I have implemented basic shadow mapping for the first time in OpenGL using shaders and I'm facing some problems. Below you can see an example of my rendered scene:

The process of the shadow mapping I'm following is that I render the scene to the framebuffer using a View Matrix from the light point of view and the projection and model matrices used for normal rendering.

In the second pass, I send the above MVP matrix from the light point of view to the vertex shader which transforms the position to light space. The fragment shader does the perspective divide and changes the position to texture coordinates.

Here is my vertex shader,

#version 150 core

uniform mat4 ModelViewMatrix;
uniform mat3 NormalMatrix;
uniform mat4 MVPMatrix;
uniform mat4 lightMVP;

uniform float scale;

in vec3 in_Position;
in vec3 in_Normal;
in vec2 in_TexCoord;

smooth out vec3 pass_Normal;
smooth out vec3 pass_Position;
smooth out vec2 TexCoord;
smooth out vec4 lightspace_Position;

void main(void){
    pass_Normal = NormalMatrix * in_Normal; 
    pass_Position = (ModelViewMatrix * vec4(scale * in_Position, 1.0)).xyz;
    lightspace_Position = lightMVP * vec4(scale * in_Position, 1.0);
    TexCoord = in_TexCoord;

    gl_Position = MVPMatrix * vec4(scale * in_Position, 1.0);
}

And my fragment shader,

#version 150 core

struct Light{
    vec3 direction;
};

uniform Light light;
uniform sampler2D inSampler;
uniform sampler2D inShadowMap;

smooth in vec3 pass_Normal;
smooth in vec3 pass_Position;
smooth in vec2 TexCoord;
smooth in vec4 lightspace_Position;

out vec4 out_Color;


float CalcShadowFactor(vec4 lightspace_Position){

    vec3 ProjectionCoords = lightspace_Position.xyz / lightspace_Position.w;

    vec2 UVCoords;
    UVCoords.x = 0.5 * ProjectionCoords.x + 0.5;
    UVCoords.y = 0.5 * ProjectionCoords.y + 0.5;

    float Depth = texture(inShadowMap, UVCoords).x;
    if(Depth < (ProjectionCoords.z + 0.001)) return 0.5;
    else return 1.0;
}

void main(void){

    vec3 Normal = normalize(pass_Normal);
    vec3 light_Direction = -normalize(light.direction);
    vec3 camera_Direction = normalize(-pass_Position);
    vec3 half_vector = normalize(camera_Direction + light_Direction);

    float diffuse = max(0.2, dot(Normal, light_Direction));
    vec3 temp_Color = diffuse * vec3(1.0);

    float specular = max( 0.0, dot( Normal, half_vector) );

    float shadowFactor = CalcShadowFactor(lightspace_Position);

    if(diffuse != 0 && shadowFactor > 0.5){
        float fspecular = pow(specular, 128.0);
        temp_Color += fspecular;
    }

    out_Color = vec4(shadowFactor * texture(inSampler, TexCoord).xyz * temp_Color, 1.0);
}

One of the problems is self shadowing as you can see in the picture, the crate has its own shadow cast on itself. What I have tried is enabling polygon offset (i.e. glEnable(POLYGON_OFFSET_FILL), glPolygonOffset(GLfloat, GLfloat) ) but it didn't change much. As you see in the fragment shader, I have put a static offset value of 0.001 but I have to change the value depending on the distance of the light to get more desirable effects , which not very handy. I also tried using front face culling when I render to the framebuffer, that didn't change much too.

The other problem is that pixels outside the Light's view frustum get shaded. The only object that is supposed to be able to cast shadows is the crate. I guess I should pick more appropriate projection and view matrices, but I'm not sure how to do that. What are some common practices, should I pick an orthographic projection?

From googling around a bit, I understand that these issues are not that trivial. Does anyone have any easy to implement solutions to these problems. Could you give me some additional tips?

Please ask me if you need more information on my code.

Here is a comparison with and without shadow mapping of a close-up of the crate. The self-shadowing is more visible.