Region-based information compaction as for digital images

Abstract

A method and apparatus for preserving the dynamic range of a relatively high dynamic range information stream, illustratively a high resolution video signal, subjected to a relatively low dynamic range encoding and/or transport process(es). A relatively high dynamic range information stream is subjected to a segmentation and remapping process whereby each segment is remapped to the relatively low dynamic range appropriate to the encoding and/or transport process(es) utilized. An auxiliary information stream includes segment and associated remapping information such that the initial, relatively high dynamic range information stream may be recovered in a post-encoding (i.e. decoding) or post-transport (i.e., receiving) process.

Description

This application is a
TP=└OP*(256/1024)+0.5┘  (eq. 2)
TP=└OP*(1024/256)+0.5┘  (eq. 3)

Using equation 2, an OP of 525 will result in a TP of 131. Using equation 3, an OP of 131 will result in a TP of 524. It can be seen that the process of linear remapping from a 10-bit dynamic range to an 8-bit dynamic range and back to the 10-bit dynamic range results in a loss of information due to quantization errors.

The above equations 1-3 mathematically illustrate the quantization error inherent in present remapping functions. By contrast, the below described remapping equations 4 and 5 are suitable for use in, respectively, the region map and scale unit 10 and inverse region map and scale unit 60 of FIG. 1.

In one embodiment of the invention a linear remapping function, such as the exemplary linear remapping function of equation 4, is utilized, where TP=Target Pixel; OP=Original Pixel; TR=Target Range; MAX=maximum parameter value and MIN=minimum parameter value. In the case of a minimum of a 10-bit system having a regional minimum of 400 and a regional maximum of 600, equation 4 becomes equation 5.
TP=└(OP−MIN)*(TR/(MAX−MIN))+0.5┘  (eq. 4)
TP=└(OP−400)*(TR/(600−400))+0.5┘  (eq. 5)

Within the context of the invention, a function such as equation 4 will be able to preserve the relatively high dynamic range of the original pixel parameter as long as the difference between the maximum and minimum parameter values does not exceed a range defined by the ratio of the original dynamic range and the target dynamic range. That is, in the case of a 10-bit original dynamic range and an 8-bit target dynamic range where the ratio is 1023:255 (i.e., 4:1), the difference between the maximum and minimum values must not be greater than one fourth of the original dynamic range. Thus, a threshold level of dynamic range for each region is established that determines if the full, original dynamic range of the parameter will be preserved by the invention. Since, in equation 5, the difference between the maximum (600) and minimum (400) is less than one fourth of the 10-bit dynamic range (256), full 10-bit dynamic range will be preserved.

It must be noted that equations 4 and 5 should not in any way be construed as limiting the scope of the invention. Rather, equations 4 and 5 are presented as only one of a plurality of linear functions suitable for use in the invention. The invention may also be practiced using non-linear functions (such as gamma correction and companding functions). Moreover, the invention may be practiced using a combination of linear and non-linear functions to optimize data compaction. The linear and/or non-linear functions selected will vary depending on the type of information stream being processed, the typical distribution of parameters of interest within the information elements of that stream, the amount of dynamic range allowed for a given application, the processing constraints of the encoder and/or decoder operating on the information streams and other criteria.

To help ensure that the difference between the maximum and minimum values remains below the threshold level, it is desirable to reduce the size of the regions. However, a reduction in region size necessarily results in additional maximum and minimum information that must be identified and processed, though this overhead may not be significant as will now be demonstrated.

The above-described method advantageously provides substantially full dynamic range preservation of selected information element parameters in an information frame. The cost, in terms of extra bits necessary to implement the invention, e.g., the overhead due to the use of minimum and maximum pixel values for each region of a picture, will now be briefly discussed. Specifically, the additional number of bits to be transported by, e.g., the communications network 35 of FIG. 1 will be discussed.

Consider the case of preserving the 10-bit dynamic range of the luminance parameter of a video information stream processed according to an 8-bit dynamic range process. Assume that a small region size is selected, such as a 16×16 block of 8-bit pixels (monochrome). The 16×16 block of 8-bit pixels is represented by 256*8 bits=2048 bits. Adding two 10-bit values, a minimum and a maximum, to this block increases the number of bits by 20 to 2068 bits, or an increase of about 1%. In return for this, the pixel intensity resolution is never worse than 8 bits, and may be as high as 10 bits, a factor of four improvement in the intensity depth resolution.

Consider the case of a 10-bit digital video stream according to the well known 4:4:4 format. In this case the luminance (Y) and color difference (U, V) signals each have 10-bit dynamic range. Again, assuming that a small region size is selected, such as a 16×16 block of 8-bit pixels. The 8-bit pixels are represented by 256*8*3 bits=6144 bits. In this case also, adding six 10-bit values, a minimum and a maximum for each of the luminance (Y) and color difference (U, V) signals, to this block increases the number of bits by 60 to 6204 bits, or an increase of about 1%. In return for this, each of the luminance (Y) and color difference (U, V) signals are never worse than 8 bits, and may be as high as 10 bits, a factor of four improvement in the respective intensity and color depth resolutions.

Returning now to the first case, if all the pixels were to be represented by 10 bits, then the total number of bits would be 256*10=2560 bits. In other words, full 10-bit representation would require 24% more bits than the regional coding described here. Thus, the method provides a substantial improvement in dynamic range without a correspondingly substantial increase in bit count. Moreover, by utilizing the method within the context of mass-produced encoder/decoder chipsets, such as the various implementations of the MPEG and MPEG-like compression standards, JPEG and JPEG-like compression standards (and other known techniques) the method leverages the cost-savings of existing 8-bit chipsets to provide a 10-bit (or higher) effective dynamic range.

The above-described embodiments of the invention achieve the desired result using linear compaction methods. However, in some applications it is desirable to process information using non-linear methods. For example, analog video signals are non-linearly processed (i.e., “gamma corrected”) to compensate for non-linearity in, e.g., picture tubes in television sets. Non-linear mapping methods according to the invention may be used to implement gamma correction and other functions while preserving the dynamic range of the underlying signal. Moreover, linear and non-linear methods may be used together.

Another scenario appropriate for non-linear processing in the mapping function occurs when there is a loss of accuracy because the original range and the target range are too far apart, even with the above-described intensity compaction methods. In this case, non-linear mapping is used to preserve the original pixel values (i.e., dynamic range) over some part of the range. This situation is depicted below with respect to FIGS. 4A and 4B, where the information located within a lower bit range (e.g., 0-131) is illustratively deemed to be more important than the information located within an upper bit range (e.g., 132-1023).

FIG. 4A depicts a diagram 4 illustrative of a non-linear encoding function. The diagram comprises an original dynamic range 410A of 1024 bits and a target dynamic range 420A of 255 bits. A signal 430A, 440A having a 1024 bit dynamic range is remapped into the 255 bit dynamic range space in two segments. The first segment 430A utilizes a substantially linear transfer function, while the second segment 440A utilizes a compressed transfer function. That is, the range of 0-131 in the original map is retained in the target map, while the range of 132 to 1023 in the original map is compressed into the 132-255 range of the target map.

FIG. 4B depicts a diagram illustrative of a non-linear decoding function associated with the encoding function of FIG. 4A. Thus, to retrieve, at a decoder, the information signal encoded according to a remapping function having the transfer function depicted in FIG. 4A, the decoder implements a remapping function having the transfer function depicted in FIG. 4B.

FIG. 5 depicts a high level function block diagram of an encoding and decoding method and apparatus according to the invention. Specifically, the encoding and decoding method and process comprises a function mapper 530, which is responsive to an information stream S1 received from, illustratively, a pixel source 510. The function mapper remaps the information stream S1 according to various function criteria f_c provided by a function criteria source 520 to produce a remapped information stream S3 and an associated map information stream S4.

The remapped information stream S3 is coupled to an encoder 540 that encodes the remapped information stream S3 to produce an encoded information stream S5. The encoded information stream S5 and the map information stream S4 are transported to, respectively, a decoder 550 and an inverse function mapper 560.

The decoder 550 decodes the transported and encoded information stream to retrieve an information stream substantially corresponding to the initial remapped information stream.

The inverse function mapper 560 performs, in accordance with the transported map information stream S4, an inverse function mapping operation on the retrieved stream to produce an information stream substantially corresponding to the original information stream. It must be noted that the information stream produced by the inverse function mapper 560 may advantageously include linear and/or non-linear modifications in furtherance of the specific application (e.g., gamma correction and the like).

It should be noted that the function mapper 530 and inverse function mapper 560 may be operated in substantially the same manner as the region map and scale unit 10 and inverse region map and scale unit 60 depicted in FIG. 1.

In one embodiment of the invention, the remapping function performed by, e.g., the function mapper 530 or region map and scale unit 10 performs a remapping function according to an arbitrary function. An arbitrary function remapping of, e.g., pixel luminance or chrominance parameters from an original dynamic range to a target dynamic range may be represented by equation 6, where TP=Target Pixel; OP=Original Pixel; TR=Target Range; OR=original Range; MAX=maximum value; MIN=minimum value; and F=the arbitrary function.
TP=F(OP,MAX,MIN,TR)  (eq. 6)

It is important to note that the function F may take a number of forms and be implemented in a number of ways. For example, the function F may implement: 1) a simple linear function such as described above with respect to FIGS. 1-2; 2) a gamma correction function that varies input video intensity levels such that they correspond to intensity response levels of a display device; 3) an arbitrary polynomial; or 4) a tabulated function (i.e., a function purely described in terms of a lookup table, where each input bit addresses a table to retrieve the contents stored therein.

In the case of remapping using a fixed gamma correction function, a function of the following form may be implemented:
TP=└F[(OP−MIN)^γ*TR/(MAX−MIN)^γ]+0.5┘  (eq. 7)

In the case of remapping using a polynomial segment, illustratively a parabola (X²+X), a function of the following form may be implemented, assuming that the polynomial segment is never be less than 0 nor greater than the target range:
TP=└[(OP−MIN)²+(OP−MIN)*TR/[(MAX−MIN)²+(MAX−MIN)]+0.5┘  (eq. 8)

In the case of remapping using a tabulated function, the table comprises an indexable array of values, where the index values are the original range and the values in the table are included in the target range. This allows any arbitrary mapping between the two ranges. Unless, like gamma correction, that mapping is one-way only (i.e., the remapping is not intended to be “unmapped”), then there an inverse table at the decoder 550 or inverse map and scale unit 60 will restore the original information values.

It should be noted that the terms dynamic range enhancement stream and map region identification stream are used in substantially the same manner to describe information streams carrying auxiliary or other data suitable for use in recovering at least a portion of the dynamic range of an information stream processed according to the invention.

Although various embodiments that incorporate the teachings of the present invention have been shown and described in detail herein, those skilled in the art can readily devise many other varied embodiments that still incorporate these teachings.

Claims

1. A method for encoding an information frame, comprising the steps of:

dividing said information frame into a plurality of information regions, at least one of said information regions comprising at least one information parameter having associated with it a plurality of intra-region values bounded by upper and lower value limits defining a dynamic range of said information parameter;

determining, for each of said at least one of said information regions, a respective maximal value and a minimal value of said at least one information parameter;

remapping, for each of said at least one of said information regions and according to a single manipulation of the respective determined maximal and minimal values, at least one respective of said plurality of intra-region values of said at least one information parameter; and

encoding each of said information regions;

wherein said information frame comprises an image frame and said at least one information parameter comprises at least one of a luminance parameter and a chrominance parameter;

wherein said steps of encoding and determining produce, respectively, an encoded image stream and an associated dynamic range enhancement stream; and

wherein said step of remapping is performed in accordance with the following linear equation:

TP=[OP*(TR/OR)+0.5]

where: TP=Target Pixel; OP=Original Pixel; TR=Target range; and OR=Original range.

2. A method for encoding an information frame, comprising the steps of:

dividing said information frame into a plurality of information regions, at least one of said information regions comprising at least one information parameter having associated with it a plurality of intra-region values bounded by upper and lower value limits defining a dynamic range of said information parameter;

determining, for each of said at least one of said information regions, a respective maximal value and a minimal value of said at least one information parameter;

remapping, for each of said at least one of said information regions and according to a single manipulation of the respective determined maximal and minimal values, at least one respective of said plurality of intra-region values of said at least one information parameter; and

encoding each of said information regions; wherein said plurality of information frames comprise image frames, said at least one information parameter comprises at least one of a luminance parameter and a chrominance parameter;

wherein said steps of encoding and determining produce, respectively, an encoded image stream and an associated dynamic range enhancement stream;

wherein said steps of dividing, determining, remapping and encoding are repeated for each of a plurality of information frames; and

wherein said step of remapping is performed in accordance with the following linear equation:

TP=[OP*(TR/OR)+0.5]

where: TP=Target Pixel; OP=Original Pixel; TR=Target range; and OR=Original range.

3. A method for decoding an encoded information frame represented by a plurality of encoded information regions within an encoded information stream, where at least one of said plurality of encoded information regions comprises at least one information parameter having associated with it a plurality of remapped intra-region values, said method comprising the steps of:

decoding each of said plurality of encoded information regions to form a corresponding plurality of decoded information regions, said decoded information regions representing a decoded information frame;

extracting, from a dynamic range enhancement stream associated with said encoded information stream, respective maximal and minimal values for each of said at least one information parameter having associated with it a plurality of remapped intra-region values;

inverse remapping, according to a single manipulation of said respective maximal and minimal values, each of said at least one information parameter of said at least one information regions having associated with it a respective plurality of remapped intra-region values; and

demultiplexing a transport stream to recover said encoded information stream and said dynamic range enhancement stream; wherein said encoded information stream and said dynamic range enhancement stream associated with said encoded information stream comprises respective portions of said transport stream;

wherein said step of demultiplexing comprises retrieving, from a private data section of said transport stream, said dynamic range enhancement stream;

wherein said decoded information frame comprises an image frame and said at least one information parameter comprises at least one of a luminance parameter and a chrominance parameter; and

wherein said step of inverse remapping is performed in accordance with the following linear equation:

TP=[OP*(TR/OR)+0.5]

where: TP=Target Pixel; OP=Original Pixel; TR=Target range; and OR=Original range.

4. A method for decoding an encoded information frame represented by a plurality of encoded information regions within an encoded information stream, where at least one of said plurality of encoded information regions comprises at least one information parameter having associated with it a plurality of remapped intra-region values, said method comprising the steps of:

decoding each of said plurality of encoded information regions to form a corresponding plurality of decoded information regions, said decoded information regions representing a decoded information frame;

extracting, from a dynamic range enhancement stream associated with said encoded information stream, respective maximal and minimal values for each of said at least one information parameter having associated with it a plurality of remapped intra-region values; and

inverse remapping, according to a single manipulation of said respective maximal and minimal values, each of said at least one information parameter of said at least one information regions having associated with it a respective plurality of remapped intra-region values;

wherein said steps of decoding, extracting and inverse remapping are repeated for each of a plurality of information frames within said encoded information stream and said dynamic range enhancement stream associated with said encoded information stream;

wherein said decoding step comprises compression decoding said encoded information frame of said information stream; and

wherein said step of inverse remapping is performed in accordance with the following linear equation:

TP=[OP*(TR/OR)+0.5]

where: TP=Target Pixel; OP=Original Pixel; TR=Target range; and OR=Original range.

5. An apparatus for decoding an encoded information frame represented by a plurality of encoded information regions within an encoded information stream, where at least one of said plurality of encoded information regions comprises at least one information parameter having associated with it a plurality of remapped intra-region values, said apparatus comprising:

a decoder, for decoding each of said plurality of encoded information regions to form a corresponding plurality of decoded information regions, said decoded information regions representing a decoded information frame; and

an inverse map and scale unit, for extracting, from a dynamic range enhancement stream associated with said encoded information stream, respective maximal and minimal values for each of said at least one information parameter having associated with it a plurality of remapped intra-region values, and for inverse remapping, according to a single manipulation of said respective maximal and minimal values, each of said at least one information parameter of said at least one of said plurality of encoded information regions having associated with it said plurality of remapped intra-region values;

wherein said decoded information frame comprises an image frame and said at least one information parameter comprises at least one of a luminance parameter and a chrominance parameter; and

wherein said inverse map and scale unit extracts said respective maximal and minimal values in accordance with the following linear equation:

TP=[OP*(TR/OR)+0.5]

where: TP=Target Pixel; OP=Original Pixel; TR=Target range; and OR=Original range.

6. An apparatus for decoding an encoded information frame represented by a plurality of encoded information regions within an encoded information stream, where at least one of said plurality of encoded information regions comprises at least one information parameter having associated with it a plurality of remapped intra-region values, said apparatus comprising:

a decoder, for decoding each of said plurality of encoded information regions to form a corresponding plurality of decoded information regions, said decoded information regions representing a decoded information frame; and

an inverse map and scale unit, for extracting, from a dynamic range enhancement stream associated with said encoded information stream, respective maximal and minimal values for each of said at least one information parameter having associated with it a plurality of remapped intra-region values, and for inverse remapping, according to a single manipulation of said respective maximal and minimal values, each of said at least one information parameter of said at least one of said plurality of encoded information regions having associated with it said plurality of remapped intra-region values;

wherein said encoded information stream and said dynamic range enhancement stream associated with said encoded information stream represent a plurality of encoded information frames;

wherein said apparatus processes each of said plurality of encoded information frames to produce a corresponding plurality of decoded information frames;

wherein said plurality of decoded information frames comprise image frames and said at least one information parameter comprises at least one of a luminance parameter and a chrominance parameter; and

wherein said inverse map and scale unit extracts said respective maximal and minimal values in accordance with the following linear equation:

TP=[OP*(TR/OR)+0.5]

where: TP=Target Pixel; OP=Original Pixel; TR=Target range; and OR=Original range.

7. A method for remapping a value of a parameter of an information element from a first representation system to a second representation system, comprising:

dividing a collection of information elements into regions;

determining a region of the regions, wherein the information element is within the region;

determining, among the information elements within the region, a maximum value of the parameter;

determining, among the information elements within the region, a minimum value of the parameter;

subtracting the minimum value from the value of the parameter of the information element to yield a first difference;

subtracting the minimum value from the maximum value to yield a second difference;

dividing a dynamic range of values of the parameter associated with the second representation system by the second difference to yield a quotient;

multiplying the quotient by the first difference to yield a product;

adding the product to one half to yield a sum; and

rounding the sum to a nearest integer that is less than or equal to the sum to yield a remapped value of the parameter of the information element.

8. The method of claim 7, wherein the second difference is raised to a power before the dividing the dynamic range of values of the parameter associated with the second representation system by the second difference, the first difference is raised to the power before the multiplying the quotient by the first difference, a function is determined before the adding the product to one half to yield the sum, and the product is a variable of the function.

9. The method of claim 7, wherein the information element is a pixel.

10. The method of claim 9, wherein the collection of information elements is an image frame.

11. The method of claim 7, further comprising performing, for each remaining information element within the region, the subtracting the minimum value from the value of the parameter of the information element, the subtracting the minimum value from the maximum value, the dividing the dynamic range of values of the parameter associated with the second representation system by the second difference, the multiplying the quotient by the first difference, the adding, and the rounding.

12. The method of claim 11, further comprising performing, for each remaining region of the regions, the determining, among the information elements within the region, the maximum value of the parameter and the determining, among the information elements within the region, the minimum value of the parameter to yield a collection of maximum values and a collection of minimum values.

13. The method of claim 12, further comprising performing, for each information element within the each remaining region, the subtracting the minimum value from the value of the parameter of the information element, the subtracting the minimum value from the maximum value, the dividing the dynamic range of values of the parameter associated with the second representation system by the second difference, the multiplying the quotient by the first difference, the adding, and the rounding.

14. The method of claim 13, further comprising encoding the regions to yield encoded regions.

15. The method of claim 14, further comprising transporting the encoded regions, the collection of maximum values, and the collection of minimum values.

16. The method of claim 13, further comprising decoding the regions.

17. A method for remapping a value of a parameter of an information element from a first representation system to a second representation system, comprising:

dividing a dynamic range of values of the parameter associated with the second representation system by a dynamic range of values of the parameter associated with the first representation system to yield a quotient;

multiplying the quotient by the value of the parameter of the information element to yield a product;

adding the product to one half to yield a sum; and

rounding the sum to a nearest integer that is less than or equal to the sum to yield a remapped value of the parameter of the information element.

18. An apparatus for remapping a value of a parameter of an information element from a first representation system to a second representation system, comprising:

a receiver configured to receive a collection of information elements; and

a processor configured to divide the collection of information elements into regions, to determine a region of the regions, wherein the information element is within the region, to determine, among the information elements within the region, a maximum value of the parameter, to determine, among the information elements within the region, a minimum value of the parameter, to subtract the minimum value from the value of the parameter of the information element to yield a first difference, to subtract the minimum value from the maximum value to yield a second difference, to divide a dynamic range of values of the parameter associated with the second representation system by the second difference to yield a quotient, to multiply the quotient by the first difference to yield a product, to add the product to one half to yield a sum, and to round the sum to a nearest integer that is less than or equal to the sum to yield a remapped value of the parameter of the information element.

19. The apparatus of claim 18, wherein the processor is further configured to raise the second difference to a power before dividing the dynamic range of values of the parameter associated with the second representation system by the second difference, to raise the first difference to the power before multiplying the quotient by the first difference, and to determine a function before adding the product to one half to yield the sum, wherein the product is a variable of the function.

20. The apparatus of claim 18, further comprising an encoder configured to encode the regions to yield encoded regions.

21. The apparatus of claim 20, further comprising a transporter configured to transport the encoded regions, a collection of maximum values, and a collection of minimum values.

22. The apparatus of claim 18, further comprising a decoder configured to decode the regions.

23. An apparatus for remapping a value of a parameter of an information element from a first representation system to a second representation system, comprising a processor configured to divide a dynamic range of values of the parameter associated with the second representation system by a dynamic range of values of the parameter associated with the first representation system to yield a quotient, to multiply the quotient by the value of the parameter of the information element to yield a product, to add the product to one half to yield a sum, and to round the sum to a nearest integer that is less than or equal to the sum to yield a remapped value of the parameter of the information element.

24. In a system for encoding an information stream, a method comprising:

dividing the information stream into a plurality of information regions;

identifying, for at least one of the plurality of information regions, a maximal value and a minimal value of at least one information element associated with the least one of the plurality of information regions;

mapping the at least one information element according to the identified maximal and minimal values associated with the respective information region, wherein said mapping comprises determining a target information element as a function of an original information element, a target range, and said maximal and minimal values, wherein the difference between the maximal and minimal values is used to map the original information element to the target range to generate a mapped information element;

encoding the mapped information element to produce an encoded information stream; and

associating said identified maximal and minimal values with the at least one of the plurality of information regions to produce a map identification stream, wherein said map identification stream includes information sufficient to substantially recover said identified maximal and minimal values associated with said mapped at least one information element.

25. The method of claim 24, wherein said mapping further comprises using a combination of linear and non-linear functions.

26. The method of claim 24, wherein said mapping further comprises using a polynomial segment.

27. The method of claim 24, wherein said mapping further comprises using a tabulated function comprising an indexable array of values.

28. A method for encoding an information stream, said method comprising:

receiving a plurality of information frames, each of said plurality of information frames comprising a plurality of information elements, each of said plurality of information elements associated with at least one information element parameter, each of the at least one information element parameters having a value between a lower limit and an upper limit, said upper and lower limits defining a dynamic range of said at least one information element parameter;

defining, for each of the plurality of information frames, a plurality of information regions, each of the plurality of information regions being associated with one or more respective information elements;

identifying, for each of the plurality of information regions, a maximal value and a minimal value of at least one information element parameter associated with said plurality of information elements of said region;

remapping, for each of said plurality of information regions, the at least one information element parameter of each of said plurality of information elements of said region according to the identified maximal and minimal value parameters associated with the respective region, wherein the difference between the maximal and minimal value parameters is used to remap the at least one of said plurality of information element parameters to a target range;

encoding each remapped information region of said plurality of information regions to produce an encoded information stream; and

associating said identified maximal and minimal values with each remapped information region of said plurality of information regions to produce a map identification stream, wherein said map identification stream includes information sufficient to substantially recover said identified maximal and minimal value parameters associated with said remapped at least one information element parameter for each of said plurality of information regions.

29. A method for encoding an information stream comprising a plurality of information frames, said method comprising:

receiving the plurality of information frames;

for at least one of the plurality of information frames, dividing the at least one of the plurality of information frames into regions according to at least one criterion for rescaling a range of one or more parameters of interest for at least one of the regions as compared to a range of the one or more parameters of interest of the at least one of the plurality of information frames;

identifying a target range for said one or more parameters of interest, the target range being different from the range of the one or more parameters of interest; and

remapping, for at least one of said regions, the one or more parameters of interest to the identified target range, the remapped parameters being bounded by the target range.

30. The method of claim 29, further comprising encoding the remapped regions to produced a compressed information stream.

31. The method of claim 30, further comprising multiplexing the encoded remapped regions with information for recovering the regions.