 18, a was considered. In Figure 18-b the same point, under the same
conditions, but taking into account radial attenuation of a trajectory signal
according to a formula (7).

At the accounting of
radial attenuation, the so-called system of automatic adjustment of
strengthening which after correlation processing increased amplitude of
responses according to remoteness of a point from a trajectory of synthesizing
of an aperture was provided in the program of calculation.

Comparing Figure 18-a
and Figure 18-b can see how at the accounting of radial attenuation, transfer
function of a point in section the horizontal plane takes a form of the eight
which is characteristic of linear aperture synthesis [2,3]. This result is
obvious as for the point remote from the center and brought closer to a
trajectory, at the accounting of radial attenuation, effective synthesizing of
an aperture happens only on a piece of an arch of a circle, approximate to a
point which can be interpreted as a piece of a rectilinear trajectory. That is
transition to rectilinear synthesis of an aperture is observed.

**7. Ring Ranges of Transfer Functions**

Interesting property of
transfer functions is noticed. Let's take two-dimensional transformation of
Fourier from transfer function of Figure 15. As a result of such transformation
ring ranges turn out [8]. The ring spectrum is shown in Figure 19-a. Figure 19-b
shows a top view of the circular spectrum depicted in Figure 19-a. Figure 19-c
shows the phase portrait of the ring spectrum.

They are a little
studied and very few people apply them to processing of transfer functions.
Ring ranges are received in 80-x years [4]. Dean Mensah registered ring ranges,
and received transfer function by the return two-dimensional transformation of
Fourier from a ring range.

**8.
****Conclusion**

Aperture synthesis at
small range opens great opportunities, in particular in a tomography. The monochromatic
signal allows to avoid dispersive distortions at reconstruction of the image of
objects. Resolution of a method allows to reach Rayleigh's limit, that is λ/5.
Changing wavelength, it is possible to operate resolution of the tomograph.

To use these remarkable
qualities of aperture synthesis it is necessary to overcome many problems.
Problems arise constantly in process of advance in realization of a laboratory
sample. For example, it is possible to face a problem of reflection of the
ultrasonic probing signal from the superficial wave left by the sonar antennas
moving in water. The solution is considered in [8].

The arising problems are
overcome in process of their identification. Help the solution of problems of
achievement in technologies, and development of the computer equipment.

**Figure 1:** Ultrasonography image of a gall bladder.

**Figures 2(a-c):** **a)** the Geometrical and function chart
of data collection (the top view in the equatorial plane), **b)** The directional pattern of piezo elements 1 and 2 in the
meridional plane, **c)** a design of piezo
elements 1 and 2. If it is required to realize circular synthesis of an
aperture, then fix Tomography object, for example a delay in motionless
situation, and begin to rotate the tank with water with a constant speed. At
the same time piezo elements, (that is ultrasonic antennas) the tank walls
attached to internal with water will make circular movement around Tomography
object, being in the water environment.

**Figure 3:** Ultrasonic experimental installation for Tomography of
objects in water. The white square from the material absorbing ultrasound on a
tank sidewall from the inside contains two piezoceramic elements which carry out function of ultrasonic
antennas of a Doppler locator.

**Figure 4:**
Geometry, the Tomography explaining process.