Abstract:
First image data at a lower sampling frequency is up-sampled in a sampling ratio N:M to a higher sampling frequency in an up-sampling filter; and, second image data at the said higher sampling frequency is down-sampled in a sampling ratio M:N to the said lower sampling frequency in a down-sampling filter where the combination of the up-sampling filter and the down-sampling filter is substantially transparent and every filtered sample is formed from a weighted sum of at least two input samples.

Description:
FIELD OF INVENTION 
       [0001]    This invention concerns re-sampling of images. 
       BACKGROUND OF THE INVENTION 
       [0002]    The increasing diversity of image display formats has resulted in considerable interest in converting images from one sampling structure to another. In particular the increased use of “high-definition” (HD) television formats (i.e. formats where the vertical spatial sampling frequency is of the order of 1,000 samples per picture height) has led to the development of production equipment which can operate on both HD and standard definition (SD) formats. In some equipment, processing is carried out at one particular high-resolution sampling structure, and inputs can be received, and outputs can be delivered, either with the structure used for processing, or at a different sampling structure. 
         [0003]    In such equipment images may need to be converted to a higher or lower rate before or after processing. It is now common for two, complementary conversions to be needed: one from the input sampling structure to the processing structure; and another, from the processing structure back to the original input structure. It is highly desirable for these conversions to be transparent to the user, and, provided the processing structure has higher sampling frequencies in all dimensions, the cascaded up- and down-sampling can be made mathematically transparent. 
         [0004]    Typically, input sample values from adjacent sample locations are combined as a weighted sum in a filter aperture so as to derive new sample values at locations other than the input sample locations. The filter may have several “phases” so that different weightings are used to obtain new values for sample sites having different spatial relationships to the input sample locations. Often the filter is one-dimensional so that the filter aperture comprises a set of horizontally contiguous samples or a set of vertically contiguous samples. However, two, and three-dimensional filters are also known, in which the filter aperture comprises a set of samples in a two-dimensional image region, or a set of two dimensional regions from images sampled at different times. 
         [0005]    In multi-standard video processing systems it is desirable that complementary up- and down-conversion filters should be available with similar levels of complexity—i.e. aperture size and coefficient magnitudes. Symmetrical filters are easier to implement. Both types of filter should have high stop-band attenuation, so as to reduce aliasing; and, these features should be combined with the property of reversibility. Such filters are required for conversion between the commonly-used horizontal sample structures of 720, 960 and 1280 samples per line; and vertical sampling structure of 480, 486, 576, 720 and 1080 lines per picture. In these conversions the relative phasing of the input and output sampling grid may be required to be offset so as to avoid a shift in the position of the image centre when converting. 
         [0006]    U.S. Pat. No. 6,760,379 (Werner)—which is hereby incorporated by reference—describes how, given a linear up-conversion filter, a complementary down-conversion filter can be derived, so that the cascading of the two complementary processes is mathematically transparent. It is helpful to review the example filters shown in this prior patent so as to clarify the improvements provided by the current invention. 
         [0007]      FIG. 1  illustrates up-conversion by a factor of 3:4 followed by down-conversion by 4:3 according to a mutually reversible pair of filters. The set of low-resolution samples L 0  to L 4  is up-converted to the (more-numerous) set of high-resolution samples H 0  to H 5 , which are then down-converted to the low-resolution samples L′ 0  to L′ 4 . The up-conversion filter is a bilinear interpolator and the down-conversion filter is the filter ‘undo-bil’ described in the Werner patent. 
         [0008]    In  FIG. 1  the filter coefficients are shown by arrows. Following the notation of Werner, up-conversion filter coefficients are denoted by g(x) and down-conversion filter coefficients are denoted by h(x), where x is the phase offset between the input and output samples expressed in units of the pitch of an intermediate sampling grid which includes both sets of samples. In the Figure each down-conversion filter coefficient is only shown once, but up-conversion filter coefficients are shown for all samples which contribute to any samples contributing to the shown down-conversion coefficients. 
         [0009]    The coefficients of the  FIG. 1  filters are shown in  FIG. 2  in the form of filter aperture functions. The filter aperture ( 20 ) corresponds to the bilinear up-conversion filter g(x), and the filter aperture ( 21 ) corresponds to the filter h(x), which is the filter ‘undo-bil’ described in the Werner patent. 
         [0010]    As can be seen from  FIGS. 1 and 2 , the down-conversion filter ‘undo-bil’ is asymmetric. The prior patent describes how such an asymmetric filter can be converted to a symmetric filter by averaging the filter aperture function with a mirror-image-reversed version of the aperture function, without losing the property of reversibility.  FIG. 3  shows the same up- and down-conversion process as  FIG. 1 , but using a symmetric down-conversion filter derived from the filter ‘undo-bil’. (Note that in  FIG. 3  zero filter coefficients are not shown.) The corresponding down-conversion filter aperture is the function ( 41 ) shown in  FIG. 4 . 
         [0011]    It can be verified from the coefficient values shown in  FIG. 3  that the set of down-converted samples L′ o  to L′ 2  are identical to the original samples L o  to L 2 , i.e. the up-conversion has been reversed transparently: 
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         [0012]    Similarly, the contributions to L′ 2  are a mirror image of the contributions to L′ 1 , and so L′ 2  is equal to L 2 . 
         [0013]    And, all other down-converted samples are derived by contributions analogous to those used to derive L′ 0 , L′ 1  and L′ 2 . 
         [0014]    The Werner patent describes a second pair of complementary up- and down conversion filters, referred to as ‘def’ and ‘undo-def’, where the up-conversion has an improved frequency response, as compared to bilinear up-conversion. The aperture functions of these prior art filters are shown in  FIG. 5 , in which the up-conversion aperture is designated ( 50 ), and the down-conversion aperture is designated ( 51 ). 
         [0015]    Note that the Figure only shows the central portion of the down-conversion filter aperture ( 51 ) which contains the largest coefficients; the full aperture extends over x values in the range −8 to +20. The coefficients of the full aperture are given in Table 4, which also includes all the down-conversion apertures described in this specification. (Table 3 lists the coefficients of all the up-conversion apertures described in this specification.) The down-conversion filter aperture ( 51 ) can be made symmetrical by the same method as described above, and the resulting filter aperture is shown as the aperture ( 61 ) in  FIG. 6 . Again, only the central portion of the aperture is shown in the figure, the full aperture extends from −20 to +20. It can be seen that the complementary down-conversion filter apertures ( 51 ) and ( 61 ) are much wider than the corresponding up-conversion filter aperture ( 50 ) (also shown as ( 60 )). 
         [0016]    It is helpful to examine the frequency responses of these prior-art filters, and they are shown in  FIGS. 7 to 10 . The responses have been calculated by converting the relevant filter apertures, which, of course, are the respective filter impulse responses, from the time domain to the frequency domain in the well-known manner. 
         [0017]    The frequency scales of  FIGS. 7 to 10  are in units of 12 times the frequency of the notional sampling grid that contains both the lower and higher sampling frequencies; i.e. the lower sampling frequency (f low ) corresponds to three units, and the higher sampling frequency (f high ) corresponds to four units. The Figures show the magnitudes of the responses; however, for the symmetrical filters, for which the response is always real (i.e. its imaginary component is zero), phase-inverted responses are shown as having negative amplitude. 
         [0018]      FIG. 7  shows the response ( 71 ) of the ‘undo-bil’ down-conversion filter of the Werner patent; and, the response ( 70 ) of a bilinear up-conversion filter. 
         [0019]      FIG. 8  shows the response ( 81 ) of a symmetric filter derived from the ‘undo-bil’ down-conversion filter by the method described in the Werner patent; and, the response ( 80 ) of a bilinear up-conversion filter (identical to the response ( 70 )). 
         [0020]      FIG. 9  shows the response ( 91 ) of the ‘undo-def’ down-conversion filter of the Werner patent; and, the response ( 90 ) of the ‘def’ up-conversion filter also described in the Werner patent. 
         [0021]      FIG. 10  shows the response ( 101 ) of a symmetric filter derived from the ‘undo-def’ down-conversion filter by the method described in the Werner patent; and, the response ( 100 ) of the ‘def’ up-conversion filter (identical to the response ( 90 )). 
         [0022]    It can be seen that the frequency responses of these prior art down-conversion filters are less than optimum. They all have poor stop-band attenuation (above 1.5 frequency units); in particular there is significant response at the lower sampling frequency f low  (3 frequency units). Any signal energy at this frequency would be aliased to DC. This alias is likely to prove particularly troublesome when down-converting material which has not been previously up-converted. It can also be seen that none of the filter pass-bands are particularly flat. 
         [0023]    The up- and down-conversion filters shown in  FIGS. 1 to 6  are what the Werner patent describes as “Nyquist” filters for which a regular sub-set of the output samples are equal to input samples. The samples in this sub-set are formed by copying the values of respective, spatially-aligned filter input samples. For example, in  FIG. 1  the up-conversion filter g(x) and the down-conversion filter h(x) both give outputs copied from their respective inputs when the input to output phase offset x is zero. 
       SUMMARY OF THE INVENTION 
       [0024]    The invention consists in one aspect in a method and apparatus for digital image processing in which first image data at a lower sampling frequency is up-sampled to a higher sampling frequency in an up-sampling filter; and, second image data at the said higher sampling frequency is down-sampled to the said lower sampling frequency in a down-sampling filter where the combination of the up-sampling filter and the down-sampling filter is substantially transparent, characterised in that every filtered sample is formed from a weighted sum of at least two input samples. 
         [0025]    Suitably, at least part the said first and second image data represent the same portrayed object. 
         [0026]    Advantageously the amplitude of the response of the said up-conversion filter and the amplitude of the response said down-conversion filter have substantially equal magnitudes at a frequency of half the said lower sampling frequency. 
         [0027]    In a preferred embodiment the frequency response of the said up-conversion filter and the frequency response said down-conversion filter are substantially identical. 
         [0028]    And, the reconstruction error due to the combination of the up-sampling filter and the down-sampling filter is smaller than one least-significant bit of the said digital image processing. 
         [0029]    The invention consists in another aspect in a method of digital image processing in which first image data at a lower sampling frequency is up-sampled in a sampling ratio N:M to a higher sampling frequency in an up-sampling filter; and, second image data at the said higher sampling frequency is down-sampled in a sampling ratio M:N to the said lower sampling frequency in a down-sampling filter where the combination of the up-sampling filter and the down-sampling filter is substantially transparent characterised in that every filtered sample is formed from a weighted sum of at least two input samples, where N and M are integers, where 1&lt;N&lt;M and where M is not a multiple of N. 
         [0030]    Preferably, which the combination of the up-sampling filter and the down-sampling filter is substantially transparent in the sense that the reconstruction error due to the combination of the up-sampling filter and the down-sampling filter is smaller than one least-significant bit of the said digital image processing. 
         [0031]    The invention consists in another aspect in a method of digital image processing for achieving a transparent cascade on up conversion in the sampling rate ratio N:M where N and M are integers and subsequent M:N down conversion, where the up and down conversion ratios N:M and M:N respectively are rational numbers and the integers N and M satisfy the condition 1&lt;N&lt;M, wherein the up conversion filter operates on a sampled signal S input  and is chosen to take the form S up (n)=ΣS input (k)·g(Nn−Mk) where k is the running integer over which the sum is taken and wherein a corresponding down conversion filter operates on the up converted signal S up  and is chosen to take the form S down (n)=ΣS up (k)·h(Mn−Nk); the pair (g, h) of up and down conversion filters being chosen so that Σh(Mn−Nk)·g(Nk−Mm) is equal to unity if n=m and is otherwise equal to zero, and wherein every filtered sample is formed from a weighted sum of at least two input samples. 
         [0032]    The invention consists in another aspect in a method of digital image processing for achieving a transparent cascade on up conversion in the sampling rate ratio N:M where N and M are integers and subsequent M:N down conversion, where the up and down conversion ratios N:M and M:N respectively are rational numbers and the integers N and M satisfy the condition 1&lt;N&lt;M, wherein the up conversion filter operates on a sampled signal S input  and is chosen to take the form S up (n)=ΣS input (k)·g(Nn−Mk) where k is the running integer over which the sum is taken and wherein a corresponding down conversion filter operates on the up converted signal S up  and is chosen to take the form S down (n)=ΣS up (k)·h(Mn−Nk); the pair (g, h) of up and down conversion filters being chosen so that Σh(Mn−Nk)·g(Nk−Mm) is equal to unity if n=m and is otherwise equal to zero, and wherein the frequency response of the up conversion filter is substantially the same as the frequency response of the down conversion filter. 
         [0033]    The invention consists in another aspect in a method of digital image processing for achieving a transparent cascade on up conversion in the sampling rate ratio N:M where N and M are integers and subsequent M:N down conversion, where the up and down conversion ratios N:M and M:N respectively are rational numbers and the integers N and M satisfy the condition 1&lt;N&lt;M, wherein the up conversion filter operates on a sampled signal S input  and is chosen to take the form S up (n)=ΣS input (k)·g(Nn−Mk) where k is the running integer over which the sum is taken and wherein a corresponding down conversion filter operates on the up converted signal S up  and is chosen to take the form S down (n)=ΣS up (k)·h(Mn−Nk); the pair (g, h) of up and down conversion filters being chosen so that Σh(Mn−Nk)·g(Nk-Mm) is equal to unity if n=m and is otherwise equal to zero, and wherein the frequency response of the down converter at the sampling frequency of the sampled signal S input  is less than 10%, preferably less than 5% and more preferably less than 1% of the frequency response at DC. 
         [0034]    The invention consists in another aspect in method of digital image processing in which first image data at a lower sampling frequency is up-sampled in a sampling ratio N:M to a higher sampling frequency in an up-sampling filter; and, second image data at the said higher sampling frequency is down-sampled in a sampling ratio M:N to the said lower sampling frequency in a down-sampling filter where the combination of the up-sampling filter and the down-sampling filter, where N and M are integers, where 1&lt;N&lt;M and where M is not a multiple of N, wherein a frequency response is optimised to minimise the reconstruction error due to the combination of the up-sampling filter and the down-sampling filter and wherein the filter aperture of the up-sampling filter and the filter aperture of the down-sampling filter are each constructed from said optimised frequency response. 
         [0035]    Preferably, in said optimised frequency response the response at the lower sampling frequency is less than 10%, preferably less than 5% and more preferably less than 1% of the frequency response at DC. 
         [0036]    Suitably, the reconstruction error is minimised in the sense of being smaller than one least-significant bit of the said digital image processing. 
         [0037]    The invention consists in another aspect in a method of digital image processing in which first image data at a lower sampling frequency is up-sampled in a sampling ratio N:M to a higher sampling frequency in an up-sampling filter; and, second image data at the said higher sampling frequency is down-sampled in a sampling ratio M:N to the said lower sampling frequency in a down-sampling filter where the combination of the up-sampling filter and the down-sampling filter is substantially transparent characterised, where N and M are integers, where 1&lt;N&lt;M and where M is not a multiple of N and wherein the frequency response of the down converter at the lower sampling frequency is less than 10%, preferably less than 5% and more preferably less than 1% of the frequency response at DC. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0038]      FIG. 1  shows a representation of a prior art transparent up- and down-conversion process using an asymmetric down-conversion filter. 
           [0039]      FIG. 2  shows the filter aperture functions of the prior art process of  FIG. 1 . 
           [0040]      FIG. 3  shows a representation of a prior art transparent up- and down-conversion process using a symmetrical down-conversion filter. 
           [0041]      FIG. 4  shows the filter aperture functions of the prior art process of  FIG. 3 . 
           [0042]      FIG. 5  shows the filter aperture functions of an alternative prior-art transparent up- and down-conversion using an asymmetric down-conversion filter. 
           [0043]      FIG. 6  shows the filter aperture functions of an alternative prior-art transparent up- and down-conversion using a symmetrical down-conversion filter. 
           [0044]      FIG. 7  shows the frequency responses of the  FIG. 2  filter aperture functions. 
           [0045]      FIG. 8  shows the frequency responses of the  FIG. 4  filter aperture functions. 
           [0046]      FIG. 9  shows the frequency responses of the  FIG. 5  filter aperture functions. 
           [0047]      FIG. 10  shows the frequency responses of the  FIG. 6  filter aperture functions. 
           [0048]      FIG. 11  shows the filter aperture functions of a transparent up- and down-conversion using an improved down-conversion filter. 
           [0049]      FIG. 12  shows a representation of the up- and down-conversion process using the filters of  FIG. 11 . 
           [0050]      FIG. 13  shows the frequency responses of the  FIG. 11  filter aperture functions. 
           [0051]      FIG. 14  shows the filter aperture functions of a transparent up- and down-conversion process using an improved down-conversion filter having null response at three frequency units. 
           [0052]      FIG. 15  shows the frequency responses of the  FIG. 14  filter aperture functions. 
           [0053]      FIG. 16  shows the filter aperture functions of a transparent up- and down-conversion using an alternative up-conversion filter and an improved down-conversion filter. 
           [0054]      FIG. 17  shows the frequency responses of the  FIG. 16  filter aperture functions. 
           [0055]      FIG. 18  shows the filter aperture functions of a transparent up- and down-conversion using an alternative up-conversion filter and an improved down-conversion filter having null response at three frequency units. 
           [0056]      FIG. 19  shows the frequency responses of the  FIG. 18  filter aperture functions. 
           [0057]      FIGS. 20   a  to  20   d  show representations of the phases of an up-conversion process according to an embodiment of the invention. 
           [0058]      FIGS. 21   a  to  21   c  show representations of the phases of a down-conversion process according to an embodiment of the invention. 
           [0059]      FIG. 22  shows exemplary up- and down-conversion filter apertures according to an embodiment the invention. 
           [0060]      FIG. 23  shows the frequency responses of the  FIG. 22  and  FIG. 24  filter aperture functions. 
           [0061]      FIG. 24  shows exemplary up- and down-conversion filter apertures according to an alternative embodiment the invention. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0062]    The Werner patent states that, when it is required to design a transparent up- and down-conversion process in which the down-conversion filter response reverses a given up-conversion filter response, the coefficients of the up-conversion filter g(x) and the coefficients of the down-conversion filter h(x) cannot be chosen independently. The prior patent goes on to say that, for given coefficients of g(x), the mathematical definition of reversibility leads to a system of linear equations for the coefficients of h(x). 
         [0063]    The present inventor has appreciated that this equation system represents an under-defined problem (i.e. there are more equations than there are unknowns) and thus the designer has a choice of many possible down-conversion filters. As explained above, the prior-art filter aperture ( 41 ), shown in  FIGS. 3 and 4 , reverses a bilinear up-conversion filter (as shown ( 40 ) in  FIG. 4 ). However, other down-conversion filters exists that transparently reverse that up-conversion; for example the aperture ( 111 ) shown in  FIG. 11 . The filter contributions for this alternative conversion are shown in  FIG. 12 . It can be seen that this alternative aperture is simpler to implement because it has fewer coefficients. The  FIG. 3  down-conversion filter requires the storage of five coefficient values (excluding the unity centre coefficient, there are five non-zero contributions in each symmetrical half aperture), whereas the  FIG. 12  down-conversion filter only needs two non-unity values to be stored. 
         [0064]    The frequency responses of the  FIG. 12  filters are shown in  FIG. 13 , in which the down-conversion filter response is designated ( 131 ), and the bilinear up-conversion filter response is designated ( 130 ). Comparison with  FIG. 8 , which shows the frequency responses of the  FIG. 3  process, shows that both the pass-band and the stop-band of the alternative down-conversion filter ( 131 ) is flatter. 
         [0065]    A further improved down-conversion filter can be obtained by summing the aperture of  FIG. 4  and the aperture of  FIG. 11  with weighting factors of ¼ and ¾ respectively. This gives an aperture ( 141 ) as shown in  FIG. 14 , and its frequency response ( 151 ) is shown in  FIG. 15 . This improved filter has zero response at f low  and therefore will suppress the DC alias due to any energy at this frequency. 
         [0066]    Improved filters which reverse the Werner ‘def’ up-conversion filter (as shown at ( 60 ) in  FIG. 6 ) are also possible. A simpler, symmetrical down-conversion filter aperture ( 161 ) is shown in  FIG. 16 , and its frequency response ( 171 ) is shown in  FIG. 17 . (The frequency response ( 170 ) of the ‘def’ up-conversion filter is also shown.) The improved aperture requires the storage of seven non-unity coefficient values, whereas the  FIG. 6  filter needs 11 non-unity values to be stored. Comparison of  FIGS. 10 and 17  shows that the improved filter has flatter pass- and stop-bands. 
         [0067]    Once again the improved down-conversion filter of  FIG. 16  can be combined with the prior down-conversion filter of  FIG. 6  by forming a weighted sum of the filter apertures, and choosing the weights to obtain a null in the frequency response at f low . The resulting aperture ( 181 ) and frequency response ( 191 ) are shown in  FIGS. 18 and 19  respectively. 
         [0068]    Although these improvements are helpful, the up-conversion filters have narrower pass-bands than the down-conversion filters, and the stop-band response is still unsatisfactory. The inventor has appreciated that there is no need for differences between the frequency responses of the up- and down-conversion filters. The frequency response of a down-converter must avoid, and an up-converter must remove, aliasing of lower-definition material. Both these requirement depend on the lower of the two sampling frequencies, and are met when the cut-off frequency of the respective filter approximates to the Nyquist limit for the lower resolution sampling process (i.e. half of f low ). The inventor has also appreciated that it is possible to optimise the frequency response of a single filter aperture to obtain reversibility. This single aperture is used to define the sample contribution values for both the up- and down-conversion filters. 
         [0069]    The optimised frequency response is used to determine the filter aperture function (i.e. the filter impulse response); however, values of this function are only required at times which correspond to filter input and output samples. Thus the frequency response need only be defined at a number of points equal to the width of the aperture (in units of the oversampling pitch that includes all input and output sample positions); this number defines the size of Fourier transform that converts between the time-domain aperture function and the frequency response. 
         [0070]    The width of the filter aperture obviously determines the number of contributions to the up- and down-filters. Because these two filters have different input sampling frequencies they will use different numbers of contributions even though they have the same aperture.  FIGS. 20   a  to  20   d  show up-conversion filter contributions for an aperture width of ±16. The four figures show all the possible phases of the up-conversion filter (f up ). Similarly  FIGS. 21   a  to  20   c  show the contributions to all possible phases of the down-conversion filter (f down ) having an aperture width of ±16. 
         [0071]    It can be seen that there are more contributions to the down-conversion filter than to the up-conversion filter; this is because it has a higher input sampling frequency. And, the number of contributions to the up-conversion filter varies between 8 and 9, depending on the phase, whereas the down-conversion filter always has 11 contributions. 
         [0072]    Suitable up- and down-filter aperture functions are shown in  FIGS. 22   a  and  22   b  respectively; the values of the contributions are also shown in Tables 3 and 4 to six decimal places. These functions are defined for input to output phase differences in the range ±16. (Outside this range the impulse responses of the filters are zero.) Although two functions are illustrated, they differ only in their amplitude scaling, which differs in inverse proportion to the number of respective filter contributions, so as to obtain unity DC gain for both filters. Because they represent differently-scaled versions of the same function they are designated f up  and f down  respectively. Note that neither of the zero-phase contributions is unity, and every filtered sample is formed from a weighted sum of at least two filter-input samples. 
         [0073]    The frequency response (identical, of course, for both the up- and down-conversion filters) of the apertures shown in  FIG. 22  is shown as the curve ( 230 ) in  FIG. 23 . 
         [0074]    The single aperture was created by optimising the reversibility of a candidate aperture by adjusting three (out of 32) frequency response parameters that control the shape the transition band of the filter frequency response. The parameters are amplitudes of the frequency response at particular frequencies; the parameters which were not optimised were set to either unity, for points in the pass-band (well below 1.5 frequency units), or zero, for points in the filter stop-band (well above 1.5 frequency units), respectively. As explained previously, the number of frequency parameters was chosen to equal the width of the filter aperture and the well-known Fourier transform was used to convert between the frequency response and the filter aperture. The 32 frequency response points from which the  FIG. 22  aperture functions are derived are given in Table 1. 
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 3-Parameter-Optimised Frequency Response 
               
             
          
           
               
                 Point 
                   
                   
               
               
                 Number 
                 Frequency 
                 Response 
               
               
                   
               
             
          
           
               
                 0 
                 0.0000 
                 1 
               
               
                 1 
                 0.1875 
                 1 
               
               
                 2 
                 0.3750 
                 1 
               
               
                 3 
                 0.5625 
                 1 
               
               
                 4 
                 0.7500 
                 1 
               
               
                 5 
                 0.9375 
                 1 
               
               
                 6 
                 1.1250 
                 1 
               
               
                 7 
                 1.3125 
                 0.973439 
               
               
                 8 
                 1.5000 
                 0.707653 
               
               
                 9 
                 1.6875 
                 0.227062 
               
               
                 10 
                 1.8750 
                 0 
               
               
                 11 
                 2.0625 
                 0 
               
               
                 12 
                 2.2500 
                 0 
               
               
                 13 
                 2.4375 
                 0 
               
               
                 14 
                 2.6250 
                 0 
               
               
                 15 
                 2.8125 
                 0 
               
               
                 16 
                 3.0000 
                 0 
               
               
                 17 
                 3.1875 
                 0 
               
               
                 18 
                 3.3750 
                 0 
               
               
                 19 
                 3.5625 
                 0 
               
               
                 20 
                 3.7500 
                 0 
               
               
                 21 
                 3.9375 
                 0 
               
               
                 22 
                 4.1250 
                 0 
               
               
                 23 
                 4.3125 
                 0 
               
               
                 24 
                 4.5000 
                 0 
               
               
                 25 
                 4.6875 
                 0 
               
               
                 26 
                 4.8750 
                 0 
               
               
                 27 
                 5.0625 
                 0 
               
               
                 28 
                 5.2500 
                 0 
               
               
                 29 
                 5.4375 
                 0 
               
               
                 30 
                 5.6250 
                 0 
               
               
                 31 
                 5.8125 
                 0 
               
               
                   
               
             
          
         
       
     
         [0075]    Although the cascaded up- and down-sampling processes using these optimised, identical filters is very close to being reversible, some small reconstruction errors do result. But, the largest of these (as evaluated on a unit impulse) is less than one third of the amplitude of one least-significant-bit in a ten-bit system. Any practical filter will use quantised signals and thus such errors will usually be eliminated by rounding inherent in the digital processing. 
         [0076]    It is possible to achieve even smaller reconstruction errors by allowing more points in the filter frequency response to be changed in the optimisation. An optimisation in which seven of the frequency response parameters were varied resulted in the alternative up- and down-filter apertures shown in  FIG. 24  (and listed in Tables 3 and 4). As before, each is a differently-scaled version of the same function. The corresponding frequency response is shown at ( 231 ) in  FIG. 23 , and it can be seen that improved reversibility has been achieved at the price of a less-sharp response, which will allow more aliasing. Again, every filtered sample is formed from a weighted sum of at least two filter-input samples, and neither zero-phase contribution is equal to unity. 
         [0077]    The frequency response parameters used to derive the  FIG. 24  aperture functions are shown in Table 2. 
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 7-Parameter-Optimised Frequency Response 
               
             
          
           
               
                 Point 
                   
                   
               
               
                 Number 
                 Frequency 
                 Response 
               
               
                   
               
             
          
           
               
                 0 
                 0.0000 
                 1 
               
               
                 1 
                 0.1875 
                 1 
               
               
                 2 
                 0.3750 
                 1 
               
               
                 3 
                 0.5625 
                 1 
               
               
                 4 
                 0.7500 
                 1 
               
               
                 5 
                 0.9375 
                 0.999977 
               
               
                 6 
                 1.1250 
                 0.996351 
               
               
                 7 
                 1.3125 
                 0.936209 
               
               
                 8 
                 1.5000 
                 0.707130 
               
               
                 9 
                 1.6875 
                 0.351339 
               
               
                 10 
                 1.8750 
                 0.085753 
               
               
                 11 
                 2.0625 
                 0.003074 
               
               
                 12 
                 2.2500 
                 0 
               
               
                 13 
                 2.4375 
                 0 
               
               
                 14 
                 2.6250 
                 0 
               
               
                 15 
                 2.8125 
                 0 
               
               
                 16 
                 3.0000 
                 0 
               
               
                 17 
                 3.1875 
                 0 
               
               
                 18 
                 3.3750 
                 0 
               
               
                 19 
                 3.5625 
                 0 
               
               
                 20 
                 3.7500 
                 0 
               
               
                 21 
                 3.9375 
                 0 
               
               
                 22 
                 4.1250 
                 0 
               
               
                 23 
                 4.3125 
                 0 
               
               
                 24 
                 4.5000 
                 0 
               
               
                 25 
                 4.6875 
                 0 
               
               
                 26 
                 4.8750 
                 0 
               
               
                 27 
                 5.0625 
                 0 
               
               
                 28 
                 5.2500 
                 0 
               
               
                 29 
                 5.4375 
                 0 
               
               
                 30 
                 5.6250 
                 0 
               
               
                 31 
                 5.8125 
                 0 
               
               
                   
               
             
          
         
       
     
         [0078]    The reconstruction errors due to the  FIG. 24  apertures (evaluated on unit impulses as before) are very much smaller than for the  FIG. 22  apertures, and amount to less than one tenth of the LSB of a ten-bit system. 
         [0079]    For a given aperture width there is therefore a trade-off between sharpness of cut and reversibility (i.e. absence of reconstruction errors). However, if larger reconstruction errors are acceptable, perhaps because fewer bits are used to represent the signal and therefore larger errors will be eliminated by rounding, then a narrower filter aperture can be used. 
         [0080]    The filters described so far have phase coincidence between suitable sub-sets of the input and output sampling structures. But it is also possible to derive oversampled filter apertures which can be used to define the contributions of filters having phase-shifted output samples on a corresponding oversampled structure. This is simply achieved by increasing the number of stop-band points at which the frequency response is defined (i.e. defining the response up to a higher oversampling frequency, greater than the 12 frequency units in the example shown) and thus obtaining more impulse response values from the inverse Fourier transform. 
         [0081]    The invention has been described in the context of 3:4 up-conversion followed by 4:3 down-conversion. It is equally applicable when the down-conversion precedes the up-conversion. The skilled person will be able to apply the invention to other conversion ratios by: determining a candidate frequency response having a transition band in the region of half the lower sampling frequency, defined at a convenient number of frequencies equal to a practical aperture width; and, optimising the frequency response by adjusting some of the defined response values so as to minimise the reconstruction errors obtained in a test (either simulated or practical) of cascaded conversions of filters defined by the aperture function represented by an inverse Fourier transform of the frequency response. 
         [0082]    The apertures shown in  FIGS. 22 and 24  were optimised to minimise the reconstruction errors due to impulse inputs. This filter selection criterion may be unrealistically stringent if it is known that the spectrum of the input signal to the cascaded conversion processes has previously been limited. It may be preferable to optimise the frequency response parameters that define the up- and down-conversion filter apertures using reconstruction errors due to signals representative of those that will typically be processed. 
         [0083]    Although it is normally desirable for the responses of the up- and down-conversion filters to be similar, this may not always be the case. If so, it is possible to optimise the reconstruction errors due the cascade of a pair of filters which are different from each other in some desired way. 
         [0084]    Filters according to the principles which have been described may be incorporated into multi-standard video processing equipment—such as vision mixers or “production switchers” as they are known in some territories. Such an equipment may allow the user to input material at different sampling resolutions and to choose between different output sampling resolutions, or provide simultaneous outputs of the same material at different sampling resolutions. The processing can be carried out at the highest expected sampling resolution and lower resolution inputs up-converted to that resolution, and lower resolution outputs down-converted from that resolution. Inputs of different resolution can thus be combined in a process operating at the higher resolution and output with or without down-conversion. 
         [0000]    
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Up-Conversion Filters 
               
             
          
           
               
                   
                 g(x) 
                   
                 f up (x) 
                   
               
             
          
           
               
                   
                 FIGS. 
                 FIGS. 
                 FIGS. 
                 FIGS. 
               
               
                   
                 1 2 3 4 
                 5 6 9 10 
                 20 
                 20 22 
               
               
                   
                 7 8 11 12 
                 16 17 18 
                 22 
                 23 
               
               
                 x 
                 13 14 15 
                 19 
                 23 
                 24 
               
               
                   
               
             
          
           
               
                 −16 
                   
                   
                 0.000448 
                 −0.000084 
               
               
                 −15 
                   
                   
                 0.000028 
                 −0.000092 
               
               
                 −14 
                   
                   
                 −0.002800 
                 0.000204 
               
               
                 −13 
                   
                   
                 0.001344 
                 0.000076 
               
               
                 −12 
                   
                   
                 0.008236 
                 0.000744 
               
               
                 −11 
                   
                   
                 −0.007048 
                 −0.000508 
               
               
                 −10 
                   
                   
                 −0.016364 
                 −0.005232 
               
               
                 −9 
                   
                   
                 0.020200 
                 0.004996 
               
               
                 −8 
                   
                   
                 0.025956 
                 0.015628 
               
               
                 −7 
                   
                 −0.004832 
                 −0.045064 
                 −0.021308 
               
               
                 −6 
                   
                 −0.030332 
                 −0.035548 
                 −0.032036 
               
               
                 −5 
                   
                 −0.054860 
                 0.090568 
                 0.063384 
               
               
                 −4 
                   
                 0 
                 0.043680 
                 0.051036 
               
               
                 −3 
                 1/4 
                 0.204804 
                 −0.188420 
                 −0.166500 
               
               
                 −2 
                 1/2 
                 0.530332 
                 −0.049112 
                 −0.066500 
               
               
                 −1 
                 3/4 
                 0.854892 
                 0.628388 
                 0.619960 
               
               
                 0 
                 1 
                 1 
                 1.051020 
                 1.072480 
               
               
                 1 
                 3/4 
                 0.854892 
                 0.628388 
                 0.619960 
               
               
                 2 
                 1/2 
                 0.530332 
                 −0.049112 
                 −0.066500 
               
               
                 3 
                 1/4 
                 0.204804 
                 −0.188420 
                 −0.166500 
               
               
                 4 
                   
                 0 
                 0.043680 
                 0.051036 
               
               
                 5 
                   
                 −0.054860 
                 0.090568 
                 0.063384 
               
               
                 6 
                   
                 −0.030332 
                 −0.035548 
                 −0.032036 
               
               
                 7 
                   
                 −0.004832 
                 −0.045064 
                 −0.021308 
               
               
                 8 
                   
                   
                 0.025956 
                 0.015628 
               
               
                 9 
                   
                   
                 0.020200 
                 0.004996 
               
               
                 10 
                   
                   
                 −0.016364 
                 −0.005232 
               
               
                 11 
                   
                   
                 −0.007048 
                 −0.000508 
               
               
                 12 
                   
                   
                 0.008236 
                 0.000744 
               
               
                 13 
                   
                   
                 0.001344 
                 0.000076 
               
               
                 14 
                   
                   
                 −0.002800 
                 0.000204 
               
               
                 15 
                   
                   
                 0.000028 
                 −0.000092 
               
               
                 16 
                   
                   
                 0.000448 
                 −0.000084 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 4 
               
             
             
               
                   
               
               
                 Down-Conversion Filters 
               
             
          
           
               
                   
                 h(x) 
                 f down (x) 
               
             
          
           
               
                   
                 FIGS. 
                 FIGS. 
                 FIGS. 
                 FIGS. 
                 FIGS. 
                 FIGS. 
                 FIGS. 
                 FIGS. 
                 FIGS. 
                 FIGS. 
               
               
                   
                 1 2 
                 3 4 
                 5 
                 6 
                 11 12 
                 14 
                 16 
                 18 
                 21 22 
                 21 23 
               
               
                 x 
                 7 
                 8 
                 9 
                 10 
                 13 
                 15 
                 17 
                 19 
                 23 
                 24 
               
               
                   
               
             
          
           
               
                 −20 
                   
                   
                   
                 −0.000009 
                   
                   
                   
                 −0.000003 
                   
                   
               
               
                 −19 
                   
                   
                   
                 0 
                   
                   
                   
                 0 
               
               
                 −18 
                   
                   
                   
                 0 
                   
                   
                   
                 0 
               
               
                 −17 
                   
                   
                   
                 0 
                   
                   
                   
                 0 
               
               
                 −16 
                   
                   
                   
                 0.000090 
                   
                   
                 0.000018 
                 0.000016 
                 0.000336 
                 D.000063 
               
               
                 −15 
                   
                   
                   
                 0 
                   
                   
                 0 
                 0 
                 0.000021 
                 −0.000069 
               
               
                 −14 
                   
                   
                   
                 −0.000302 
                   
                   
                 0 
                 −0.000091 
                 −0.002100 
                 0.000153 
               
               
                 −13 
                   
                   
                   
                 0 
                   
                   
                 0 
                 0 
                 0.001008 
                 0.000057 
               
               
                 −12 
                   
                   
                   
                 0 
                   
                   
                 0 
                 0 
                 0.006177 
                 0.000558 
               
               
                 −11 
                   
                   
                   
                 −0.002919 
                   
                   
                 0 
                 −0.000880 
                 −0.005286 
                 −0.000381 
               
               
                 −10 
                   
                   
                   
                 0.000302 
                   
                   
                 0.000604 
                 0.000513 
                 −0.012273 
                 −0.003924 
               
               
                 −9 
                   
                   
                   
                 0 
                   
                   
                 0 
                 0 
                 0.015150 
                 0.003747 
               
               
                 −8 
                 1/3 
                 1/6 
                 0.003449 
                 0.141700 
                   
                 0.041670 
                 0.003449 
                 0.045120 
                 0.019467 
                 0.011721 
               
               
                 −7 
                 0 
                 0 
                 0 
                 0.002919 
                   
                 0 
                 −0.222900 
                 0.004959 
                 −0.033798 
                 −0.015981 
               
               
                 −6 
                 0 
                 0 
                 0 
                 0 
                   
                 0 
                 0 
                 0 
                 −0.026661 
                 −0.024027 
               
               
                 −5 
                 −4/3   
                 −2/3   
                 0 
                 −0.549600 
                   
                 −0.166700 
                 0 
                 −0.165600 
                 0.067926 
                 0.047538 
               
               
                 −4 
                 −1/3   
                 −1/6   
                 0.053745 
                 −0.084550 
                 −1/3   
                 −0.291700 
                 −0.222900 
                 −0.181200 
                 0.032760 
                 0.038277 
               
               
                 −3 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 −0.141315 
                 −0.124875 
               
               
                 −2 
                 2 
                 1 
                 0.113710 
                 0.942800 
                 0 
                 0.250000 
                 0.113700 
                 0.363500 
                 −0.036834 
                 −0.049875 
               
               
                 −1 
                 4/3 
                 2/3 
                 0 
                 0.549600 
                 4/3 
                 1.166667 
                 1.099000 
                 0.933600 
                 0.471291 
                 0.464970 
               
               
                 0 
                 1 
                 1 
                 1 
                 1 
                 1 
                 1 
                 1 
                 1 
                 0.788265 
                 0.804360 
               
               
                 1 
                   
                 2/3 
                 1.099232 
                 0.549600 
                 4/3 
                 1.166667 
                 1.099000 
                 0.933600 
                 0.471291 
                 0.464970 
               
               
                 2 
                   
                 1 
                 1.771901 
                 0.942800 
                 0 
                 0.250000 
                 0.113700 
                 0.363500 
                 −0.036834 
                 −0.049875 
               
               
                 3 
                   
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 −0.141315 
                 −0.124875 
               
               
                 4 
                   
                 −1/6   
                 −0.222855 
                 −0.084550 
                 −1/3   
                 −0.291700 
                 −0.222900 
                 −0.181200 
                 0.032760 
                 0.038277 
               
               
                 5 
                   
                 −2/3   
                 −1.099232 
                 −0.549600 
                   
                 −0.166700 
                 0 
                 −0.165600 
                 0.067926 
                 0.047538 
               
               
                 6 
                   
                 0 
                 0 
                 0 
                   
                 0 
                 0 
                 0 
                 −0.026661 
                 −0.024027 
               
               
                 7 
                   
                 0 
                 0.005838 
                 0.002919 
                   
                 0 
                 −0.222900 
                 0.004959 
                 −0.033798 
                 −0.015981 
               
               
                 8 
                   
                 1/6 
                 0.280050 
                 0.141700 
                   
                 0.041670 
                 0.003449 
                 0.045120 
                 0.019467 
                 0.011721 
               
               
                 9 
                   
                   
                 0 
                 0 
                   
                   
                 0 
                 0 
                 0.015150 
                 0.003747 
               
               
                 10 
                   
                   
                 0.000604 
                 0.000302 
                   
                   
                 0.000604 
                 0.000513 
                 −0.012273 
                 −0.003924 
               
               
                 11 
                   
                   
                 −0.005838 
                 −0.002919 
                   
                   
                 0 
                 −0.000880 
                 −0.005286 
                 −0.000381 
               
               
                 12 
                   
                   
                 0 
                 0 
                   
                   
                 0 
                 0 
                 0.006177 
                 0.000558 
               
               
                 13 
                   
                   
                 0 
                 0 
                   
                   
                 0 
                 0 
                 0.001008 
                 0.000057 
               
               
                 14 
                   
                   
                 −0.000604 
                 −0.000302 
                   
                   
                 0 
                 −0.000091 
                 −0.002100 
                 0.000153 
               
               
                 15 
                   
                   
                 0 
                 0 
                   
                   
                 0 
                 0 
                 0.000021 
                 −0.000069 
               
               
                 16 
                   
                   
                 0.000018 
                 0.000090 
                   
                   
                 0.000018 
                 0.000016 
                 0.000336 
                 −0.000063 
               
               
                 17 
                   
                   
                 0 
                 0 
                   
                   
                   
                 0 
               
               
                 18 
                   
                   
                 0 
                 0 
                   
                   
                   
                 0 
               
               
                 19 
                   
                   
                 0 
                 0 
                   
                   
                   
                 0 
               
               
                 20 
                   
                   
                 −0.000018 
                 −0.000009 
                   
                   
                   
                 −0.000003