Author: claimsbot

1. A method performed by a server for synchronizing first data stored to a mobile device and second data stored to a remote storage device, the method comprising steps of:
receiving a schema that comprises group-matching date representative of groups of instances of the first data;
generating corresponding group-matching data for groups of instances of the second data using the schema;
determining group-level differences between respective group-matching data for the first and second data;
matching respective instances of the first and second data in response to the group-level differences; and
updating at least one of the first and second data in response to the differences, wherein the group-matching data for the first data and group-matching data for the second data comprise respective hash data of corresponding groups of the first data and the second data as well as record counts comprising a number of records for each group of instances of the first and second data, respectively.
2. The method of claim 1 comprising maintaining the synchronization of the first and second data in response to the matching.
3. The method of claim 2 wherein maintaining comprises: determining changes to the second data and transmitting at least some of the changes to update the first data in accordance with an over the air session-oriented protocol; and receiving changes to the first data in accordance with an over the air session-oriented protocol and updating the second data in response.
4. The method of claim 3 wherein said transmitting is responsive to a progressive mode and a batch mode.
5. The method of claim 2 wherein said receiving, matching, determining and updating define a slow synchronization phase and said maintaining comprises a synchronization phase.
6. The method of claim 5 comprising performing said slow synchronization phase in response to a change in the schema.
7. The method of claim 1 comprising requesting instance matching data for the first data in response to the group-level differences; generating corresponding instance matching data for the second data using the schema; determining instance-level differences between respective instance matching data for the first data and the second data and wherein matching the respective instances of the first and second data is in response to the instance-level differences.
8. The method of claim 7 wherein the instance matching data for the first data and instance matching data for the second data comprises respective hash data of instances of the first data and second data from corresponding groups of the first data and second data.
9. The method of claim 1 wherein the first and second data are stored to corresponding tables of respective databases, said schema defining a database schema for said respective databases and said respective instances of the first and second data comprising records of said respective databases.
10. The method of claim 1 wherein updating comprises communicating at least some of the differences to update the first data in accordance with an over the air session-oriented protocol.
11. The method of claim 1 wherein updating comprises receiving at least some of the differences to update the second data in accordance with an over the air session-oriented protocol.
12. The method of claim 1 wherein the steps of receiving an updating are performed using a wireless network coupling the server and mobile device for communications.
13. A method performed by a mobile device for synchronizing first data stored to the mobile device and second data stored to a remote storage device, the method comprising steps of:
storing at the mobile device a schema that comprises group-matching data representative of groups of instances of the first data;
communicating the schema to a server adapted to generate corresponding group-matching data for groups of instances of the second data using the schema and to match respective instances of the first and second data using the schema by determining group-level differences between respective group-matching data for the first and second data; and
updating at least one of the first and second data in response to the differences determined by the server, wherein the group-matching data for the first data and group-matching data for the second data comprise respective hash data of corresponding groups of the first data and the second data as well as record counts comprising a number of records for each group of instances of the first and second data respectively.
14. The method of claim 13 comprising maintaining the synchronization of the first and second data in response to the matching.
15. The method of claim 14 wherein maintaining comprises: determining changes to the first data and transmitting at least some of the changes to update the second data in accordance with an over the air session-oriented protocol; and receiving changes to the second data in accordance with an over the air session-oriented protocol and updating the first data in response.
16. The method of claim 15 wherein said transmitting is responsive to a progressive mode and a batch mode.
17. The method of claim 14 wherein said defining, communicating and updating define a slow synchronization phase and said maintaining comprises a synchronization phase.
18. The method of claim 17 comprising initiating said slow synchronization phase on response to a change in the schema.
19. The method of claim 13 comprising providing instance matching data for the first data in response to a request from the server for generating corresponding instance matching data for the second data using the schema; determining instance-level differences between respective instance matching data for the first data and the second data and matching the respective instances of the first and second data in response to the instance-level differences.
20. The method of claim 19 wherein the instance matching data for the first data and instance matching data for the second data comprises respective hash data of instances of the first data and second data from corresponding groups of the first data and second data.
21. The method of claim 13 wherein the first and second data are stored to corresponding tables of respective databases, said schema defining a database schema for said respective databases and said respective instances of the first and second data comprising records of said respective databases.
22. The method of claim 13 wherein updating comprises receiving from the server at least some of the differences to update the first data in accordance with an over the air session-oriented protocol.
23. The method of claim 13 wherein updating comprises transmitting to the server at least some of the differences to update the second data in accordance with an over the air session-oriented protocol.
24. The method of claim 13 wherein the steps of communicating and updating are performed using a wireless network coupling the server and mobile device for communications.
25. A server for synchronizing first data stored to a mobile device and second data stored to a remote storage device, the server comprising:
a communications system for transmitting to and receiving messages from the mobile device;
a processor coupled to the communication system for processing received messages and messages for transmitting; and
a memory coupled to the processor for storing instructions to configure the processor to: receive a schema from the mobile device, the schema comprising group-matching data representative of groups of instances of the first data; generate corresponding group-matching data for groups of instances of the second data using the schema; match respective instances of the first and second data in response to group-level differences between respective group-matching data for the first and second data; and update at least one of the first and second data in response to the differences, wherein the group-matching data for the first data and group-matching data for the second data comprise respective hash data of corresponding groups of the first data and the second data as well as record counts comprising a number of records for each group of instances of the first and second data, respectively.
26. The server of claim 25 wherein the instructions configure the processor to maintain the synchronization of the first and second data in response to the match of respective instances.
27. The server of claim 26 wherein the first and second data are stored to corresponding tables of respective databases, said schema defining a database schema for said respective databases and said respective instances of the first and second data comprising records of said respective databases.
28. The method of claim 26 wherein the instructions to configure the processor to maintain comprise instructions to configure the processor to determine changes to the second data and transmit at least some of the changes to update the first data in accordance with an over the air session-oriented protocol; and receive changes to the first data in accordance with an over the air session-oriented protocol and update the second data in response.
29. The server of claim 28 wherein said the instructions to configure the processor to transmit are responsive to a progressive mode and a batch mode.
30. The server of claim 26 wherein the instructions to configure the processor to receive, match, determine and update a slow synchronization phase capability and said instructions to configure the processor to maintain define a synchronization phase capability.
31. The server of claim 30, comprising instructions to perform said slow synchronization phase capability in response to a change in the schema.
32. The server of claim 25 wherein the instructions configure the processor to request instance matching data for the first data in response to the group-level differences; generate corresponding instance matching data for the second data using the schema; determine instance-level differences between respective instances matching data for the first data and the second data and wherein the instructions to match the respective instances of the first and second data are response to the instance-level differences.
33. The server of claim 32 wherein the instance matching data for the first data and instance matching data for the second data comprises respective hash data of instances of the first data and second data from corresponding groups of the first data and second data.
34. The server of claim 25 wherein the instructions to configure the processor to update comprise instructions to configure the processor to transmit to the mobile device at least some of the differences to update the first data in accordance with an over the air session-oriented protocol.
35. The server of claim 25 wherein the instructions to configure the processor to update comprise instructions to configure the processor to receive from the mobile device at least some of the differences to update the second data in accordance with an over the air session-oriented protocol.
36. A mobile device for synchronizing first data stored to the mobile device and second data store to a remote storage device, the mobile device comprising:
a communications system for transmitting to and receiving messages from the mobile device;
a processor coupled to the communication system for processing received messages and messages for transmitting; and
a memory coupled to the processor for storing instructions to configure the processor to: define at the mobile device a schema, the schema comprising group-matching data representative of groups of instances of the first data; transmit the schema to a server adapted to match respective instances of the first and second data by using the schema to generate corresponding group-matching data for groups of instances of the second data in order to then determine group-level differences between respective group-matching data for the first and second data; and update at least one of the first and second data in response to the differences determined by the server, wherein the group-matching data for the first data and group-matching data for the second data comprise respective hash data of corresponding groups of the first data and the second data as well as record counts comprising a number of records for each group of instances of the first and second data, respectively.
37. The mobile device of claim 36 comprising instructions to configure the processor to maintain the synchronization of the first and second data in response to the match of respective instances.
38. The mobile device of claim 37 wherein the instructions to configure the processor to maintain comprising instructions to determine changes to the first data and transmit at least some of the changes to update the second data in accordance with an over the air session oriented protocol.
39. The mobile device of claim 38 wherein said instructions to transmit are responsive to a progressive mode and a batch mode.
40. The mobile device or claim 37 wherein the instructions to configure the processor to define, transmit and update define a slow synchronization phase capability and wherein the instructions to configure the processor to maintain define a synchronization phase capability.
41. The mobile device of claim 40 comprising instructions to configure the processor to initiate said slow synchronization phase in response to a change in the schema.
42. The mobile device of claim 36 comprising instructions to configure the processor to provide instance matching data for the first data in response to a request from the server to generate corresponding instance matching data for the second using the schema; determine instance-level differences between respective instance matching data for the first data and the second data and match the respective instances of the first and second data in response to the instance-level differences.
43. The mobile device of claim 42 wherein the instance matching data for the first data and instance matching data for the second data comprises respective hash data of instances of the first data and second data from corresponding groups of the first data and second data.
44. The mobile device of claim 36 wherein the first and second data are stored to corresponding tables of respective databases, said schema defining a database schema for said respective databases and said respective instances of the first and second data comprising records of said respective databases.
45. The mobile device of claim 36 wherein the instructions to configure the processor to update comprise instruction for receiving from the server at least some of the differences to update the first data in accordance with an over the air session-oriented protocol.
46. The mobile device of claim 36 wherein the instructions to configure the processor to update comprise instruction to configure the processor to transmit to the server at least some of the differences to the first data to update the second data in accordance with an over the air session-oriented protocol.

The claims below are in addition to those above.
All refrences to claim(s) which appear below refer to the numbering after this setence.

1. A printing apparatus comprising:
a print head configured to eject ink onto a print medium;
a heating unit configured to heat the print medium onto which the print head has ejected ink, the heating unit includes a heat-pump mechanism with a channel through which a refrigerant passes, and
a conveying unit configured to convey the print medium downstream along a conveying direction,
wherein the heat-pump mechanism comprises a compressor, a condenser, an expansion valve, and an evaporator each provided along the channel,
the heating unit heating the print medium by transferring heat generated by the condenser when the refrigerant is condensed by the condenser, and
the refrigerant in the channel flowing into the condenser at a downstream side thereof in the conveying direction and out from the condenser at an upstream side thereof in the conveying direction.
2. The printing apparatus according to claim 1, wherein the condenser is separated into at least two portions along the conveying direction.
3. The printing apparatus according to claim 2, wherein the condenser is separated into two portions, and one of the two portions which is located downstream of the other portion in the conveying direction of the print medium has a smaller thermal capacity than the other portion.
4. The printing apparatus according to claim 1, further comprising a temperature-measuring unit configured to measure the temperature of the condenser, and a control unit configured to control the compressor based on an output of the temperature-measuring unit, in such a manner that the temperature of the condenser at the downstream side thereof is higher than the condensing temperature of the refrigerant.
5. The printing apparatus according to claim 1, wherein the channel in the condenser is arranged in a meandering configuration.

1. A laminated coil component comprising:
a magnetic section including stacked magnetic layers; and
a conductor section having a plurality of conductor pattern layers arranged between the magnetic layers, and the conductor pattern layers being interconnected in a coiled shape to pass through the magnetic layers, the conductor section being buried in the magnetic section,
the conductor section including a conductor containing silver,
the magnetic section including a sintered ferrite material containing Fe, Ni, Zn, and Cu, and
a ratio of Cu content in CuO in a near-conductor section region of the magnetic section to Cu content in CuO in a central region of the magnetic section is 0.2 to 0.5.
2. The laminated coil component according to claim 1, wherein the ratio of Cu content in CuO in the near-conductor section region of the magnetic section to Cu content in CuO in the central region of the magnetic section is 0.2 to 0.3.
3. The laminated coil component according to claim 1, wherein the Cu content in CuO in the central region of the magnetic section is 0.2 to 3 weight %.

The claims below are in addition to those above.
All refrences to claim(s) which appear below refer to the numbering after this setence.

1. A method for producing a high frame rate, high resolution and high contrast image, comprising:
a) transmitting a group of signals of energy weighted by single spatial frequency but may be of different phases or linear time delay toward an object to be imaged;
b) weighting receive signals from the object with multiple spatial frequencies, or by performing a spatial Fourier transform;
c) reconstructing a two- or three-dimensional image data set from the group of the transmitted signals weighted by the single spatial frequency or linear time delay, and the receive signals weighted with the multiple spatial frequencies or processed by the spatial Fourier transform; and,
d) reconstructing the high frame rate, high resolution and high contrast image from the image data set of step c.
2. A method for producing a high frame rate, high resolution and high contrast velocity vector image of an object where at least a part of the object is moving, comprising:
a) transmitting two or more groups of signals of energy weighted by single spatial frequency but may be of different phases or linear time delay-toward the object,
b) weighting receive signals from the object with multiple spatial frequencies or by performing a spatial Fourier transform;
c) reconstructing two- or three-dimensional image data sets from the groups of the transmitted signals weighted by the single spatial frequency or linear time delay, and the receive signals weighted with the multiple spatial frequencies or processed by the spatial Fourier transform;
d) using the image data sets to reconstruct:
a first set of flow velocity component images in a first direction, and
a second set of flow velocity component images in a second direction that is different from the first direction; and,

e) reconstructing the velocity vector image from the two sets of velocity component images.
3. The method of claim 1, wherein step a) each group may contain one or more signals, each of which is produced with one transmission.
4. The method of claim 1, in which the transmit group comprises one or more limited diffraction beams.
5. The method of claim 1, in which the received signal for echoes returned from all random scatterers within the object f( r0) is a linear superposition of those echo signals from individual point scatterers as follows:
R
k
x

+

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

z
T
\ue8a0

(
t
)
=

\ue89e
1

2
\ue89e
\u03c0
\ue89e
\u222b

–
\u221e

\u221e

\ue89e
A
\ue8a0

(
k
)
\ue89e

T
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
c

\xd7
\u222b
V

\ue89e
f
\ue8a0

(
r
\u21c0

0

)
\ue89e

\uf74d
\uf74e
\ue8a0

(
k
x

+

k

x
T
)
\ue89e

x
0
+
\uf74e
\ue8a0

(
k
y

+

k

y
T
)
\ue89e

y
0
+
\uf74e
\ue8a0

(
k
z

+

k

z
T
)
\ue89e

z
0
\ue89e
\ue89e

\uf74c
r
\u21c0

0
\ue89e

\uf74d
–
i

\ue89e
\ue89e
\u03c9
\ue89e
\ue89e
t
\ue89e

\uf74c
k

\ue89e

1

2
\ue89e
\u03c0
\ue89e
\u222b

–
\u221e

\u221e

\ue89e
A
\ue8a0

(
k
)
\ue89e

T
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
c

\ue89e

F
\ue8a0

(
k
x

+

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

z
T
)
\ue89e

\uf74d
–
i

\ue89e
\ue89e
\u03c9
\ue89e
\ue89e
t
\ue89e
\ue89e
\uf74c
k

.
(
8
)
6. The method of claim 5, in which the temporal Fourier transform (spectrum) of the received signal obtained as follows:
R
~
k
x

+

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

z
T
\ue8a0

(
\u03c9
)
=
A
\ue8a0

(
k
)
\ue89e

T
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
c
2
\xd7
\u222b
V

\ue89e
f
\ue8a0

(
r
\u21c0

0

)
\ue89e

\uf74d
\uf74e
\ue8a0

(
k
x

+

k

x
T
)
\ue89e

x
0
+
\uf74e
\ue8a0

(
k
y

+

k

y
T
)
\ue89e

y
0
+
\uf74e
\ue8a0

(
k
z

+

k

z
T
)
\ue89e

z
0
\ue89e
\ue89e

\uf74c
r
\u21c0

0
\ue89e
\ue89e
or
\ue89e
\ue89e
R
~
k
x

+

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

x
T
\ue8a0

(
\u03c9
)
=
A
\ue8a0

(
k
)
\ue89e

T
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
c
2
\xd7

F
\ue8a0

(
k
x

+

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

z
T
)
\ue89e
\ue89e
or
\ue89e
\ue89e
F
BL

\ue8a0

(
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
)
=
c
2

\ue89e

H
\ue8a0

(
k
)
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
\ue8a0

(
\u03c9
)
.
(
10
)
7. The method of claim 6, in which a 2D Fourier transform of the echo signals in terms of both x1 and y1 over a transducer surface is as follows:
R
~
k
x

+

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

z
T
\ue8a0

(
\u03c9
)
=
A
\ue8a0

(
k
)
\ue89e

T
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
c
2
\xd7
\u222b
V
\ue89e
f
\ue8a0

(
r
\u21c0

0

)
\ue89e

\uf74d
\uf74ek
x

\ue89e

x
0
+
\uf74ek
y

\ue89e

y
0
+
\uf74ek
z

\ue89e

z
0
\ue89e

\uf74d
\uf74ek

x
T
\ue89e

x
0
+
\uf74ek

y
T
\ue89e

y
0
+
\uf74ek

z
T
\ue89e

z
0
\ue89e
\ue89e

\uf74c
r
\u21c0

0
=
A
\ue8a0

(
k
)
c

\ue89e
\u222b
V

\ue89e
f
\ue8a0

(
r
\u21c0

0

)
\ue8a0
T
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
c

\ue89e

\uf74d
\uf74ek
x

\ue89e

x
0
+
\uf74ek
y

\ue89e

y
0
+
\uf74ek
z

\ue89e

z
0
\ue89e

\uf74d
\uf74ek

x
T
\ue89e

x
0
+
\uf74ek

y
T
\ue89e

y
0
+
\uf74ek

z
T
\ue89e

z
)
\ue89e
\ue89e

\uf74c
r
\u21c0

0
=
A
\ue8a0

(
k
)
c

\ue89e
\u222b
V

\ue89e
f
\ue8a0

(
r
\u21c0

0

)
\ue8a0
\u03a6
~

Array
R

\ue8a0

(
r
\u21c0

0

,
\u03c9

)
\ue89e

\uf74d
\uf74ek

x
T
\ue89e

x
0
+
\uf74ek

y
T
\ue89e

y
0
+
\uf74ek

z
T
\ue89e

z
)
\ue89e
\ue89e

\uf74c
r
\u21c0

0
=
A
\ue8a0

(
k
)
c

\ue89e
\u222b
V

\ue89e
f
\ue8a0

(
r
\u21c0

0

)
\ue8a0
\ue529
x
1

,

y
1
\ue89e

{
E
~

\ue8a0

(
x
1

,
y
1

;
r
\u21c0

0

;
\u03c9
)
}
\ue89e
\ue89e

\uf74d
\uf74ek

x
T
\ue89e

x
0
+
\uf74ek

y
T
\ue89e

y
0
+
\uf74ek

z
T
\ue89e

z
0
\ue89e

\uf74c
r
\u21c0

0
=
\ue529
x
1

,

y
1
\ue89e

{
\u222b
V

\ue89e
f
\ue8a0

(
r
\u21c0

0

)
\ue89e
A
\ue8a0

(
k
)
c

\ue89e

\uf74d
\uf74ek

x
T
\ue89e

x
0
+
\uf74ek

y
T
\ue89e

y
0
+
\uf74ek

z
T
\ue89e

z
0
\ue89e
\ue89e
E
~

\ue8a0

(
x
1

,
y
1

;
r
\u21c0

0

;
\u03c9
)
\ue89e

\uf74c
r
\u21c0

0
}
.
(
13
)
8. The method of claim 1, wherein four limited-diffraction array beams are transmitted (fix both kxT and kyT) in each group as follows:
\u03a6

Array
\ue8a0

(
1
)
T

(
r
->

0

,
t

)

=
1

2
\ue89e
\ue89e
\u03c0
\ue89e
\u222b

–
\u221e

\u221e

\ue89e
A
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
\ue89e

cos
\ue8a0

(
k

x
T
\ue89e

x
0
)
\ue89e

cos
\ue8a0

(
k

y
T
\ue89e

y
0
)
\ue89e

\uf74d

\uf74e
\ue89e
\ue89e

k
z
T

\ue89e

z
0
\ue89e

\uf74d

\uf74e\u03c9
\ue89e
\ue89e
t
\ue89e

\uf74c
k
,
(
26
)
\u03a6

Array
\ue8a0

(
2
)
T

(
r
->

0

,
t

)

=
1

2
\ue89e
\ue89e
\u03c0
\ue89e
\u222b

–
\u221e

\u221e

\ue89e
A
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
\ue89e

cos
\ue8a0

(
k

x
T
\ue89e

x
0
)
\ue89e

sin
\ue8a0

(
k

y
T
\ue89e

y
0
)
\ue89e

\uf74d

\uf74e
\ue89e
\ue89e

k
z
T

\ue89e

z
0
\ue89e

\uf74d

\uf74e\u03c9
\ue89e
\ue89e
t
\ue89e

\uf74c
k
,
(
27
)
\u03a6

Array
\ue8a0

(
3
)
T

(
r
->

0

,
t

)

=
1

2
\ue89e
\ue89e
\u03c0
\ue89e
\u222b

–
\u221e

\u221e

\ue89e
A
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
\ue89e

sin
\ue8a0

(
k

x
T
\ue89e

x
0
)
\ue89e

cos
\ue8a0

(
k

y
T
\ue89e

y
0
)
\ue89e

\uf74d

\uf74e
\ue89e
\ue89e

k
z
T

\ue89e

z
0
\ue89e

\uf74d

\uf74e\u03c9
\ue89e
\ue89e
t
\ue89e

\uf74c
k
,
\ue89e
and
(
28
)
\u03a6

Array
\ue8a0

(
4
)
T

(
r
->

0

,
t

)

=
1

2
\ue89e
\ue89e
\u03c0
\ue89e
\u222b

–
\u221e

\u221e

\ue89e
A
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
\ue89e

sin
\ue8a0

(
k

x
T
\ue89e

x
0
)
\ue89e

sin
\ue8a0

(
k

y
T
\ue89e

y
0
)
\ue89e

\uf74d

\uf74e
\ue89e
\ue89e

k
z
T

\ue89e

z
0
\ue89e

\uf74d

\uf74e\u03c9
\ue89e
\ue89e
t
\ue89e

\uf74c
k
,
(
29
)
wherein four coverage areas are obtained in a spatial Fourier space of f( r0) from combinations of the four echo signals, and wherein denoting the Fourier transform of the four echo signals as {tilde over (R)}k\u2032x,k\u2032y,k\u2032z(1)(\u03c9), {tilde over (R)}k\u2032x,k\u2032y,k\u2032z(2)(\u03c9), {tilde over (R)}k\u2032x,k\u2032y,k\u2032z(3)(\u03c9), and {tilde over (R)}k\u2032x,k\u2032y,k\u2032z(4)(\u03c9), corresponding to (26)-(29), respectively, provides:
F
BL

\ue8a0

(
k
x

+

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

z
T
)
=
c
2

\ue89e

H
\ue8a0

(
k
)
\ue89e

(
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
1
)
\ue8a0

(
\u03c9
)
+

\uf74e
\ue89e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
2
)
\ue8a0

(
\u03c9
)
+

\uf74e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
3
)
\ue8a0

(
\u03c9
)
–
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
4
)
\ue8a0

(
\u03c9
)
)
,
(
30
)
F
BL

\ue8a0

(
k
x

+

k

x
T
,
k
y

–

k

y
T
,
k
z

+

k

z
T
)
=
c
2

\ue89e

H
\ue8a0

(
k
)
\ue89e

(
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
1
)
\ue8a0

(
\u03c9
)
–

\uf74e
\ue89e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
2
)
\ue8a0

(
\u03c9
)
+

\uf74e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
3
)
\ue8a0

(
\u03c9
)
+
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
4
)
\ue8a0

(
\u03c9
)
)
,
(
31
)
F
BL

\ue8a0

(
k
x

–

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

z
T
)
=
c
2

\ue89e

H
\ue8a0

(
k
)
\ue89e

(
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
1
)
\ue8a0

(
\u03c9
)
+

\uf74e
\ue89e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
2
)
\ue8a0

(
\u03c9
)
–

\uf74e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
3
)
\ue8a0

(
\u03c9
)
+
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
4
)
\ue8a0

(
\u03c9
)
)
,
(
32
)

.
F
BL

\ue8a0

(
k
x

–

k

x
T
,
k
y

–

k

y
T
,
k
z

+

k

z
T
)
=
c
2

\ue89e

H
\ue8a0

(
k
)
\ue89e

(
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
1
)
\ue8a0

(
\u03c9
)
–

\uf74e
\ue89e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
2
)
\ue8a0

(
\u03c9
)
–

\uf74e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
3
)
\ue8a0

(
\u03c9
)
–
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
4
)
\ue8a0

(
\u03c9
)
)
,
(
33
)

.
9. The method of claim 1, further comprising:
using (10) and (11) to directly give a relationship between the 3D Fourier transform of measured echo signals at a transducer surface and the 3D spatial Fourier transform of the object function for a steered plane wave transmission with a fixed Axicon angle (steering angle for plane waves), \u03b6T, of X wave and azimuthal angle, \u03b8T,
obtaining the spatial Fourier transform of the object function, and
using (17) to reconstruct images with an inverse 3D Fourier transform; wherein, for steered plane waves, the relationship of the parameters between the Fourier transform of the echoes and the object function is obtained:
{
k
x
\u2032

=
k
x

+

k
\ue89e
\ue89e
sin
\ue89e
\ue89e

\u03b6
T

\ue89e
cos
\ue89e
\ue89e

\u03b8
T
k
y
\u2032

=
k
y

+

k
\ue89e
\ue89e
sin
\ue89e
\ue89e

\u03b6
T

\ue89e
sin
\ue89e
\ue89e

\u03b8
T
k
z
\u2032

=
k
z

+

k
\ue89e
\ue89e
cos
\ue89e
\ue89e

\u03b6
T
=
k
2

–

k
x
2

–

k
y
2
+

k
\ue89e
\ue89e
cos
\ue89e
\ue89e

\u03b6
T
\u2265
0
.
(
34
)
10. The method of claim 1, wherein, for a 2D image reconstruction, using a 2D imaging formula:
FBL(k\u2032x,k\u2032z)=c2H(k){tilde over (R)}k\u2032x,k\u2032z(\u03c9),\u2003\u2003(35)

where
{
k
x
\u2032

=
k
x

+

k

x
T
k
z
\u2032

=
k
z

+

k

z
T
=
k
2

–

k
x
2
+
k
2

–

k

x
T

2
\u2265
0
.
(
36
)
11. The method of claim 1, wherein step c) includes Fourier transforming along a time domain of one or more of: i) the weighted transmitted signal, or ii) the spatial Fourier transform, whereby a multi-dimensional k-space data set is formed.
12. The method of claim 1, wherein step c) includes:
i) interpolating a multi-dimensional k-space data set to produce rectilinear multi-dimensional k-space data sets; and,
ii) performing inverse Fourier transformations of the interpolated rectilinear multi-dimensional k-space data sets along each of its dimensions to produce the image data set.
13. The method of claim 12, wherein the k-space data sets have two- or three-dimensions, and the inverse Fourier transformation is performed along each of the two- or three-dimensions to produce a two- or three-dimensional image data set.
14. The method of claim 1, or wherein the image data set of step c) is reconstructed using the formula kz\u2032=k+kzT.
15. The method of claim 1, wherein the image data set of step c) is reconstructed using
{
k
x
\u2032

=
k
x

+

k

x
T
k
y
\u2032

=
k
y

+

k

y
T
k
z
\u2032

=
k
z

+

k

z
T
=
k
2

–

k
x
2

–

k
y
2
+
k
2

–

k

x
T

2

–

k

y
T

2
\u2265
0
.
16. The method of claim 1, wherein the image data set of step c) is reconstructed using
{
k
x

=
k
x
\u2032

–

k

x
T
k
=
(
k
z
\u20322

+

k

x
T

2

–
(
k
x
\u2032

–

k

x
T
)

2
)

2

+

4
\ue89e
\ue89e
k
z
\u20322

\ue8a0

(
k
x
\u2032

–

k

x
T
)
2
2
\ue89e

k
z
\u2032
,
for producing a two-dimensional image.
17. The method of claim 1, wherein steps a)-c) are performed a plurality of times.
18. The method of claim 1, wherein the spatial frequency is non-uniform.
19. The method of claim 1, further including combining a plurality of the single transmit spatial frequency signals and the multiple receive spatial frequency signals to increase signal-to-noise ratio, image resolution, image contrast, and reduce sidelobes for the image.
20. The method of claim 2, further including combining a plurality of the multiple transmit spatial frequency signals and the multiple receive spatial frequency signals to increase signal-to-noise ratio, image resolution, image contrast, and reduce sidelobes for the image.
21. The method of claim 1, in which the step a) and step b) are performed using the same transducer array.
22. The method of claim 1, in which the multiple receive signals are Fourier transformed over a single transducer aperture.
23. The method of claim 1, in which the multiple receive signals are superposed coherently with corresponding or other transmission weightings or steered angles to enhance resolutions and contrast, and to reduce sidelobes.
24. The method of claim 1, in which the multiple receive signals are superposed incoherently with corresponding or other transmission weightings or steered angles to reduce speckle formation.
25. The method of claim 1, in which one group of transmitted signals is used to reconstruct an image.
26. The method of claim 2, in which at least two groups of transmitted signals are used to reconstruct an image.
27. The method of claim 1, in which a single transmitter is used to produce weightings for different transducer elements.
28. The method of claim 1, in which more than one transmitter is used to produce weightings for different transducer elements.
29. The method of claim 1, in which harmonic andor elastic images are produced.
30. The method of claim 1, in which a physiological functional image is reproduced.
31. The method of claim 1, in which at least a part of the object to be imaged is moving.
32. The method of claim 2, in which the first direction is perpendicular with respect to a surface transmitting the signals.
33. The method of claim 2, in which a single image is used to construct the first and second sets of flow velocity component images by rotating the single image and interpolating data from the rotated image.
34. The method of claim 2, in which pulse Doppler or color flow Doppler method is used to reconstruct images.
35. The method of claim 1, in which the transmitted signals comprise sine and cosine spatially weighted signals.
36. An apparatus for producing a high frame rate, high resolution and high contrast image of an object, comprising:
a) a device configured to:
i) transmit one or more groups of signals of energy weighted by single spatial frequency but may be of different phases or linear time delay toward an object to be imaged; and
ii) receive by weighting receive signals from the object with multiple spatial frequencies, or by performing a spatial Fourier transform;
b) a device configured to reconstruct a two- or three-dimensional image data set from the group of the transmitted signals weighted by the single spatial frequency or linear time delay, and the receive signals weighted with the multiple spatial frequencies or processed by the spatial Fourier transform; and,
c) a device configured to reconstruct the high frame rate, high resolution and high contrast image from the image data set.
37. The apparatus of claim 36, wherein the transmitreceive device comprises a transducer array configured to form one or more limited diffraction transmitted beams.
38. The apparatus of claim 37, wherein the transmitreceive device comprises a transducer array configured to form one or more steered transmitted beams.
39. The apparatus of claim 36, wherein the device b) is configured to: i) Fourier transform the weighted echo signals to form at least a first multi-dimensional k-space data set.
40. The apparatus of claim 39 wherein device c) is further configured to: ii) interpolate one or more multi-dimensional k-space data sets to produce rectilinear multi-dimensional k-space data sets; and, iii) perform inverse Fourier transformations of the interpolated k-space data sets along each of its dimensions, whereby the image data set is produced.
41. The apparatus of claim 36, wherein the transmitreceive device comprising separate elements in a transducer array arranged in a one- or two-dimensional array.
42. The apparatus of claim 36, wherein the transmitreceive device is configured to steer one or more plane waves at one or more different angles.
43. The apparatus of claim 36, wherein a single transmitreceive device is configured to produce weightings for different transducer elements.
44. The apparatus of claim 36, wherein the transmitreceive device comprises a transducer having a single aperture, and wherein the receive signals are Fourier transformed over the single transducer aperture.
45. The apparatus of claim 36, wherein the transmitreceive device transmits more than one beam at the same spatial frequency or steering angle in order to obtain the velocity component image and to improve signal-to-noise ratio.
46. The method of claim 2, wherein step a) each group may contain one or more signals, each of which is produced with one transmission.
47. The method of claim 2, in which the transmit group comprises one or more limited diffraction beams.
48. The method of claim 2, in which the received signal for echoes returned from all random scatterers within the object f( r0) is a linear superposition of those echo signals from individual point scatterers as follows:
R
k
x

+

k

x
T
,
k
y

+

k
y
T

,
k
z

+

k

z
T
\ue8a0

(
t
)
=
1

2
\ue89e
\u03c0
\ue89e
\u222b

–
\u221e

\u221e

\ue89e
A
\ue8a0

(
k
)
\ue89e

T
\ue8a0

(
k
)
\ue89e
H
\ue89e

(
k
)
c

\xd7
\u222b
V

\ue89e
f
\ue8a0

(
r
->

0

)
\ue89e
\uf74d
\uf74e
\ue8a0

(
k
x

+

k

x
T
)
\ue89e

x
0
+
\uf74e
\ue8a0

(
k
y

+

k

y
T
)
\ue89e

y
0
+
\uf74e
\ue8a0

(
k
z

+

k

z
T
)
\ue89e

z
0
\ue8a0

(
\uf74c
r

_

)
0
\ue89e

\uf74d
–
\uf74e\u03c9

\ue89e
\ue89e
t
\ue89e

\uf74c
k
\ue89e
=
1

2
\ue89e
\u03c0
\ue89e
\u222b

–
\u221e

\u221e

\ue89e
A
\ue8a0

(
k
)
\ue89e

T
\ue8a0

(
k
)
\ue89e
H
\ue89e

(
k
)
c

\ue89e

F
\ue8a0

(
k
x

+

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

z
T
)
\ue89e

\uf74d
–
\uf74e\u03c9

\ue89e
\ue89e
t
\ue89e

\uf74c
k
(
8
)
49. The method of claim 48, in which the temporal Fourier transform (spectrum) of the received signal obtained as follows:
R
~
k
x

+

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

z
T
\ue8a0

(
\u03c9
)
=
A
\ue8a0

(
k
)
\ue89e

T
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
c
2
\xd7
\u222b
V

\ue89e
f
(
r
->

0

)

\ue89e

\uf74d
\uf74e
\ue8a0

(
k
x

+

k

x
T
)
\ue89e

x
0
+
\uf74e
\ue8a0

(
k
y

+

k

y
T
)
\ue89e

y
0
+
\uf74e
\ue8a0

(
k
z

+

k

z
T
)
\ue89e

z
0
\ue89e

\uf74c
r
->

0
\ue89e
\ue89e
or
\ue89e
\ue89e
R
~
k
x

+

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

z
T
\ue8a0

(
\u03c9
)
=
A
\ue8a0

(
k
)
\ue89e

T
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
c
2
\xd7

F
\ue8a0

(
k
x

+

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

z
T
)
\ue89e
\ue89e
or
\ue89e
\ue89e
F
BL

\ue8a0

(
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
)
=
c
2

\ue89e

H
\ue8a0

(
k
)
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
\ue8a0

(
\u03c9
)
.
(
10
)
50. The method of claim 49, in which a 2D Fourier transform of the echo signals in terms of both x1 and y1 over a transducer surface is as follows:
R
~
k
x

+

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

z
T
\ue8a0

(
\u03c9
)
=
A
\ue8a0

(
k
)
\ue89e

T
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
c
2
\xd7
\u222b
V

\ue89e
f
(
r
->

0

)

\ue89e

\uf74d
\uf74e
\ue89e
\ue89e

k
x

\ue89e

x
0
+

\uf74e
\ue89e
\ue89e

k
y

\ue89e

y
0
+

\uf74e
\ue89e
\ue89e

k
z

\ue89e

z
0
\ue89e

\uf74d
\uf74e
\ue89e
\ue89e

k

x
T
\ue89e

x
0
+

\uf74e
\ue89e
\ue89e

k

y
T
\ue89e

y
0
+

\uf74e
\ue89e
\ue89e

k

z
T
\ue89e

z
0
\ue89e

\uf74c
r
->

0
\ue89e
=
A
\ue8a0

(
k
)
c

\ue89e
\u222b
V

\ue89e
f
(
r
->

0

)

\ue8a0
T
\ue8a0

(
k
)
\ue89e
H
\ue89e

(
k
)
c

\ue89e

\uf74d
\uf74e
\ue89e
\ue89e

k
x

\ue89e

x
0
+

\uf74e
\ue89e
\ue89e

k
y

\ue89e

y
0
+

\uf74e
\ue89e
\ue89e

k
z

\ue89e

z
0
\ue89e

\uf74c

\uf74d
\uf74e
\ue89e
\ue89e

k

x
T
\ue89e

x
0
+

\uf74e
\ue89e
\ue89e

k

y
T
\ue89e

y
0
+

\uf74e
\ue89e
\ue89e

k

z
T
\ue89e

z
0
\ue89e

\uf74c
r
->

0
\ue89e
=
A
\ue8a0

(
k
)
c

\ue89e
\u222b
V

\ue89e
f
(
r
->

0

)
\u03a6
~

Array
R

\ue8a0

(
r
->

0

,
\u03c9

)
\ue89e

\uf74d
\uf74e
\ue89e
\ue89e

k
x

\ue89e

x
0
+

\uf74e
\ue89e
\ue89e

k
y

\ue89e

y
0
+

\uf74e
\ue89e
\ue89e

k
z

\ue89e

z
0
\ue89e

\uf74c
r
->

0
\ue89e
=
A
\ue8a0

(
k
)
c

\ue89e
\u222b
V

\ue89e
f
(
r
->

0

)
\ue569
x
1

,

y
1
\ue89e

{
E
~

\ue8a0

(
x
1

,
y
1

;
r
->

0

;
\u03c9
)
}
\ue89e

\uf74d
\uf74e
\ue89e
\ue89e

k

x
T
\ue89e

x
0
+

\uf74e
\ue89e
\ue89e

k

y
T
\ue89e

y
0
+

\uf74e
\ue89e
\ue89e

k

z
T
\ue89e

z
0
\ue89e

\uf74c
r
->

0
\ue89e
=
\ue569
x
1

,

y
1
\ue89e
{
\u222b
V

\ue89e
f
\ue8a0

(
r
->

0

)
\ue89e
A
\ue8a0

(
k
)
c

\ue89e

\uf74d
\uf74e
\ue89e
\ue89e

k

x
T
\ue89e

x
0
+

\uf74e
\ue89e
\ue89e

k

y
T
\ue89e

y
0
+

\uf74e
\ue89e
\ue89e

k

z
T
\ue89e

z
0
\ue89e
E
~

\ue8a0

(
x
1

,
y
1

;
r
_

0

;
\u03c9
)
\ue89e

\uf74c
r
->

0
}

.
(
13
)
51. The method of claim 2, wherein four limited-diffraction array beams are transmitted (fix both kxT and kyT) in each group as follows:
\u03a6

Array
\ue8a0

(
1
)
T

(
r
->

0

,
t

)

=
1

2
\ue89e
\ue89e
\u03c0
\ue89e
\u222b

–
\u221e

\u221e

\ue89e
A
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
\ue89e

cos
\ue8a0

(
k

x
T
\ue89e

x
0
)
\ue89e

cos
\ue8a0

(
k

y
T
\ue89e

y
0
)
\ue89e

\uf74d

\uf74e
\ue89e
\ue89e

k
z
T

\ue89e

z
0
\ue89e

\uf74d

\uf74e\u03c9
\ue89e
\ue89e
t
\ue89e

\uf74c
k
,
(
26
)
\u03a6

Array
\ue8a0

(
2
)
T

(
r
->

0

,
t

)

=
1

2
\ue89e
\ue89e
\u03c0
\ue89e
\u222b

–
\u221e

\u221e

\ue89e
A
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
\ue89e

cos
\ue8a0

(
k

x
T
\ue89e

x
0
)
\ue89e

sin
\ue8a0

(
k

y
T
\ue89e

y
0
)
\ue89e

\uf74d

\uf74e
\ue89e
\ue89e

k
z
T

\ue89e

z
0
\ue89e

\uf74d

\uf74e\u03c9
\ue89e
\ue89e
t
\ue89e

\uf74c
k
,
(
27
)
\u03a6

Array
\ue8a0

(
3
)
T

(
r
->

0

,
t

)

=
1

2
\ue89e
\ue89e
\u03c0
\ue89e
\u222b

–
\u221e

\u221e

\ue89e
A
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
\ue89e

sin
\ue8a0

(
k

x
T
\ue89e

x
0
)
\ue89e

cos
\ue8a0

(
k

y
T
\ue89e

y
0
)
\ue89e

\uf74d

\uf74e
\ue89e
\ue89e

k
z
T

\ue89e

z
0
\ue89e

\uf74d

\uf74e\u03c9
\ue89e
\ue89e
t
\ue89e

\uf74c
k
,
\ue89e
and
(
28
)
\u03a6

Array
\ue8a0

(
4
)
T

(
r
->

0

,
t

)

=
1

2
\ue89e
\ue89e
\u03c0
\ue89e
\u222b

–
\u221e

\u221e

\ue89e
A
\ue8a0

(
k
)
\ue89e

H
\ue8a0

(
k
)
\ue89e

sin
\ue8a0

(
k

x
T
\ue89e

x
0
)
\ue89e

sin
\ue8a0

(
k

y
T
\ue89e

y
0
)
\ue89e

\uf74d

\uf74e
\ue89e
\ue89e

k
z
T

\ue89e

z
0
\ue89e

\uf74d

\uf74e\u03c9
\ue89e
\ue89e
t
\ue89e

\uf74c
k
,
(
29
)
wherein four coverage areas are obtained in a spatial Fourier space of f( r0) from combinations of the four echo signals, and wherein denoting the Fourier transform of the four echo signals as {tilde over (R)}k\u2032x,k\u2032y,k\u2032z(1)(\u03c9), {tilde over (R)}k\u2032x,k\u2032y,k\u2032z(2)(\u03c9), {tilde over (R)}k\u2032x,k\u2032y,k\u2032z(3)(\u03c9), and {tilde over (R)}k\u2032x,k\u2032y,k\u2032z(4)(\u03c9), and corresponding to (26)-(29), respectively, provides:
F
BL

\ue8a0

(
k
x

+

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

z
T
)
=
c
2

\ue89e

H
\ue8a0

(
k
)
\ue89e

(
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
1
)
\ue8a0

(
\u03c9
)
+

\uf74e
\ue89e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
2
)
\ue8a0

(
\u03c9
)
+

\uf74e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
3
)
\ue8a0

(
\u03c9
)
–
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
4
)
\ue8a0

(
\u03c9
)
)
,
(
30
)
F
BL

\ue8a0

(
k
x

+

k

x
T
,
k
y

–

k

y
T
,
k
z

+

k

z
T
)
=
c
2

\ue89e

H
\ue8a0

(
k
)
\ue89e

(
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
1
)
\ue8a0

(
\u03c9
)
–

\uf74e
\ue89e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
2
)
\ue8a0

(
\u03c9
)
+

\uf74e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
3
)
\ue8a0

(
\u03c9
)
+
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
4
)
\ue8a0

(
\u03c9
)
)
,
(
31
)
F
BL

\ue8a0

(
k
x

–

k

x
T
,
k
y

+

k

y
T
,
k
z

+

k

z
T
)
=
c
2

\ue89e

H
\ue8a0

(
k
)
\ue89e

(
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
1
)
\ue8a0

(
\u03c9
)
+

\uf74e
\ue89e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
2
)
\ue8a0

(
\u03c9
)
–

\uf74e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
3
)
\ue8a0

(
\u03c9
)
+
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
4
)
\ue8a0

(
\u03c9
)
)
,
(
32
)

.
F
BL

\ue8a0

(
k
x

–

k

x
T
,
k
y

–

k

y
T
,
k
z

+

k

z
T
)
=
c
2

\ue89e

H
\ue8a0

(
k
)
\ue89e

(
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
1
)
\ue8a0

(
\u03c9
)
–

\uf74e
\ue89e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
2
)
\ue8a0

(
\u03c9
)
–

\uf74e
\ue89e
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
3
)
\ue8a0

(
\u03c9
)
–
R
~
k
x
\u2032

,

k
y
\u2032

,

k
z
\u2032
(
4
)
\ue8a0

(
\u03c9
)
)
,
(
33
)

.
52. The method of claim 2, further comprising:
using (10) and (11) to directly give a relationship between the 3D Fourier transform of measured echo signals at a transducer surface and the 3D spatial Fourier transform of the object function for a steered plane wave transmission with a fixed Axicon angle (steering angle for plane waves), \u03b6T, of X wave and azimuthal angle, \u03b8T,
obtaining the spatial Fourier transform of the object function, and
using (17) to reconstruct images with an inverse 3D Fourier transform; wherein, for steered plane waves, the relationship of the parameters between the Fourier transform of the echoes and the object function is obtained:
{
k
x
\u2032

=
k
x

+

k
\ue89e
\ue89e
sin
\ue89e
\ue89e

\u03b6
T

\ue89e
cos
\ue89e
\ue89e

\u03b8
T
k
y
\u2032

=
k
y

+

k
\ue89e
\ue89e
sin
\ue89e
\ue89e

\u03b6
T

\ue89e
sin
\ue89e
\ue89e

\u03b8
T
k
z
\u2032

=
k
z

+

k
\ue89e
\ue89e
cos
\ue89e
\ue89e

\u03b6
T
=
k
2

–

k
x
2

–

k
y
2
+

k
\ue89e
\ue89e
cos
\ue89e
\ue89e

\u03b6
T
\u2265
0
.
(
34
)
53. The method of claim 2, wherein, for a 2D image reconstruction, using a 2D imaging formula:
FBL(k\u2032x,k\u2032z)=c2H(k){tilde over (R)}k\u2032x,k\u2032z(\u03c9),\u2003\u2003(35)

where
{
k
x
\u2032

=
k
x

+

k

x
T
k
z
\u2032

=
k
z

+

k

z
T
=
k
2

–

k
x
2
+
k
2

–

k

x
T

2
\u2265
0
.
(
36
)
54. The method of claim 2, wherein step c) includes Fourier transforming along a time domain of one or more of: i) the weighted transmitted signal, or ii) the spatial Fourier transform, whereby a multi-dimensional k-space data set is formed.
55. The method of claim 2, wherein step c) includes:
i) interpolating a multi-dimensional k-space data set to produce rectilinear multi-dimensional k-space data sets; and,
ii) performing inverse Fourier transformations of the interpolated rectilinear multi-dimensional k-space data sets along each of its dimensions to produce the image data set.
56. The method of claim 55, wherein the k-space data sets have two- or three-dimensions, and the inverse Fourier transformation is performed along each of the two- or three-dimensions to produce a two- or three-dimensional image data set.
57. The method of claim 2, wherein the image data set of step c) is reconstructed using the formula kz\u2032=k+kzT.
58. The method of claim 2, wherein the image data set of step c) is reconstructed using
{
k
x
\u2032

=
k
x

+

k

x
T
k
y
\u2032

=
k
y

+

k

y
T
k
z
\u2032

=
k
z

+

k

z
T
=
k
2

–

k
x
2

–

k
y
2
+
k
2

–

k

x
T

2

–

k

y
T

2
\u2265
0
.
59. The method of claim 2, wherein the image data set of step c) is reconstructed using
{
k
x

=
k
x
\u2032

–

k

x
T
k
=
(
k
z
\u20322

+

k

x
T

2

–
(
k
x
\u2032

–

k

x
T
)

2
)

2

+

4
\ue89e
\ue89e
k
z
\u20322

\ue8a0

(
k
x
\u2032

–

k

x
T
)
2
2
\ue89e

k
z
\u2032
,
for producing a two-dimensional image.
60. The method of claim 2, wherein steps a)-c) are performed a plurality of times.
61. The method of claim 2, wherein the spatial frequency is non-uniform.
62. The method of claim 2, in which the step a) and step b) are performed using the same transducer array.
63. The method of claim 2, in which the multiple receive signals are Fourier transformed over a single transducer aperture.
64. The method of claim 2, in which the multiple receive signals are superposed coherently with corresponding or other transmission weightings or steered angles to enhance resolutions and contrast, and to reduce sidelobes.
65. The method of claim 2, in which the multiple receive signals are superposed incoherently with corresponding or other transmission weightings or steered angles to reduce speckle formation.
66. The method of claim 2, in which a single transmitter is used to produce weightings for different transducer elements.
67. The method of claim 2, in which more than one transmitter is used to produce weightings for different transducer elements.
68. The method of claim 2, in which harmonic andor elastic images are produced.
69. The method of claim 2, in which a physiological functional image is reproduced.
70. The method of claim 2, in which at least a part of the object to be imaged is moving.
71. The method of claim 2, in which the transmitted signals comprise sine and cosine spatially weighted signals.