Udacity cs344-Introduction to Parallel Programming学习笔记-第二单元


1、parallel communication patterns   并行通信模式

Map:映射,在特定的位置读取和写入。

Gather:收集,从多个不同的位置读入,写入一个位置。

Scatter:分发,写入多个位置。


第一个quiz答案:Scatter

有点费解,一开始我以为是Map,后来有人提出一个这样的理解,貌似能说的通。先附在下方,留待日后回过头来理解。

MAP:

output[i] = function(input[i]);

SCATTER:

output[ scatterLoc[i] ] = input[i];

GATHER:

output[i] = input[ gatherLoc[i] ];

我自己的理解:map代表一对一,gather代表多对一,scatter代表一对多


2、stencil:tasks read input from a fixed neighborhood in an array.

第二个quiz答案:5,9,7


3、Transpose:Tasks re-order data elements in memory

AOS: array of structures

SOA:structures of array


4、what kind of communication pattern

第三个quiz的答案:A、E、C、B

对最后一个选B做一下解释,因为如果选stencil的话,就是对每一个元素都进行取平均值的操作,可是,代码的上方有一个if语句,它只对奇数进行这样的操作,所以不满足stencil的定义,故只能是属于scatter。


5、各种通信模式总结:

Map:one to one

Transpose:one to one

gather:many to one

scatter:one to many

stencil:several to one(一种特殊的收集)

reduce:all to one

scan/sor:all to all


6、第四个quiz答案:1、2、4

解答:一个线程块不能在多个SM上执行;不同线程块之间不能合作解决同一个问题,因为无法通信


7、第五个quiz答案:1、2


8、第六个quiz答案:上述均错


9、第七个quiz答案:16!16个线程块以随机的方式执行,根据排列组合,有A16种可能,即16的阶乘。


10、cuda可以保证:

-所有在同一个线程块中的线程运行在同一个SM上。

-在一个内核中的所有块,在下个内核中任意块启动前都已完成。


11、共享内存是在一个块的线程之间共享。


12、第八个quiz答案:全对


13、第九个quiz答案:3

array[idx] = threadIdx.x下方需要一个同步,保证所有数组元素都赋值完成。

为了实现题目要求,需要对代码进行改写,如下所示

int temp = array[idx+1];
array[idx] = temp;
需要在上述两句代码中间和结尾各加一句同步语句,这样才能防止乱序的情况发生。比如a[1]本来是存放a[2]的值,可是a[2]已经存了a[3],这样就会导致a[1]中最后是a[3]的值,这显然是错误的,所以需要同步语句。


14、第十个quiz答案:B


15、第十一个quiz答案:3124,较简单。


16、第十二个quiz答案:1、3、4、5、6,注意存取连续内存即可。


17、第十三个quiz答案:第1、2、4、5可以实现,3会出错。

然后再来看一下各自花的时间排序,我们知道原子操作是更花费时间的,所以最快的肯定是第三项,接下来是第一项,在1000000个元素上进行加一操作,这个比其他原子操作更快,再下来是第四项操作,因为对比第二项,它访问的元素更少,故比第二项快,最慢的是第五项,因为线程数目是其他几项的十倍,所以这个是最慢的。

按时间排序将是2、4、1、3、5

18、避免线程发散 


19、第十四个quiz答案:


原始代码如下:

// Homework 2
// Image Blurring
//
// In this homework we are blurring an image. To do this, imagine that we have
// a square array of weight values. For each pixel in the image, imagine that we
// overlay this square array of weights on top of the image such that the center
// of the weight array is aligned with the current pixel. To compute a blurred
// pixel value, we multiply each pair of numbers that line up. In other words, we
// multiply each weight with the pixel underneath it. Finally, we add up all of the
// multiplied numbers and assign that value to our output for the current pixel.
// We repeat this process for all the pixels in the image.

// To help get you started, we have included some useful notes here.

//****************************************************************************

// For a color image that has multiple channels, we suggest separating
// the different color channels so that each color is stored contiguously
// instead of being interleaved. This will simplify your code.

// That is instead of RGBARGBARGBARGBA... we suggest transforming to three
// arrays (as in the previous homework we ignore the alpha channel again):
//  1) RRRRRRRR...
//  2) GGGGGGGG...
//  3) BBBBBBBB...
//
// The original layout is known an Array of Structures (AoS) whereas the
// format we are converting to is known as a Structure of Arrays (SoA).

// As a warm-up, we will ask you to write the kernel that performs this
// separation. You should then write the "meat" of the assignment,
// which is the kernel that performs the actual blur. We provide code that
// re-combines your blurred results for each color channel.

//****************************************************************************

// You must fill in the gaussian_blur kernel to perform the blurring of the
// inputChannel, using the array of weights, and put the result in the outputChannel.

// Here is an example of computing a blur, using a weighted average, for a single
// pixel in a small image.
//
// Array of weights:
//
//  0.0  0.2  0.0
//  0.2  0.2  0.2
//  0.0  0.2  0.0
//
// Image (note that we align the array of weights to the center of the box):
//
//    1  2  5  2  0  3
//       -------
//    3 |2  5  1| 6  0       0.0*2 + 0.2*5 + 0.0*1 +
//      |       |
//    4 |3  6  2| 1  4   ->  0.2*3 + 0.2*6 + 0.2*2 +   ->  3.2
//      |       |
//    0 |4  0  3| 4  2       0.0*4 + 0.2*0 + 0.0*3
//       -------
//    9  6  5  0  3  9
//
//         (1)                         (2)                 (3)
//
// A good starting place is to map each thread to a pixel as you have before.
// Then every thread can perform steps 2 and 3 in the diagram above
// completely independently of one another.

// Note that the array of weights is square, so its height is the same as its width.
// We refer to the array of weights as a filter, and we refer to its width with the
// variable filterWidth.

//****************************************************************************

// Your homework submission will be evaluated based on correctness and speed.
// We test each pixel against a reference solution. If any pixel differs by
// more than some small threshold value, the system will tell you that your
// solution is incorrect, and it will let you try again.

// Once you have gotten that working correctly, then you can think about using
// shared memory and having the threads cooperate to achieve better performance.

//****************************************************************************

// Also note that we've supplied a helpful debugging function called checkCudaErrors.
// You should wrap your allocation and copying statements like we've done in the
// code we're supplying you. Here is an example of the unsafe way to allocate
// memory on the GPU:
//
// cudaMalloc(&d_red, sizeof(unsigned char) * numRows * numCols);
//
// Here is an example of the safe way to do the same thing:
//
// checkCudaErrors(cudaMalloc(&d_red, sizeof(unsigned char) * numRows * numCols));
//
// Writing code the safe way requires slightly more typing, but is very helpful for
// catching mistakes. If you write code the unsafe way and you make a mistake, then
// any subsequent kernels won't compute anything, and it will be hard to figure out
// why. Writing code the safe way will inform you as soon as you make a mistake.

// Finally, remember to free the memory you allocate at the end of the function.

//****************************************************************************

#include "reference_calc.cpp"
#include "utils.h"

__global__
void gaussian_blur(const unsigned char* const inputChannel,
                   unsigned char* const outputChannel,
                   int numRows, int numCols,
                   const float* const filter, const int filterWidth)
{
  // TODO
  
  // NOTE: Be sure to compute any intermediate results in floating point
  // before storing the final result as unsigned char.

  // NOTE: Be careful not to try to access memory that is outside the bounds of
  // the image. You'll want code that performs the following check before accessing
  // GPU memory:
  //
  // if ( absolute_image_position_x >= numCols ||
  //      absolute_image_position_y >= numRows )
  // {
  //     return;
  // }
  
  // NOTE: If a thread's absolute position 2D position is within the image, but some of
  // its neighbors are outside the image, then you will need to be extra careful. Instead
  // of trying to read such a neighbor value from GPU memory (which won't work because
  // the value is out of bounds), you should explicitly clamp the neighbor values you read
  // to be within the bounds of the image. If this is not clear to you, then please refer
  // to sequential reference solution for the exact clamping semantics you should follow.
}

//This kernel takes in an image represented as a uchar4 and splits
//it into three images consisting of only one color channel each
__global__
void separateChannels(const uchar4* const inputImageRGBA,
                      int numRows,
                      int numCols,
                      unsigned char* const redChannel,
                      unsigned char* const greenChannel,
                      unsigned char* const blueChannel)
{
  // TODO
  //
  // NOTE: Be careful not to try to access memory that is outside the bounds of
  // the image. You'll want code that performs the following check before accessing
  // GPU memory:
  //
  // if ( absolute_image_position_x >= numCols ||
  //      absolute_image_position_y >= numRows )
  // {
  //     return;
  // }
}

//This kernel takes in three color channels and recombines them
//into one image.  The alpha channel is set to 255 to represent
//that this image has no transparency.
__global__
void recombineChannels(const unsigned char* const redChannel,
                       const unsigned char* const greenChannel,
                       const unsigned char* const blueChannel,
                       uchar4* const outputImageRGBA,
                       int numRows,
                       int numCols)
{
  const int2 thread_2D_pos = make_int2( blockIdx.x * blockDim.x + threadIdx.x,
                                        blockIdx.y * blockDim.y + threadIdx.y);

  const int thread_1D_pos = thread_2D_pos.y * numCols + thread_2D_pos.x;

  //make sure we don't try and access memory outside the image
  //by having any threads mapped there return early
  if (thread_2D_pos.x >= numCols || thread_2D_pos.y >= numRows)
    return;

  unsigned char red   = redChannel[thread_1D_pos];
  unsigned char green = greenChannel[thread_1D_pos];
  unsigned char blue  = blueChannel[thread_1D_pos];

  //Alpha should be 255 for no transparency
  uchar4 outputPixel = make_uchar4(red, green, blue, 255);

  outputImageRGBA[thread_1D_pos] = outputPixel;
}

unsigned char *d_red, *d_green, *d_blue;
float         *d_filter;

void allocateMemoryAndCopyToGPU(const size_t numRowsImage, const size_t numColsImage,
                                const float* const h_filter, const size_t filterWidth)
{

  //allocate memory for the three different channels
  //original
  checkCudaErrors(cudaMalloc(&d_red,   sizeof(unsigned char) * numRowsImage * numColsImage));
  checkCudaErrors(cudaMalloc(&d_green, sizeof(unsigned char) * numRowsImage * numColsImage));
  checkCudaErrors(cudaMalloc(&d_blue,  sizeof(unsigned char) * numRowsImage * numColsImage));

  //TODO:
  //Allocate memory for the filter on the GPU
  //Use the pointer d_filter that we have already declared for you
  //You need to allocate memory for the filter with cudaMalloc
  //be sure to use checkCudaErrors like the above examples to
  //be able to tell if anything goes wrong
  //IMPORTANT: Notice that we pass a pointer to a pointer to cudaMalloc

  //TODO:
  //Copy the filter on the host (h_filter) to the memory you just allocated
  //on the GPU.  cudaMemcpy(dst, src, numBytes, cudaMemcpyHostToDevice);
  //Remember to use checkCudaErrors!

}

void your_gaussian_blur(const uchar4 * const h_inputImageRGBA, uchar4 * const d_inputImageRGBA,
                        uchar4* const d_outputImageRGBA, const size_t numRows, const size_t numCols,
                        unsigned char *d_redBlurred, 
                        unsigned char *d_greenBlurred, 
                        unsigned char *d_blueBlurred,
                        const int filterWidth)
{
  //TODO: Set reasonable block size (i.e., number of threads per block)
  const dim3 blockSize;

  //TODO:
  //Compute correct grid size (i.e., number of blocks per kernel launch)
  //from the image size and and block size.
  const dim3 gridSize;

  //TODO: Launch a kernel for separating the RGBA image into different color channels

  // Call cudaDeviceSynchronize(), then call checkCudaErrors() immediately after
  // launching your kernel to make sure that you didn't make any mistakes.
  cudaDeviceSynchronize(); checkCudaErrors(cudaGetLastError());

  //TODO: Call your convolution kernel here 3 times, once for each color channel.

  // Again, call cudaDeviceSynchronize(), then call checkCudaErrors() immediately after
  // launching your kernel to make sure that you didn't make any mistakes.
  cudaDeviceSynchronize(); checkCudaErrors(cudaGetLastError());

  // Now we recombine your results. We take care of launching this kernel for you.
  //
  // NOTE: This kernel launch depends on the gridSize and blockSize variables,
  // which you must set yourself.
  recombineChannels<<<gridSize, blockSize>>>(d_redBlurred,
                                             d_greenBlurred,
                                             d_blueBlurred,
                                             d_outputImageRGBA,
                                             numRows,
                                             numCols);
  cudaDeviceSynchronize(); checkCudaErrors(cudaGetLastError());
}


//Free all the memory that we allocated
//TODO: make sure you free any arrays that you allocated
void cleanup() {
  checkCudaErrors(cudaFree(d_red));
  checkCudaErrors(cudaFree(d_green));
  checkCudaErrors(cudaFree(d_blue));
}

填充后完整代码如下:

// Homework 2
// Image Blurring
//
// In this homework we are blurring an image. To do this, imagine that we have
// a square array of weight values. For each pixel in the image, imagine that we
// overlay this square array of weights on top of the image such that the center
// of the weight array is aligned with the current pixel. To compute a blurred
// pixel value, we multiply each pair of numbers that line up. In other words, we
// multiply each weight with the pixel underneath it. Finally, we add up all of the
// multiplied numbers and assign that value to our output for the current pixel.
// We repeat this process for all the pixels in the image.

// To help get you started, we have included some useful notes here.

//****************************************************************************

// For a color image that has multiple channels, we suggest separating
// the different color channels so that each color is stored contiguously
// instead of being interleaved. This will simplify your code.

// That is instead of RGBARGBARGBARGBA... we suggest transforming to three
// arrays (as in the previous homework we ignore the alpha channel again):
//  1) RRRRRRRR...
//  2) GGGGGGGG...
//  3) BBBBBBBB...
//
// The original layout is known an Array of Structures (AoS) whereas the
// format we are converting to is known as a Structure of Arrays (SoA).

// As a warm-up, we will ask you to write the kernel that performs this
// separation. You should then write the "meat" of the assignment,
// which is the kernel that performs the actual blur. We provide code that
// re-combines your blurred results for each color channel.

//****************************************************************************

// You must fill in the gaussian_blur kernel to perform the blurring of the
// inputChannel, using the array of weights, and put the result in the outputChannel.

// Here is an example of computing a blur, using a weighted average, for a single
// pixel in a small image.
//
// Array of weights:
//
//  0.0  0.2  0.0
//  0.2  0.2  0.2
//  0.0  0.2  0.0
//
// Image (note that we align the array of weights to the center of the box):
//
//    1  2  5  2  0  3
//       -------
//    3 |2  5  1| 6  0       0.0*2 + 0.2*5 + 0.0*1 +
//      |       |
//    4 |3  6  2| 1  4   ->  0.2*3 + 0.2*6 + 0.2*2 +   ->  3.2
//      |       |
//    0 |4  0  3| 4  2       0.0*4 + 0.2*0 + 0.0*3
//       -------
//    9  6  5  0  3  9
//
//         (1)                         (2)                 (3)
//
// A good starting place is to map each thread to a pixel as you have before.
// Then every thread can perform steps 2 and 3 in the diagram above
// completely independently of one another.

// Note that the array of weights is square, so its height is the same as its width.
// We refer to the array of weights as a filter, and we refer to its width with the
// variable filterWidth.

//****************************************************************************

// Your homework submission will be evaluated based on correctness and speed.
// We test each pixel against a reference solution. If any pixel differs by
// more than some small threshold value, the system will tell you that your
// solution is incorrect, and it will let you try again.

// Once you have gotten that working correctly, then you can think about using
// shared memory and having the threads cooperate to achieve better performance.

//****************************************************************************

// Also note that we've supplied a helpful debugging function called checkCudaErrors.
// You should wrap your allocation and copying statements like we've done in the
// code we're supplying you. Here is an example of the unsafe way to allocate
// memory on the GPU:
//
// cudaMalloc(&d_red, sizeof(unsigned char) * numRows * numCols);
//
// Here is an example of the safe way to do the same thing:
//
// checkCudaErrors(cudaMalloc(&d_red, sizeof(unsigned char) * numRows * numCols));
//
// Writing code the safe way requires slightly more typing, but is very helpful for
// catching mistakes. If you write code the unsafe way and you make a mistake, then
// any subsequent kernels won't compute anything, and it will be hard to figure out
// why. Writing code the safe way will inform you as soon as you make a mistake.

// Finally, remember to free the memory you allocate at the end of the function.

//****************************************************************************

#include "reference_calc.cpp"
#include "utils.h"

__global__
void gaussian_blur(const unsigned char* const inputChannel,
                   unsigned char* const outputChannel,
                   int numRows, int numCols,
                   const float* const filter, const int filterWidth)
{
  // TODO
  
  // NOTE: Be sure to compute any intermediate results in floating point
  // before storing the final result as unsigned char.

  // NOTE: Be careful not to try to access memory that is outside the bounds of
  // the image. You'll want code that performs the following check before accessing
  // GPU memory:
  //
  // if ( absolute_image_position_x >= numCols ||
  //      absolute_image_position_y >= numRows )
  // {
  //     return;
  // }
  
  // NOTE: If a thread's absolute position 2D position is within the image, but some of
  // its neighbors are outside the image, then you will need to be extra careful. Instead
  // of trying to read such a neighbor value from GPU memory (which won't work because
  // the value is out of bounds), you should explicitly clamp the neighbor values you read
  // to be within the bounds of the image. If this is not clear to you, then please refer
  // to sequential reference solution for the exact clamping semantics you should follow.
    int c = blockIdx.x * blockDim.x + threadIdx.x;  //cols num
    int r = blockIdx.y * blockDim.y + threadIdx.y;  //rows num
    
    if(c >= numCols || r >= numRows) 
    {
        return;
    }
    
     float result = 0.f;
      //For every value in the filter around the pixel (c, r)
      for (int filter_r = -filterWidth/2; filter_r <= filterWidth/2; ++filter_r) {
        for (int filter_c = -filterWidth/2; filter_c <= filterWidth/2; ++filter_c) {
          //Find the global image position for this filter position
          //clamp to boundary of the image
          int image_r = min(max(r + filter_r, 0), static_cast<int>(numRows - 1));
          int image_c = min(max(c + filter_c, 0), static_cast<int>(numCols - 1));

          float image_value = static_cast<float>(inputChannel[image_r * numCols + image_c]);
          float filter_value = filter[(filter_r + filterWidth/2) * filterWidth + filter_c + filterWidth/2];

          result += image_value * filter_value;
        }
      }
    outputChannel[r * numCols + c] = result;
}

//This kernel takes in an image represented as a uchar4 and splits
//it into three images consisting of only one color channel each
__global__
void separateChannels(const uchar4* const inputImageRGBA,
                      int numRows,
                      int numCols,
                      unsigned char* const redChannel,
                      unsigned char* const greenChannel,
                      unsigned char* const blueChannel)
{
  // TODO
  //
  // NOTE: Be careful not to try to access memory that is outside the bounds of
  // the image. You'll want code that performs the following check before accessing
  // GPU memory:
  //
  // if ( absolute_image_position_x >= numCols ||
  //      absolute_image_position_y >= numRows )
  // {
  //     return;
  // }
  const int2 thread_2D_pos = make_int2( blockIdx.x * blockDim.x + threadIdx.x,
                                        blockIdx.y * blockDim.y + threadIdx.y);

  const int thread_1D_pos = thread_2D_pos.y * numCols + thread_2D_pos.x;
  if (thread_2D_pos.x >= numCols || thread_2D_pos.y >= numRows)
    return;
    uchar4 rgba = inputImageRGBA[thread_1D_pos];
    redChannel[thread_1D_pos] = rgba.x;
    greenChannel[thread_1D_pos] = rgba.y;
    blueChannel[thread_1D_pos] = rgba.z;
}

//This kernel takes in three color channels and recombines them
//into one image.  The alpha channel is set to 255 to represent
//that this image has no transparency.
__global__
void recombineChannels(const unsigned char* const redChannel,
                       const unsigned char* const greenChannel,
                       const unsigned char* const blueChannel,
                       uchar4* const outputImageRGBA,
                       int numRows,
                       int numCols)
{
  const int2 thread_2D_pos = make_int2( blockIdx.x * blockDim.x + threadIdx.x,
                                        blockIdx.y * blockDim.y + threadIdx.y);

  const int thread_1D_pos = thread_2D_pos.y * numCols + thread_2D_pos.x;

  //make sure we don't try and access memory outside the image
  //by having any threads mapped there return early
  if (thread_2D_pos.x >= numCols || thread_2D_pos.y >= numRows)
    return;

  unsigned char red   = redChannel[thread_1D_pos];
  unsigned char green = greenChannel[thread_1D_pos];
  unsigned char blue  = blueChannel[thread_1D_pos];

  //Alpha should be 255 for no transparency
  uchar4 outputPixel = make_uchar4(red, green, blue, 255);

  outputImageRGBA[thread_1D_pos] = outputPixel;
}

unsigned char *d_red, *d_green, *d_blue;
float         *d_filter;

void allocateMemoryAndCopyToGPU(const size_t numRowsImage, const size_t numColsImage,
                                const float* const h_filter, const size_t filterWidth)
{

  //allocate memory for the three different channels
  //original
  checkCudaErrors(cudaMalloc(&d_red,   sizeof(unsigned char) * numRowsImage * numColsImage));
  checkCudaErrors(cudaMalloc(&d_green, sizeof(unsigned char) * numRowsImage * numColsImage));
  checkCudaErrors(cudaMalloc(&d_blue,  sizeof(unsigned char) * numRowsImage * numColsImage));

  //TODO:
  //Allocate memory for the filter on the GPU
  //Use the pointer d_filter that we have already declared for you
  //You need to allocate memory for the filter with cudaMalloc
  //be sure to use checkCudaErrors like the above examples to
  //be able to tell if anything goes wrong
  //IMPORTANT: Notice that we pass a pointer to a pointer to cudaMalloc
  checkCudaErrors(cudaMalloc(&d_filter, sizeof(float) * filterWidth * filterWidth));
  //TODO:
  //Copy the filter on the host (h_filter) to the memory you just allocated
  //on the GPU.  cudaMemcpy(dst, src, numBytes, cudaMemcpyHostToDevice);
  //Remember to use checkCudaErrors!
  checkCudaErrors(cudaMemcpy(d_filter,h_filter,sizeof(float) * filterWidth * filterWidth,cudaMemcpyHostToDevice));
}

void your_gaussian_blur(const uchar4 * const h_inputImageRGBA, uchar4 * const d_inputImageRGBA,
                        uchar4* const d_outputImageRGBA, const size_t numRows, const size_t numCols,
                        unsigned char *d_redBlurred, 
                        unsigned char *d_greenBlurred, 
                        unsigned char *d_blueBlurred,
                        const int filterWidth)
{
  //TODO: Set reasonable block size (i.e., number of threads per block)
  const dim3 blockSize(32,32);

  //TODO:
  //Compute correct grid size (i.e., number of blocks per kernel launch)
  //from the image size and and block size.
  const dim3 gridSize((numCols+31)/32,(numRows+31)/32);

  //TODO: Launch a kernel for separating the RGBA image into different color channels
  separateChannels<<<gridSize,blockSize>>>(d_inputImageRGBA,numRows,numCols,d_red,d_green,d_blue);
  // Call cudaDeviceSynchronize(), then call checkCudaErrors() immediately after
  // launching your kernel to make sure that you didn't make any mistakes.
  cudaDeviceSynchronize(); checkCudaErrors(cudaGetLastError());

  //TODO: Call your convolution kernel here 3 times, once for each color channel.
  gaussian_blur<<<gridSize,blockSize>>>(d_red,d_redBlurred,numRows,numCols,d_filter,filterWidth);
  gaussian_blur<<<gridSize,blockSize>>>(d_green,d_greenBlurred,numRows,numCols,d_filter,filterWidth);
  gaussian_blur<<<gridSize,blockSize>>>(d_blue,d_blueBlurred,numRows,numCols,d_filter,filterWidth);
  // Again, call cudaDeviceSynchronize(), then call checkCudaErrors() immediately after
  // launching your kernel to make sure that you didn't make any mistakes.
  cudaDeviceSynchronize(); checkCudaErrors(cudaGetLastError());

  // Now we recombine your results. We take care of launching this kernel for you.
  //
  // NOTE: This kernel launch depends on the gridSize and blockSize variables,
  // which you must set yourself.
  recombineChannels<<<gridSize, blockSize>>>(d_redBlurred,
                                             d_greenBlurred,
                                             d_blueBlurred,
                                             d_outputImageRGBA,
                                             numRows,
                                             numCols);
  cudaDeviceSynchronize(); checkCudaErrors(cudaGetLastError());
}


//Free all the memory that we allocated
//TODO: make sure you free any arrays that you allocated
void cleanup() {
  checkCudaErrors(cudaFree(d_red));
  checkCudaErrors(cudaFree(d_green));
  checkCudaErrors(cudaFree(d_blue));
}



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