Abstract:
A method for producing a noise-reduced digital image captured using a digital imaging system having signal-dependent noise characteristics, comprising: capturing one or more noisy digital images of a scene, wherein said at least one noisy digital image has signal-dependent noise characteristics; defining a functional relationship to relate the noisy digital images to a noise-reduced digital image, wherein the functional relationship includes at least two sets of unknown parameters, and wherein at least one of the sets of unknown parameters relates to the signal-dependent noise characteristics; defining an energy function responsive to the functional relationship which includes at least a data fidelity term to enforce similarities between the noisy digital images and the noise-reduced digital image, and a spatial fidelity term to encourage sharp edges in the noise-reduced digital image; and using an optimization process to determine a noise-reduced image responsive to the energy function.

Description:
FIELD OF THE INVENTION 
       [0001]    This invention relates generally to the field of denoising digital images, and more particularly to a method to denoise digital images having signal-dependent noise. 
       BACKGROUND OF THE INVENTION 
       [0002]    Digital images captured by digital cameras are corrupted by “noise,” wherein noise can be defined as unwanted random variations in the digital image. Noise can arise from a number of sources such as sensor shot noise and fixed-pattern noise. The amount of noise in a digital image can depend on many factors such as sensor design, exposure level, and digital image processing applied to the image. Denoising algorithms are often used to “denoise” captured digital images (i.e., reduce the level of noise) in order to improve the signal-to-noise (SNR) ratio of the captured digital images. Noise in digital camera images generally depends on the signal level (i.e., the image pixel values); this type of noise is commonly referred to as “signal-dependent noise.” Traditional denoising algorithms assume the level of noise to be independent of the image pixel values. As a result, such algorithms are inadequate to deal with signal-dependent noise. 
         [0003]    One method described by Rudin et al., in the article “Nonlinear total variation based noise removal algorithms” (Physica D, Vol. 60, pp. 259-268, 1992) uses total variation minimization to generate denoised image. 
         [0004]    Another method taught by Foi et al., in the article “Pointwise shape-adaptive DCT for high-quality denoising and deblocking of grayscale and color images” (IEEE Transactions on Image Processing, Vol. 16, pp. 1395-1411, May 2007) analyzes noisy image using a shape-adaptive discrete cosine transform (DCT). 
         [0005]    Coifman et al., in the article “Translation-invariant de-noising” (Lecture Notes in Statistics: Wavelets and Statistics, Springer Verlag, New York, pp. 125-150, 1995) use translation invariant thresholding of the wavelet coefficients of the noisy image to produce a denoised image. 
         [0006]    All of the above denoising algorithms assume noise to be independent of the image pixel values and are therefore inadequate to deal with signal-dependent noise. 
         [0007]    Hirakawa et al., in the article “Image denoising for signal-dependent noise” (IEEE International Conference on Acoustics, Speech, and Signal Processing, Vol. 2, pp. 29-32, 2005) teaches a rather elegant signal-dependent denoising technique. In this approach, the total least square (TLS) approach is used for modeling the uncertainties in the noisy image and to reduce the signal-dependent noise. However, this approach is computationally demanding and too slow for many applications. 
         [0008]    Thus, there exists a need for an efficient signal-dependent denoising algorithm that preserves salient features of the image. 
       SUMMARY OF THE INVENTION 
       [0009]    The present invention represents a method for producing a noise-reduced digital image captured using a digital imaging system having signal-dependent noise characteristics, comprising using a digital processor to perform at least some of the steps of: 
         [0010]    a) capturing one or more noisy digital images of a scene, wherein said at least one noisy digital image has signal-dependent noise characteristics; 
         [0011]    b) defining a functional relationship to relate the one or more noisy digital images to a noise-reduced digital image estimate, wherein the functional relationship includes at least two sets of unknown parameters, and wherein at least one of the sets of unknown parameters relates to the signal-dependent noise characteristics; 
         [0012]    c) defining an energy function responsive to the functional relationship which includes at least:
       i) a data fidelity term to enforce similarities between the one or more noisy digital images and the noise-reduced digital image estimate; and   ii) a spatial fidelity term to encourage sharp edges in the noise-reduced digital image estimate; and       
 
         [0015]    d) using an optimization process to determine a noise-reduced image responsive to the energy function. 
         [0016]    It is an advantage of the present invention that it deals with signal-dependent noise in digital images, which is more realistic for digital camera applications as compared to traditional signal-independent noise models. 
         [0017]    It is an additional advantage that by using a spatial penalty term, the resulting noise-reduced image preserves the salient features with improved accuracy. 
         [0018]    It is a further advantage of the present invention that it can process multiple captures of the scene having signal-dependent noise, which enables the noise to be reduced with improved accuracy. 
         [0019]    In addition to the embodiments described above, further embodiments will become apparent by reference to the drawings and by study of the following detailed description. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0020]    The present invention will be more readily understood from the detailed description of exemplary embodiments presented below considered in conjunction with the attached drawings, of which: 
           [0021]      FIG. 1  is a high-level diagram showing the components of a system for denoising digital image according to an embodiment of the present invention; 
           [0022]      FIG. 2  is a flow diagram illustrating a method for producing a noise-reduced digital image according to an embodiment of the present invention; 
           [0023]      FIG. 3  is a block diagram showing the define energy function step of  FIG. 2  in more detail; and 
           [0024]      FIG. 4  is a flow diagram illustrating a method for producing a noise-reduced digital image according to an alternate embodiment of the present invention. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0025]    The invention is inclusive of combinations of the embodiments described herein. References to “a particular embodiment” and the like refer to features that are present in at least one embodiment of the invention. Separate references to “an embodiment” or “particular embodiments” or the like do not necessarily refer to the same embodiment or embodiments; however, such embodiments are not mutually exclusive, unless so indicated or as are readily apparent to one of skill in the art. The use of singular and/or plural in referring to the “method” or “methods” and the like is not limiting. 
         [0026]    The phrase, “digital content record”, as used herein, refers to any digital content record, such as a digital still image, a digital audio file, a digital video file, etc. 
         [0027]    It should be noted that, unless otherwise explicitly noted or required by context, the word “or” is used in this disclosure in a non-exclusive sense. 
         [0028]      FIG. 1  is a high-level diagram showing the components of a system for image denoising according to an embodiment of the present invention. The system includes a data processing system  110 , a peripheral system  120 , a user interface system  130 , and a data storage system  140 . The peripheral system  120 , the user interface system  130  and the data storage system  140  are communicatively connected to the data processing system  110 . 
         [0029]    The data processing system  110  includes one or more data processing devices that implement the processes of the various embodiments of the present invention, including the example processes of  FIGS. 2-4  described herein. The phrases “data processing device” or “data processor” are intended to include any data processing device, such as a central processing unit (“CPU”), a desktop computer, a laptop computer, a mainframe computer, a personal digital assistant, a Blackberry™, a digital camera, cellular phone, or any other device for processing data, managing data, or handling data, whether implemented with electrical, magnetic, optical, biological components, or otherwise. 
         [0030]    The data storage system  140  includes one or more processor-accessible memories configured to store information, including the information needed to execute the processes of the various embodiments of the present invention, including the example processes of  FIGS. 2-4  described herein. The data storage system  140  may be a distributed processor-accessible memory system including multiple processor-accessible memories communicatively connected to the data processing system  110  via a plurality of computers and/or devices. On the other hand, the data storage system  140  need not be a distributed processor-accessible memory system and, consequently, may include one or more processor-accessible memories located within a single data processor or device. 
         [0031]    The phrase “processor-accessible memory” is intended to include any processor-accessible data storage device, whether volatile or nonvolatile, electronic, magnetic, optical, or otherwise, including but not limited to, registers, floppy disks, hard disks, Compact Discs, DVDs, flash memories, ROMs, and RAMs. 
         [0032]    The phrase “communicatively connected” is intended to include any type of connection, whether wired or wireless, between devices, data processors, or programs in which data may be communicated. 
         [0033]    The phrase “communicatively connected” is intended to include a connection between devices or programs within a single data processor, a connection between devices or programs located in different data processors, and a connection between devices not located in data processors at all. In this regard, although the data storage system  140  is shown separately from the data processing system  110 , one skilled in the art will appreciate that the data storage system  140  may be stored completely or partially within the data processing system  110 . Further in this regard, although the peripheral system  120  and the user interface system  130  are shown separately from the data processing system  110 , one skilled in the art will appreciate that one or both of such systems may be stored completely or partially within the data processing system  110 . 
         [0034]    The peripheral system  120  may include one or more devices configured to provide digital content records to the data processing system  110 . For example, the peripheral system  120  may include digital still cameras, digital video cameras, cellular phones, or other data processors. The data processing system  110 , upon receipt of digital content records from a device in the peripheral system  120 , may store such digital content records in the data storage system  140 . 
         [0035]    The user interface system  130  may include a mouse, a keyboard, another computer, or any device or combination of devices from which data is input to the data processing system  110 . In this regard, although the peripheral system  120  is shown separately from the user interface system  130 , the peripheral system  120  may be included as part of the user interface system  130 . 
         [0036]    The user interface system  130  also may include a display device, a processor-accessible memory, or any device or combination of devices to which data is output by the data processing system  110 . In this regard, if the user interface system  130  includes a processor-accessible memory, such memory may be part of the data storage system  140  even though the user interface system  130  and the data storage system  140  are shown separately in  FIG. 1 . 
         [0037]      FIG. 2  is a flow diagram illustrating a method for producing a noise-reduced digital image according to an embodiment of the present invention from a set of one or more noisy digital images captured using a digital imaging system having signal-dependent characteristics. One or more noisy digital images  203  representing a scene and having signal-dependent noise characteristics are received in receive noisy digital images step  202 . The noisy digital images  203  can be captured by a digital camera or a scanner. Alternatively, they may be frames of a video sequence captured by a video camera. 
         [0038]    A priori information  205  is received in receive a priori information step  204 . The a priori information  205  contains the prior information about the content of a noise-reduced output image  211  that is to be produced according to the method of the present invention. The a priori information  205  will be discussed in more detail later. A define spatial mapping step  206  defines a spatial mapping  207 . The spatial mapping  207  is a functional relationship relating the one or more noisy digital images  203  to the noise-reduced output image  211 . A define energy function step  208  uses the spatial mapping  207  to compute an energy function  209 . In the art, energy functions  209  are sometimes called optimization functions or cost functions. Finally, an optimization process step  210  uses the one or more noisy digital images  203  and the energy function  209  to produce the noise-reduced output image  211 . 
         [0039]    The individual steps outlined in  FIG. 2  will now be described in greater detail. The define spatial mapping step  206  defines a spatial mapping  207 , which is a functional relationship relating the one or more noisy digital images  203  to the noise-reduced output image  211 . The spatial mapping  207  can be defined in any appropriate way known to those skilled in the art. One way to define the spatial mapping  207  according to a preferred embodiment of the present invention is using the following equation: 
         [0000]        I   k ( x,y )= I   0 ( x,y )+ K   0   +K   1   I   0 ( x,y ))η k ( x,y )  (1)
 
         [0000]    where I k (x, y) is a pixel value of the k th  noisy digital images  203  at pixel location (x,y), η k  (x, y) is a unit amplitude random noise function for the k th  noisy digital images  203  at pixel location (x,y), and I 0 (x, y), K 0  is a constant relating to the signal-independent noise amplitude, and K 1  is a constant relating to the signal-dependent noise amplitude. (This spatial mapping is similar to one defined by Hirakawa et al., in the aforementioned article “Image denoising for signal-dependent noise.)” 
         [0040]    The spatial mapping  207  given in Eq. (1) can be rearranged as follows: 
         [0000]        I   k ( x,y )=β k ( x,y ) I   0 ( x,y )+ K   0 η k ( x,y )  (2)
 
         [0000]    where β k (x, y)=1+K 1 η k  (x, y) incorporates the signal-dependent noise characteristics. The individual equations for each of the noisy digital images  203  can be combined into vector form to yield: 
         [0000]          I   ( x,y )= β ( x,y ) I   0 ( x,y )+ K   0   η ( x,y )  (3)
 
         [0000]    where  I (x, y) is a vector of the pixels from the one or more noisy digital images  203  at pixel location (x,y),  η (x, y) is a function parameter vector of the random noise functions at pixel location (x,y), and  β (x, y) is a function parameter vector relating to the signal-dependent noise characteristics The function parameter vector  β (x, y) can be expressed by the following equation: 
         [0000]        β ( x,y )= 1   +K   1   η ( x,y )  (4)
 
         [0000]    where  1  is a vector containing all 1s. 
         [0041]      FIG. 3  is a more detailed view of the define energy function step  208  shown in  FIG. 2  according to a preferred embodiment of the present invention. A define data fidelity term step  302  defines a data fidelity term  303  responsive to the spatial mapping  207 . The data fidelity term  303  enforces similarities between the noisy digital images  203  ( FIG. 2 ) and the noise-reduced output image  211  ( FIG. 2 ). The data fidelity term  303  can be defined in any appropriate way known to those skilled in the art. One way to define the data fidelity term  303  according to a preferred embodiment of the present invention can be described using the following equation: 
         [0000]      DF( x,y )=( β ( x,y ) I   0 ( x,y )−   I   ( x,y )) t ( β ( x,y ) I   0 ( x,y )−   I   ( x,y ))  (5)
 
         [0000]    where DF(x, y) is the value of the data fidelity term  303  at pixel location (x,y), and (•) t  denotes a transpose operation. 
         [0042]    A define spatial fidelity term step  304  uses the a priori information  205  to define a spatial fidelity term  305 . The spatial fidelity term  305  encourages sharp edges in the noise-reduced output image  211  ( FIG. 2 ). The spatial fidelity term  305  can be defined in any appropriate way known to those skilled in the art. One way to compute the spatial fidelity term  305  according to a preferred embodiment of the present invention is to make use of the a priori information  205  that noise-reduced images will generally contain smooth image regions separated by edges having high spatial frequency content. It is well known in the art that a spatial fidelity term can be defined in accordance with this a priori information using a total variation regularization technique. (For example, see the aforementioned article by Rudin et al., entitled “Nonlinear total variation based noise removal algorithms”) According to this approach, the spatial fidelity term  305  can be defined using the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    where SF(x, y) is the value of the spatial fidelity term  305  at pixel location (x,y), and ∇I 0 (x, y) is the gradient of the noise-reduced output image  211  ( FIG. 2 ) at pixel location (x, y). 
         [0043]    A construct energy function step  306  combines the data fidelity term  303  and the spatial fidelity term  305  to construct the energy function  209 . The construct energy function step  306  can be performed in any appropriate way known to those skilled in the art. One way to construct an energy function  209  with the construct energy function step  306  according to a preferred embodiment of the present invention can be described using the following equation: 
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         [0000]    where g(I 0 (x, y),  β (x, y), λ, α, γ) is the energy function  209 , and λ, α and γ are weighting parameters. The weighting parameter λ is used to weight the data fidelity term, and the weighting parameter α s used to weight the spatial fidelity term. The weighting parameter γ is used to weight an optional third term which encourages  β (x, y) to be smooth. 
         [0044]    The optimization process step  210  of  FIG. 2  uses the energy function  209  and the one or more noisy digital images  203  to produce the noise-reduced output image  211 . The optimization process step  210  can be performed in any appropriate way known to those skilled in the art. Conceptually, the optimization process step  210  determines a noise-reduced image by evaluating the energy function to select between candidate noise-reduced images. The candidate noise reduced image having the lowest value of the energy function is selected. In a preferred embodiment of the present invention, an alternating minimization algorithm is applied to minimize the energy function  209 , g(I 0 (x, y),  β (x, y), λ, α, γ). Alternating minimization algorithms are well-known in the optimization art. For example, see “Convergence of the alternating minimization algorithm for blind deconvolution” by Chan et al. (Linear Algebra Appl., Vol. 316, pp. 259-285, 2000). The values of the weighting parameters λ, α and γ are set to predetermined values and the alternating minimization algorithm is used to determine the  β (x, y) and I 0 (x, y) values that minimize the energy function  209 . In a preferred embodiment of the present invention, the alternating minimization algorithm uses a steepest descent optimization to determine I 0 (x, y). When the optimization process step  210  converges, the determined value of I 0 (x, y) is selected as the noise-reduced output image  211 . 
         [0045]      FIG. 4  is a flow diagram illustrating a method for producing the noise-reduced output image  211  according to an alternate embodiment of the present invention. An initialize noise-reduced image step  402  is used to initialize an intermediate noise-reduced image  403 . The initialization of the intermediate noise-reduced image  403  can be performed in any appropriate way known to those skilled in the art. In a preferred embodiment of the present invention, the intermediate noise-reduced image  403  is initialized by setting it equal to the average of the one or more noisy digital images  203 . 
         [0046]    An estimate scaled noise images step  404  uses the intermediate noise-reduced image  403  and the one or more noisy digital images  203  to produce a corresponding set of one or more scaled noise images  405 . The scaled noise images  405  can be computed in any appropriate way known to those skilled in the art. One way to compute the scaled noise images  405  according to a preferred embodiment of the present invention can be described using the following equation: 
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         [0000]    where  SN (x, y) is a vector of the pixel values of the scaled noise images  405  at pixel location (x,y), I INR (x, y) is the value of the intermediate noise-reduced image  403  at pixel location (x,y) and  I (x, y), λ, and γ have been defined earlier. This equation follows from the definition of the energy function defined in Eq. (7) using derivations that are well-known in the optimization art. It can be obtained by taking the derivative of Eq. (7) with respect to  β (x, y), then setting it equal to zero and solving for  β (x, y). 
         [0047]    An estimate noise reduced image step  406  uses the intermediate noise-reduced image  403 , the one or more scaled noise images  405  and the one or more noisy digital images  203  to produce a noise-reduced image  407 . The noise-reduced image  407  can be computed in any appropriate way known to those skilled in the art. One way to compute the noise-reduced image  407  according to a preferred embodiment of the present invention can be described using the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0000]    where I NR (x, y) and I INR (x, y) are pixel values of noise-reduced image  407  and intermediate noise-reduced image  403 , respectively, at pixel location (x,y), τ and α are empirically determined weighting parameters, “•” represents the dot product, “|•|” is the absolute value operator, and V is the gradient operator. This equation also follows from the definition of the energy function defined in Eq. (7). It can be obtained by taking the derivative of Eq. (7) with respect to I 0 (x, y), then setting the derivative equal to zero and applying a gradient descent method 
         [0048]    A define convergence criterion step  408  specifies a convergence criterion  409  to terminate the algorithm. The convergence criterion  409  can be determined in any appropriate way known to those skilled in the art. In a preferred embodiment of the present invention, the convergence criterion  409  is satisfied when the algorithm is repeated for a predetermined number of iterations. Alternate forms of convergence criteria are well known to those skilled in the art. As an example, the convergence criterion  409  can specify that the algorithm is terminated if the mean square difference between the intermediate noise-reduced image  403  and the noise-reduced image  407  is less than a predetermined threshold. Alternatively, the convergence criterion  409  can specify that the algorithm is terminated if the mean square difference between the intermediate noise-reduced image  403  and the noise-reduced image  407  is less than a predetermined threshold, but is terminated after the algorithm is repeated for a predetermined number of iterations even if the mean square difference condition is not satisfied. 
         [0049]    The convergence test  410  checks whether the convergence criterion  409  is satisfied. If the convergence criterion  409  is satisfied then the algorithms is terminated and the noise-reduced image  407  is selected as the final noise-reduced output image  211 . Otherwise, the intermediate noise-reduced image  403  is set to be equal to the noise-reduced image  407  and the entire process is repeated until the convergence criterion is satisfied. 
         [0050]    In a preferred embodiment of the present invention, a plurality of noisy digital images  203  are used to determine the noise-reduced output image  211 . This has the advantage that it provides multiple instances of the image noise, and therefore provides information that is useful to reduce the noise. The plurality of images is preferably captured sequentially during a short time interval in order to minimize any relative motion between the digital camera and objects in the scene. In the case where there may be some amount of relative motion between the times that the noisy digital images  203  are captured, the noisy digital images  203  can be spatially registered first and the proposed invention can then be applied to the registered images. The image registration can be performed in any appropriate way known to those skilled in the art. In a preferred embodiment of the present invention, the image registration is performed by selecting one of the noisy digital images  203  as a reference digital image and applying translation, rotation or scaling operations to the other noisy digital images  203  to align them with the reference digital image. In cases where the images contain moving objects, only portions of the digital images may be misaligned. In this case, one or more subsets of image pixels that are misaligned can be determined and appropriate translation, rotation or scaling operations can be applied to each subset of image pixels. 
         [0051]    In another embodiment of the present invention, only a single noisy digital image  203  is used in the determination of the noise-reduced output image  211 . The same basic methods described with reference to  FIGS. 2 and 4  can be applied even with a single noisy digital image  203 . While it is more convenient to use only a single noisy digital image  203  in many cases, the quality of the resulting noise-reduced output images  211  is generally inferior to those obtained using multiple noisy digital images  203 . 
         [0052]    It is to be understood that the exemplary embodiments disclosed herein are merely illustrative of the present invention and that many variations of the above-described embodiments can be devised by one skilled in the art without departing from the scope of the invention. It is therefore intended that all such variations be included within the scope of the following claims and their equivalents. 
       PARTS LIST 
       [0000]    
       
           110  Data processing system 
           120  Peripheral system 
           130  User interface system 
           140  Data storage system 
           202  Receive noisy digital images step 
           203  Noisy digital images 
           204  Receive a priori information step 
           205  A priori information 
           206  Define spatial mapping step 
           207  Spatial mapping 
           208  Define energy function step 
           209  Energy function 
           210  Optimization process step 
           211  Noise-reduced output image 
           302  Define data fidelity term step 
           303  Data fidelity term 
           304  Define spatial fidelity term step 
           305  Spatial fidelity term 
           306  Construct energy function step 
           402  Initialize noise-reduced image step 
           403  Intermediate noise-reduced image 
           404  Estimated scaled noise images step 
           405  Scaled noise images 
           406  Estimate noise-reduced image step 
           407  Noise-reduced image 
           408  Define convergence criterion step 
           409  Convergence criterion 
           410  Convergence test